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Economic production order quantity and quality Angus Jeang* Department of Industrial Engineering and Systems Management, Feng Chia University, P.O. Box 25-150, Taichung, Taiwan, ROC (Received 1 March 2009; final version received 10 December 2009) The purpose of this study is to integrate a conventional production-inventory management approach and a process-quality design approach so as to promote quality and reduce costs. Because the integration of these two approaches into a single system is conducted under the influence of a deterioration process, estimated costs of production must be modified. Only then can these various costs be presented as a total cost for the present integrated system. This total cost includes: a setup cost for production reordering and process resetting, quality-related costs stemming from the loss of product quality along with a tolerance-related production process cost and a failure cost for product defects, backorder costs for production shortage, carrying costs generated by the storing and handling of inventory, and material costs for determining the process mean. The Taguchi quality loss function is introduced to assess quality loss in the system. The decision variables include the initial setting (process mean), process tolerance, and production order quantity. These variables need to be determined simultaneously to minimise the average total cost for a cycle time. An example is presented to demonstrate the proposed approach. Keywords: initial setting; process tolerance; production order quantity; quality; cost; deterioration process; optimisation

1. Introduction One of the drawbacks of conventional inventory models is the idealistic assumption that produced parts are all of good quality with the production process being in perfect condition. The problem with this assumption is that it does not hold true in practical application. Quality value may fall outside product specification due to the stochastic nature of produced quality values, and some defective parts may be produced due to a deterioration process related to systemic causes. Another drawback to conventional models is that decision problems are solved by using economic criteria obtained within a company. For a life cycle application, the relevant measurement criteria in resolving the production ordering problem should not be limited to costs solely involved in the production process; rather, quality-related costs that accrue after the delivery of goods to customers should also be included. Production process related issues in inventory modelling require additional consideration as these issues affect the quality and cost of the produced product in general. Hence, inventory, production, and quality costs should be simultaneously minimised when determining the optimal levels of production

*Email: [email protected] ISSN 0020–7543 print/ISSN 1366–588X online 2010 Taylor & Francis DOI: 10.1080/00207540903555528 http://www.informaworld.com

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order quantity. The proposed model is designed to meet these requirements which have long been neglected. The conventional belief that products with quality values that meet specification requirements will not result in quality loss is simply untrue. Recent studies have shown that this narrow definition of quality greatly underestimates the costs of poor quality and leads to decisions that are less than optimal, except in particular local situations. It is now no longer appropriate to base product acceptance or rejection on specification requirements. The Taguchi quality loss function was introduced to assess the loss arising from the quality value deviating from its design target (Phadke 1989, Taguchi et al. 1989). This assessment can also evaluate quality-related costs that accrue after the delivery of goods to customers. Consequently, the minimisation of quality variability is now the main concern of process and quality engineers in terms of product and process design. There are two ways of reducing variability: one is to establish the process mean as near as possible to the design target, and the other is to tighten the tolerance value as small as possible within the limits of process capability. Because process mean and process tolerance are mutually interdependent, a concurrent determination of process mean and process tolerance in balancing tolerance cost and quality loss is essential for a process design to realise economic and quality achievement (Jeang 2001). Defective items resulting from imperfect quality production processes were originally considered by several authors (Porteus 1986, Rosenblatt and Lee 1986, Kim and Hong 1999, Salameh and Jaber 2000). Chung and Hou (2003) have generalised the work of Kim and Hong and have assumed that shortage was permitted. However, as mentioned in the work of Rahin and Al-Hajailan (2006), the common assumption of the above models is that a fixed percentage of defective items were assigned. Thus, Rahin and Al-Hajailan have further generalised the work of Chung and Hou by introducing a time varying fraction defective rate. However, none of these works considered quality-related costs after the delivery of goods to customers, which means they lack a life cycle view in regard to cost evaluation. To offset this drawback Ganeshan et al. (2001) have linked the lot-size problem with the Taguchi quality loss function in inventory modelling development. The purpose was to determine the optimal levels of inventory and production order quantity in order to minimise inventory and quality-related costs. Chen and Lai (2007) revisited this interaction between the economics of production with the Taguchi quality cost perspective in determining both process mean and product specifications. An ordinary assumption among Ganeshan et al., and Chen and Lai, is that Taguchi quality loss is independent of time spent and products produced. However, as time passes, the output values of quality characteristics may change during the production process as time passes. Consequently, quality loss should be expressed as a function of the time period in which a product is produced. Another assumption appearing in the work of Ganeshan et al., and Chen and Lai is that Taguchi quality loss is purely a function of process mean. The assumption is mistaken, as the formulation excluding the influence of process tolerance is insufficient in representing quality-related costs, particularly when deterioration phenomena exist in the production process (Jeang 2001, Jeang and Chung 2008). The proposed approach integrates a process-quality design approach with a conventional production-inventory management approach in order to view process mean and process tolerance concurrently in assessing quality loss using the Taguchi quality loss function. A process-quality design model was recently developed in which the resetting cost, Taguchi quality loss, and tolerance cost are considered as functions of decision variables

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such as initial setting (process mean), process tolerance, and cycle time under the deterioration process (Jeang and Chung 2008). A unit cost subject to deterioration process is approximated as a function of process mean and process tolerance. This cost affects the inventory carrying cost, which in turn influences production order quantity as determined in the production-inventory management system. The chain of influence extends even further as the production order quantity influences the initial setting (process mean) and process tolerance obtained in process design. The inter-influence becomes great when the production process is subject to the deterioration process. The proposed model aims to take these influences into account by integrating production-inventory management and process-quality design. Due to the deterioration process, economic production order quantity (EPQ) is a good starting-point in representing the production-inventory management system. Inventory-related costs include an ordering cost and a carrying cost. The processrelated costs include: a resetting cost, quality-related costs stemming from loss of quality, and a tolerance cost. These costs are represented in the proposed model. Quality loss is expressed by the Taguchi quality loss function, and the tolerance cost appears as a tolerance cost function. Because the costs associated with a production-inventory management approach and a process-quality design approach are characterised by periodical recurrence, a single setup with one common cycle time for production reordering and process resetting is assumed in the proposed model (Hax and Canadea 1984, Silver and Peterson 1985, Drezner and Wesolowsky 1989, Jeang and Yang 1992, Jeang 1998). Other common costs, including a backorder cost for shortage, a failure cost for defectives, and a raw material cost for process mean, are also included. Thus, the costs of production-inventory and process-quality design are combined as a total cost in the proposed model. The purpose of this model is to concurrently optimise the cycle time, initial setting (process mean), process tolerance, and production order quantity by integrating previous process-design models with the production-inventory EPQ model. To achieve this optimisation, a trade-off among the various costs pertaining to both inventory and process-design is presented via mathematical representation. The paper itself is divided into six sections. Section 1 provides an introduction to the current study. Section 2 defines the relevant assumptions and notation. Section 3 outlines the background related to this research. Section 4 describes the model development. Section 5 demonstrates the proposed model by an example. Finally, a summary is given in Section 6.

2. Assumptions and notation Assumptions: (1) Produced quality value may deteriorate during the production process. (2) The deterioration rate is a linear function of time. (3) After each cycle time, the process mean is reset to its initial value to compensate for process deterioration. (4) Common cycle time for production reordering and process resetting. (5) Setup costs include production reordering cost and process resetting cost. (6) Shortages are allowed for backorder in Cases 2 and 3. (7) The demand rate and production rate are constant and deterministic.

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(8) Unit cost does not remain fixed over the cycle time, which is the sum of material costs and process costs. Material cost is a function of process mean and process cost is tolerance cost. Notation: AC AB AF AI AL AM AR AS a0 a000 B CF CM(t) CP CR Craw CU D d F(Q/P) f(X ) f(s) G h I Imax K L(X ) L(Q/P) l(s) m P Q Q0 Q00 QL(s) q0 RC r S

consumer loss; average backorder cost per time; average failure cost per time; average inventory cost per time; average quality loss per time; average tolerance cost per time; average material cost per time; average setup cost per time; initial setting for process mean at the beginning of the cycle time; initial setting for process-quality design approach (see Equation (33)); average backorder during production period T1; associated failure cost when quality value X exceeds its limits m d; tolerance cost per unit which is a function of tolerance t; process capability index; total raw material cost for production order quantity Q; raw material cost; unit cost of a product; demand rate in units per time; design specification (design tolerance) of produced quality value X; cumulative failure cost during production period Q/P; normal probability density function of produced quality value X; failure cost at time s; þ1 or 1 which is dependent on the direction of process deterioration; inventory carrying cost per unit per time; average on-hand inventory level for a cycle time, T; maximum on-hand inventory level for a cycle time, T; quality loss coefficient, which is: AC =S2max ; quality loss of produced quality value X; cumulative quality loss for nonconforming product within production period Q/P; quality loss at time s when failure exists; design target of produced quality value X; production rate in units per time (P4D); backorder cost for shortage; production order quantity requiring determination; production order quantity for conventional EPQ model; production order quantity for process-quality design approach (see Equation (33)); expected quality loss at time s for conforming product; produced units ranging from zero to production order quantity Q; raw material cost per unit; inventory carrying rate, percent of dollar value per time; setup cost for each cycle time;

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Smax s T T1 T2 T3 T4 TP t tL t00 TC1 TC2 TC3 U U(s) U(q0 ) W Wp X X(s)

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maximum deviation from design target; time ranging from 0 to cycle time Q/D; standard deviation of produced quality value X; cycle time, when backorder exists (Cases 2 and 3), T ¼ T1 þ T2 þ T3 þ T4; production period for producing backorder (see Figure 3); production period for cumulating inventory (see Figure 3); consumption period for consuming inventory (see Figure 3); consumption period for cumulating backorder (see Figure 3); production period, TP ¼ T1 þ T2 (see Figure 3); process tolerance for nominal-the-best type quality characteristic; lower limit of process capability range; process tolerance for process-quality design approach; average total cost per time for Case 1; average total cost per time for Case 2; average total cost per time for Case 3; process mean; process mean expressed as a function of time s under a deterioration process; process mean expressed as a function of produced units q0 under a deterioration process; deterioration rate per time; deterioration rate per unit, which is equal to W/P; quality value for produced product, which is a nominal-the-best type quality characteristic; quality value produced at time s, which is a nominal-the-best type quality characteristic.

3. Quality loss, tolerance cost, variance estimation, and simultaneous determination of mean and tolerance values 3.1 Quality loss Traditionally, the view that no loss would occur as long as the product quality characteristics fell within the specifications range, was accepted. However, this view is no longer accepted, since a quality characteristic that falls within the specifications range can still lead to varying degrees of quality loss depending on its deviation from the design target (see Figure 1). The Taguchi’s nominal-the-best quality loss function is considered in this study. The loss function is (Phadke 1989, Taguchi et al. 1989): 0, if X ¼ m LðX Þ ¼ : ð1Þ K ðX mÞ2 , otherwise The expected quality loss of Equation (1) is (Phadke 1989, Taguchi et al. 1989): E½LðXÞ ¼ K ðU mÞ2 þ 2 :

ð2Þ

3.2 Tolerance cost Usually, a high tolerance cost is associated with a tight process tolerance, while a low tolerance cost results from a loose process tolerance. It is known that a tight process

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A. Jeang f(X) L(X)

K (X– m)2

σ

0

X m–d

m

U

m+d

Figure 1. Quality loss function L(X ) and probabilistic normal distribution f(X ).

tolerance results in a higher tolerance cost and a lower failure cost due to the need for additional manufacturing operations and expensive equipment and materials, whereas a loose process tolerance results in a lower tolerance cost and a higher failure cost (Jeang 1994). The tolerance cost can be formulated according to various function expressions. To evaluate the tolerance cost, this paper directly adopts the tolerance cost function as developed in the literature (Chase et al. 1990): CM ðtÞ ¼ a þ b expðc tÞ,

ð3Þ

where a, b, and c are the coefficients found by regressing the real data from the shop floor.

3.3 Variance estimation Process variance is a common feature in all manufacturing processes. However, in most cases, the process variance is unknown due to the lack of previous data, particularly for new products or new processes, unless previous data from similar processes is available. Therefore, a reasonable estimate is important to obtain. It is known that variance is a function of process tolerance. If a manufacturing process is stable after a period of production has been assumed, the product engineers or process engineers can then indirectly estimate the process variance through the following relation (Jeang 1994): t 2 : ð4Þ 2 ¼ 3CP CP is the process capability index which becomes a stable value when the production process is run for a period of time. In other words, the equality in Equation (4) becomes true. Hence, Equation (2) can be further expressed as: " # t 2 2 : ð5Þ E½LðXÞ ¼ K ðU mÞ þ 3CP

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This indicates that the expected quality loss is a function of controllable variables such as process means, U, and process tolerances, t.

3.4 Simultaneous determination of parameter and tolerance values The total quality-related cost incurred during the life cycle of a product is normally the sum of expected quality loss E[L(X )] and tolerance cost CM(t). The quality loss characterises the quality cost for customers after the production process, and the tolerance cost represents the quality cost for producers during the production process. To minimise the total cost, there must be a trade-off between quality loss E[L(X )] and tolerance cost CM(t). For example, a small t will result in a small quality loss and give rise to a greater tolerance cost; conversely, a great t will result in a great quality loss and a smaller tolerance cost. Again, when we intend to minimise the total cost, E[L(X )] þ CM(t), a small(great) quality loss coefficient, K, followed by a great(small) tolerance value results. The purpose of these examples is to emphasise that when an objective function contains decision variables, process mean U and process tolerance t, they have to be determined simultaneously for true optimisation. Another reason for simultaneous optimisation of the process mean and process tolerance is that they are interdependent by nature.

4. Proposed model development TQM programmes have increasingly focused their attention on the need for better quality, thereby increasing productivity and lowering cost (Rampersad 2001). With higher quality and a lower price, a product on the marketplace has an enhanced competitive position. The procedure for improving quality starts by taking into consideration the needs of customers. A focus on the customer is evident in the current trend toward customer-driven manufacturing and the incorporation of customer requirements in the production process, resulting in what is called ‘process-quality design’. However, all of this ignores the fact that production-inventory management is among the most important of related functions, since inventory requires a great deal of capital and affects the delivery of goods to customers. Additionally, quality level and manufacturing process cost are both closely related to the production order quantity, which is decided at the stage of production-inventory management. Thus, an exclusive process-quality design approach is not sufficient to guarantee an economic and quality product. Instead, a further incorporation is necessary. In a situation of a deterioration process various inventory levels reveal different collective quality levels for produced products. Hence, production inventory based solely on the conventional production-inventory management approach may be over-produced, due to a lack of consideration of varied quality costs at different points of time along with the production run length. For cyclical quality promotion and cost reduction, there is a motivation to extend the conventional production-inventory management approach by adding the process-quality design approach, which considers the varied quality-related costs as a function of time. One of the main objectives in the production process is to achieve greater accuracy with respect to economical products. In the production process, there are various sources of errors that result in poor, inaccurate and defective products, meaning products that deviate randomly and systematically from the design target. These deviations reflect inaccuracies resulting from the collective effects of various working conditions,

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including: material variations, inaccurate operations, environmental differences, and process deterioration. All these errors, individually and collectively, can eventually lead to quality loss. Product accuracy can be depicted by a set of controllable terms such as process mean and process tolerance. Thus, one way to achieve the accuracy of a product is to determine the best process mean and process tolerance for each operation in the production process. Fortunately, these findings are available before production processes are realised. While it is important to have the process setting and process tolerance established at the beginning of the production process, this cannot always guarantee a quality product since process deterioration or systematic error may occur during the production process (Jeang and Chung 2008). Owing to the deterioration process, a continuous change in the quality of a product may take place and the product may fail before the expiration of its intended life. This typically results in a great loss of customer satisfaction. For this reason, controllable variables affecting the quality values of a product should be incorporated into the process design. In this regard, process designers often have the process mean reset cyclically for an initial deviation from target value in order to compensate for process deterioration. In the long run, this reduces the production cost and results in a higher quality product. Production-inventory management is crucial, owing to the amount of capital it requires. For this reason, the costs arising from a production-inventory management approach should also be included, as previously mentioned. This study incorporates the classic EPQ model of a production-inventory management approach with the process-quality design approach to ensure that decisions made at the early production planning and process design stage are able to effectively reduce costs and enhance quality. Consequently, the chances of business success in a competitive environment are increased.

4.1 Having EPQ model contained with quality costs for process design under deterioration process Conventional inventory decision problems are solved by offsetting economic criteria, such as ordering cost and holding cost, to determine economical production quantity for a cycle time Q/D. However, under the deterioration process, the product quality level and process cost among Q may vary at different points of time along the cycle time Q/D. Because previous inventory models failed to consider these aspects of the deterioration process, the determined production order quantity generally exceeded the true order quantity. Other aspects ignored in conventional inventory decision problems are the variations in unit costs and quality levels resulting from process designs that differ from one another. These variations are a concern, since on-line quality improvement leading to process redesign may occur after production and inventory decisions have been made. It is known that variations in unit costs will result in different inventory holding costs and that differing quality levels will bring about dissimilar quality-related costs. These, in turn, will drive the determined production order quantity deviation even further from the true order quantity. Thus, the proposed model intends to integrate the conventional production ordering problem with a process-quality design problem for off-line application to avoid uneconomical and overproduced low-quality products. Considerable savings and quality improvement can be realised under process deterioration if the process mean, U, is reset to its initial value, a0, regularly with cycle

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time Q/D in the opposite direction of the deteriorating course to compensate for the amount of shift occurring during the production process. Each resetting involves an appreciable cost. There is the cost of process resetting itself, but also costs associated with lost production time. Thus, for the sake of quality concern and cost savings, there is a motivation to simultaneously conduct the resetting of the initial establishment for quality assurance and the ordering of the commencement preparation for economic production. A single cost is applied to collectively represent the resetting cost and the ordering cost in one setup. Other than the resetting and ordering costs involved at the beginning of the cycle time, the holding cost for inventory, quality related costs, and a tolerance cost should also be included. In this study, quality loss is the value determined by a loss function which characterises economic loss due to deviation from the target value. Tolerance cost is the production cost for quality products. When the quality loss function and tolerance cost function are integrated, the production and quality related costs can be truly reflected. Only then can producers offset these cost factors by determining the production quantity and the optimal process parameters concurrently. The optimal solutions presented in the proposed model stand for pre-eminent decisions pertaining to product cost and product quality that are in the best interests of both producers and customers.

4.2 Proposed model derivation Since process setting and process tolerance are established prior to the production process, the process mean will be close to the preselected process setting, whereas the process variance will be close to the estimated value of the process tolerance after the production process. To formulate the model, we assume that the product quality value is a random variable following a normal distribution with the process mean, U, and the process variance, 2 . Thus, the associated probability density function is: 1 ðX UÞ2 p ﬃﬃﬃﬃﬃ f ðX Þ ¼ : ð6Þ exp 2 2 2n However, with the possibility of process deterioration or systematic error being included in the production process, the process mean U in Equation (6) turns out to be time-dependent and is expressed as a function of time s: ð7Þ UðsÞ ¼ a0 þ G W s: a0 is the initial establishment, G is þ1 or 1, depending on the direction of process deterioration, and W is the process deterioration rate. This can be seen in Figure 2. The reason for having a0 established in a deviation from the target value is to provide an allowance, so that the cycle time can be extended for economic and quality considerations. For this reason also, the time element s must be included as an additional allowance, G W s, during the problem formulation. In the Sections 4.2.1 and 4.2.2, we develop a model that takes into account the costs pertaining to setup, inventory holding, and quality loss, as well as the related costs for process mean, process tolerance, and production order quantity under a deterioration process. Three cases are introduced: Case 1: Conforming product is produced with the assumption that backorder is not allowed. The corresponding total cost per time is TC1.

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m+d

U(s)

U(s2) U(s1) m a0

m–d

0

s

s2

s1

Figure 2. A time-dependent quality value for nominal-the-best case under deterioration process.

Case 2: Conforming product is produced with the assumption that backorder is allowed. The corresponding total cost per time is TC2. Case 3: Nonconforming product is produced with the assumption that backorder is allowed. The corresponding total cost per time is TC3.

4.2.1 Conforming product is produced for inventory management Case 1 and Case 2 are introduced in this section. Total raw material cost for quantity Q: Assume the raw material cost is in a linear function of process mean U(s), which varies in time because of the deterioration process. Then, the total raw material cost for production order quantity Q is: Z

Q=P

CR ¼

Craw UðsÞds, 0

where Q/P is the production period.

ð8Þ

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Raw material cost per unit: During production period Q/P, production quantity, Q, is produced. Thus, the raw material cost per unit is: Z Q=P

RC ¼ CR =Q ¼ Craw ða0 þ G W sÞds Q: ð9Þ 0

The reason for taking the average of production quantity Q is that under the deterioration process the product quality level varies at different points of time along the production period Q/P. Because Craw is independent of times S, Craw will be moved to the other side of the integral sign in the following model derivation. Unit cost CU: The unit cost consists of raw material cost per unit and production cost per unit. The latter cost is estimated from tolerance cost, CM(t): CU ¼ RC þ CM ðtÞ

Z Q=P ða0 þ G W sÞds Q þ ½a þ b expðc tÞ: ¼ Craw

ð10Þ

0

The average setup cost AS: AS ¼ S

D : Q

ð11Þ

The average material cost AR: Q D Z Q=P D ¼ Craw ða0 þ G W sÞds : Q 0

AR ¼ CR

ð12Þ

The average tolerance cost AM: The tolerance cost per unit, CM(t), is a þ b expðc tÞ. Then, the average tolerance cost is:

Q AM ¼ ½ða þ b expðc tÞÞ Q D ð13Þ ¼ ½a þ b expðc tÞ D: The average inventory cost AI (backorder does not exist): Q PD I ¼ 2 P Q PD AI ¼ h, 2 P

ð14Þ

ð15Þ

where h ¼ r CU , and r is the inventory carrying rate. That is,

Z Q=P Q PD AI ¼ ða0 þ G W sÞds Q þ ða þ b expðc tÞÞ r: ð16Þ Craw 2 P 0

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The average quality loss AL: Based on Equation (5), the expected quality loss for conforming product at time s is: " # t 2 2 QL ðsÞ ¼ K ðða0 þ G W sÞ mÞ þ : ð17Þ 3CP Thus, when the time, s, and production order quantity, Q, are connected in the formulation, the average quality loss per time for cycle time Q=D is: Z Q=P

Q QL ðsÞds AL ¼ D 0 ( ) Z Q=P D t 2 2 ¼ K ½ða0 þ G W sÞ m þ ds: ð18Þ Q 0 3CP The average total cost is the sum of the following: average setup per time, AS, average inventory cost per time, AI, average quality loss per time, AL, average raw material cost per time, AR, and average tolerance cost per time, AM, for the cycle time Q=D. Because the unit cost, RC þ CM(t), varies according to order quantity and process designs, the expressions, AR and AM, must be included in the total cost expression for the proposed model. The corresponding terms: AS, AI, AL, AR and AM can be derived from Equations (11), (16), (18), (12) and (13). To reflect the economic reasons a backorder is sometimes allowed. Thus, the total cost per time for a conforming product can be further divided into two cases: Case 1 – backorder is not allowed; Case 2 – backorder is allowed. The differences for these two cases come from the average inventory holding cost AI and the average backordering cost AB. Case 1: Backorder is not allowed for conforming product The total cost per time is: TC1 ¼ AS þ AI þ AL þ AR þ AM

ð19Þ

D Q

Z Q=P Q PD þ ða0 þ G W sÞds Q þ ða þ b expðc tÞÞ r Craw 2 P 0 " # Z Q=P D t 2 2 K ð a0 þ G W s m Þ þ þ ds Q 0 3CP Z Q=P D ða0 þ G W sÞds þ Craw Q 0

TC1 ¼ S

þ D ½a þ b expðc tÞ:

ð20Þ

Decision variables are a0 , t, and Q. The objective is to find the optimal values for decision variables, a0 , t , and Q which minimise TC1 as given in Equation (20). Obtaining optimal values of these decision variables is a typical problem in calculus. By taking the derivatives of TC1 and setting them equal to zero, the resulting expressions of a0 , t , and Q can be obtained.

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Details concerning the optimal a0 , t , and Q have been derived and are shown in Equations (A3.1), (A3.4), and (A3.8) in Appendix 3. The associated necessary and sufficient conditions are proved in Appendices 1 and 2, respectively. The optimal solutions can be found either by solving Equation (20) directly with the computer software, Mathematica (Trott 2006), or by taking the steps for solution algorithm here introduced. The steps in finding optimal solutions for a0 , t , and Q are: Step 1: Have an estimation a0 ¼ m and t ¼ tL be substituted into Equation (A3.8) to find Q initially, where m and tL are known values. Step 2:

Have the just found Q be plugged into Equations (A3.1) and (A3.4) to find a0 and t.

Step 3:

Similar to Step 1, let the found a0 and t be substituted into Equation (A3.8) to find the next Q.

Step 4: Go back to Step 2. Continue the loop until the values a0 , t, and Q are convergent to an acceptable level. These values will then be optimal solutions. It is economical to have process design with an initial establishment a0 and process tolerance t in producing quality products in order quantity Q over a production period of time Q =P. No further parts are established until the lot is depleted at time Q =D. This makes it possible to spread the setup cost of initial establishment and production preparation over a number of parts Q to offset the quality loss resulting from quality deviation and the holding cost which cumulates from inventory build-up. Consequently, the process parameters and production quantity for an economic and quality product are determined. By observing Equations (A3.1), (A3.4), and (A3.8) from Appendix 3, we can conclude: (1) The decision variables, a0 and t, which are process design related, are in a functional relationship to the production and inventory related decision variable, Q, and also to related conditions such as r, P and D. Additionally, these decision variables are influenced by the coefficient of quality loss function, K. (2) The decision variable, Q, which is production and inventory related, is in a functional relationship to the process design related decision variables, a0 and t, and to production and inventory related conditions such as r, P and D. Of course, the decision variable, Q, is also affected by the quality related costs, K value. Clearly, based on the evidence of relationship and dependence among these variables, the production-inventory management approach and the process-quality design approach must work together for optimisation of the global system. Case 2:

Backorder is allowed for conforming product

By referring to Equations (A3.12) and (A3.13) from Appendix 3, it is apparent that the average inventory holding cost AI and the average backordering cost AB can be expressed in the following. The average inventory cost AI (backorder exists): AI ¼

rCU ðP DÞðTP T1 Þ2 : 2TP

ð21Þ

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The average backordering cost AB: AB ¼

T12 ðP DÞ : 2TP

ð22Þ

As indicated in Figure 3, TP is the production period which is also equal to Q=P. Similarly, AS, AL, AR, and AM are from Equations (11), (18), (12) and (13). TC2 ¼ AS þ AI þ AB þ AL þ AR þ AM TC2 ¼ S

D Q

ðP DÞðTP T1 Þ2 þ

ð23Þ

h

. i R Q=P Craw 0 UðsÞds Q þ CM ðtÞ r 2TP

T21 ðP

DÞ 2TP ( ) Z Q=P D t 2 2 þ K ½ða0 þ G W sÞ m þ ds Q 0 3CP Z Q=P D ða0 þ G W sÞds þ Craw Q 0 þ

þ D CM ðtÞ:

ð24Þ

Decision variables are a0 , t, Q, and T1

Lot size

Inventory level

T

−D

P−D

0

Time s

T1

T2 TP

Figure 3. EPQ model with shortage.

T3

T4

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The evidence of relationship and dependence among variables in Case 1 also appears in Case 2. By referring to Equations (A3.18), (A3.21), (A3.25), and (A3.30), it is apparent that the production-inventory related decision variables, Q and T1, are correlated with process-quality decision variables, a0 and t. The process related decision variables, a0 and t, are linked with production-inventory related decision variables, Q and T1, and related conditions such as r, P and D. Obviously, both the production-inventory management approach and the process-quality design approach are allied even when a backorder exists. For this reason, it is necessary to integrate these two approaches to achieve product quality at an economical cost.

4.2.2 Nonconforming product is produced for inventory management In contrast to Cases 1 and 2, Case 3 does not assume a conforming product. Rather, a nonconforming product is assumed. Because of this relaxation of assumptions, the failure cost needs to be considered and a revision in terms of quality loss needs to be made. The expected quality loss AL and AF: Let the quality loss and failure cost at time s be: Zm Z mþd 2 l ðsÞ ¼ K ðX mÞ f ðXÞdX þ K ðX mÞ2 f ðXÞdX md

ð25Þ

m

Z

Z

md

1

f ðXÞdX þ CF

f ðsÞ ¼ CF

f ðXÞdX:

0

ð26Þ

mþd

The cumulative quality loss for imperfect product for production period Q=P is: Z Q=P l ðsÞds LðQ=PÞ ¼ 0 Z Q=P Z m Z mþd ¼ K ðX mÞ2 f ðXÞdX þ K ðX mÞ2 f ðXÞdX ds ð27Þ 0

md

Z

m

Q=P

FðQ=PÞ ¼

f ðsÞds Z Q=P Z ¼ CF 0

0

Z

md

1

f ðXÞdX þ CF 0

f ðXÞdX ds:

ð28Þ

mþd

The expected quality loss and expected failure cost for cycle time Q=D are: D LðQ=PÞ Q Z Q=P Z m Z mþd D 2 2 ¼ K ðX mÞ f ðXÞdX þ K ðX mÞ f ðXÞdX ds Q 0 md m

AL ¼

D FðQ=PÞ Q Z Q=P Z md Z1 D ¼ CF f ðXÞdX þ CF f ðXÞdX ds : Q 0 0 mþd

ð29Þ

AF ¼

ð30Þ

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Backorder is allowed for nonconforming product

Case 3:

With the exception of AL and AF, the remaining AS, AI, AB, AR, AM are the same as those in Case 2. Thus, by referring to Equations (11), (21), (22), (29), (30), (12), and (13), it is apparent that the total cost, TC3, with nonconforming product and backorder is: TC3 ¼ AS þ AI þ AB þ AL þ AF þ AR þ AM TC3 ¼ S

D Q

ðP DÞðTP T1 Þ2

h

Craw

R Q=P

þ þ þ

0

ða0 þ G W sÞds

.

ð31Þ

i Q þ CM ðtÞ r

2TP T21 ðP D Q

D þ Q

DÞ 2TP Z Q=P Z 0

Z

m

K ðX mÞ2 f ðXÞdX þ md

Q=P

Z CF

0

Z

md

Z

K ðX mÞ2 f ðXÞdX ds

m

1

f ðXÞdX þ CF 0

Z Q=P D þ Craw ða0 þ G W sÞds Q 0

mþd

f ðXÞdX ds

mþd

þ D CM ðtÞ:

ð32Þ

Decision variables are a0 , t, Q, and T1.

5. Illustrative example For the purpose of illustrating the proposed model and the themes arising from the preceding discussion, an example is provided that can be demonstrated. Let m ¼ 30 mm, K ¼ $2000/mm2, W ¼ 0.05 mm/month, G ¼ 1, CP ¼ 1, P ¼ 1200 units per month, D ¼ 1000 units per month, S ¼ $150 for each production run (setup cost), and r ¼ 10 percent of dollar value per unit time. Then, the coefficients of the tolerance function, a ¼ 20.00, b ¼ 70.00, c ¼ 30.00, and d ¼ 0.30 mm. Table 1 attempts to prove the necessity of integrating the production-inventory management approach with the process-quality design approach. The numerical analysis is

Table 1. Various process designs, quality levels, and quality costs vs. identical unit cost and production order quantity. i

U

1 2 3 4 5

29.70 29.85 30.00 30.15 30.30

t

Q

(U T )2 þ 2

Failure rate

Failure cost

0.220785 395.628 0.163595 0.1742 $2.6130 0.220396 395.628 0.095965 0.1735 $2.6025 0.219799 395.628 0.073266 0.1723 $2.5845 0.219596 395.628 0.095699 0.1719 $2.5785 0.219391 395.628 0.163130 0.1715 $2.5725 R md R1 R md R1 Note: failure rate ¼ 0 f ðxÞdx þ mþd f ðxÞdx and failure cost ¼ CF ð 0 f ðxÞdx þ mþd f ðxÞdxÞ.

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based on the conventional EPQ model with the above conditions except that the conditions of the deterioration process and quality loss are absent. By assuming a mixture of U and t for having a fixed and equal unit cost, $120, it is possible that various combinations of process mean, U, and process tolerance, t, representing alternative process designs, may result in an identical production order quantity, Q. However, aside from the systematic error resulting from process deterioration, associated quality measurements such as process variability, failure rate, and failure cost for each alternative are quite different. As a result, the possibility of process design for quality enhancement may be lost. When the assumption of a fixed unit cost is relaxed, quality improvement, along with a cost reduction in the determination of production quantity, becomes possible. Thus, a variable unit cost that takes into consideration alternative processes is absolutely necessary for model development. Because Case 3 is the most generalised case, we will focus on it for the purpose of numerical analysis. To prove the convexity of the total cost TC3 with respect to decision variables, a0 , t, Q, and T1, let us consider the following drawings. For example, when a0 is fixed at 29.9774 mm, the relationship between TC3, Q, and t gives a convex shape which is shown in Figure 4. Similarly, the convex shapes for TC3 versus any two combinations of a0 , t, Q, and T1 still hold. However, only partial combinations are shown in Figures 5 and 6 for the purpose of demonstration. Clearly, we can conclude that TC3 must be a convex set with global optimal solutions existing. The computer software, Mathematica, has been used to solve the proposed model (Trott 2006). With the conditions given in the beginning of this section, the optimal solutions are: initial setting a0 ¼ 29.9774 mm, process tolerance t ¼ 0.335040 mm, T1 ¼ 0.147935, production order quantity Q ¼ 1038.72 units, and total cost TC3 ¼ $20369.37/month.

Figure 4. TC3 versus Q and t. Note: a0 ¼ 29.9774 mm, T1 ¼ 0.147935.

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Figure 5. TC3 versus Q and a0 . Note: t ¼ 0.335040 mm, T1 ¼ 0.147935.

Figure 6. TC3 versus Q and T1. Note: a0 ¼ 29.9774, t ¼ 0.335040 mm.

A. Jeang

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International Journal of Production Research $ 800

TC3

AI + AB + AL + AR + AF

600

400

200

0

1000

Q*

2000

3000

4000

AS Q

= 1038.72

Figure 7. The partial components of TC3 such as AS and AI þ AB þ AL þ AR þ AF versus production order quantity Q.

Figure 7 is a plot of TC3 versus Q, with two components of TC3 shown separately. The first component is AS. The second component is AI þ AB þ AL þ AR þ AF. The tolerance cost CM(t), could be a þ b expðc tÞ. The reason for AM being excluded is that AM is independent of order quantity, Q. As shown in Figure 7, when Q increases, the component, AS, decreases; at the same time, the component, AI þ AB þ AL þ AR þ AF, increases; thus, one decreases while the other increases. This is precisely the trade-off between first and second components that was mentioned previously. In the case that quality-related costs AL or AF are absent, the new shape of the second component becomes flatter than the original shape. Consequently, the new Q turns out to be a greater value than the original Q. This result coincides with the observation that the conventional inventory model generally overproduces as a result of failing to take into consideration quality related costs. In fact, quality related costs are connected not only to the order quantity being produced, but also to process parameters being established. This connection becomes even tighter under a deterioration process. Thus, as mentioned in the preceding discussion on model development, it is essential to integrate production inventory management with a process quality design approach in order to truly optimise the production order quantity and process parameters for quality and cost. To confirm this thesis, a sensitivity analysis of the proposed model TC3 was carried out to study the effects of variables associated with a production inventory management approach and a process-quality design approach. These variables include: the deterioration rate, W, quality loss coefficients, K, the coefficient, b of CM(t), production rate, P, and inventory carrying rate, r. For various deterioration rate W values, the influence on the values of a0 , t , Q , T1 , TP , T, and TC3 are displayed in Table 2. From this table, the following results were obtained: (1) When the deterioration rate, W, increases, the following happens: AS increases, AI decreases, AL increases, AB increases, AF increases, AR decreases, AM decreases, and TC3 increases.

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A. Jeang Table 2. The values of a0 , t , Q , T1 , TP, T, and TC3 vs. W. Deterioration rate a0 t Q TP T1 T TC3

W ¼ 0.001

W ¼ 0.01

W ¼ 0.05

W ¼ 0.1

29.9986 29.9947 29.9774 29.9558 0.334671 0.334684 0.335040 0.33618 1041.46 1039.43 1038.72 1026.58 0.867883 0.866192 0.865600 0.855483 0.147357 0.147785 0.147935 0.150533 1.04146 1.03943 1.03872 1.02658 $20,369.14 $20,369.17 $20,369.37 $20,370.01

Note: K ¼ 2000, b ¼ 70, r ¼ 0.1, P ¼ 1200.

Table 3. The values of a0 , t , Q , T1 , TP , T, and TC3 vs. K. Deterioration rate a0 t Q TP T1 T TC3

K ¼ 1000

K ¼ 1500

K ¼ 2000

K ¼ 2500

29.9763 0.359010 1040.14 0.866783 0.147630 1.04014 $20,359.01

29.9769 0.345079 1039.94 0.866617 0.147674 1.03994 $20,364.27

29.9774 0.335040 1038.72 0.865600 0.147935 1.03872 $20,369.37

29.9776 0.327273 1028.50 0.857083 0.150123 1.02850 $20,374.12

Note: W ¼ 0.05, b ¼ 70, r ¼ 0.1, P ¼ 1200.

(2) The reason that a high deterioration rate W will end up with a small a0 at a farther distance from target, m, can be explained by the fact that a high deterioration rate W will increase the probability of the product quality value becoming poorer early on in the cycle time, while the cumulative quality loss will increase significantly as products are being produced. Therefore, a0 should be strategically located so as to ensure that the product quality value remains at an acceptable level for the remainder of the cycle time. (3) An increase in the deterioration rate, W, causes a greater t value, namely, a smaller unit cost. Consequently, a smaller AI is needed to compensate for other increased costs. (4) An increase in the deterioration rate, W, results in a reduction of Q , Tp, and T values. This may be explained as follows: when the deterioration rate increases, it is possible that the cumulative quality loss will significantly increase as products are being produced. Hence, we prefer to have a smaller production quantity to ensure that the produced product performs its desired function. Here again, smaller Tp, and T values will result. (5) An increase in the deterioration rate W causes a greater T1 value. This can be explained by the fact that AI is reduced because of shortage being held. Similarly, the influence on the values of a0 , t , Q , T1 , TP, T, and TC3 under various quality loss coefficients, K, are displayed in Table 3. The following results were

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International Journal of Production Research Table 4. The values of a0 , t , Q , T1 , TP , T, and TC3 vs. r. Deterioration rate a0 t Q TP T1 T TC3

r ¼ 0.1

r ¼ 0.2

r ¼ 0.3

r ¼ 0.4

29.9774 0.335040 1038.72 0.865600 0.147935 1.03872 $20,369.37

29.9803 0.335127 892.699 0.743916 0.182359 0.892699 $20,510.52

29.9812 0.335284 836.852 0.697377 0.197175 0.836852 $20,642.74

29.9817 0.335456 805.348 0.671123 0.205804 0.805348 $20,771.80

Note: K ¼ 2000, b ¼ 70, W ¼ 0.05, P ¼ 1200.

obtained: (1) When the quality loss coefficient, K, increases, the following happens: AS increases, AI decreases, AL increases, AB increases, AF decreases, AR increases, AM increases, and TC3 increases. (2) An increase in quality loss coefficient, K, results in a reduction of the Q value. This may be explained as follows: when K increases, the cumulative quality loss will increase considerably as units are produced. To trim this loss, we prefer frequent resetting of the production process at a quality level. Similar explanations can be found for smaller T and TP values. (3) A greater K value causes a smaller process tolerance value t . This result can be explained by the fact that a greater K value will drive a greater value of AL. Naturally, a smaller process tolerance t value is preferred. (4) A greater K value causes a greater a0 value. The reason is that the closeness of the process mean to the design target, m, reduces the cumulative quality loss for greater K values. (5) An increase in quality loss coefficients, K, causes a greater T1 value. This is because a higher shortage occurs under greater T1 values, allowing AI to be reduced significantly. The influence on the values of a0 , t , Q , T1 , TP, T, and TC3 under various inventory carrying rates, r, is listed in Table 4. The following results were obtained: (1) When the inventory carrying rate, r, increases, the following happens: AS increases, AI increases, AL decreases, AB increases, AF decreases, AR decreases, AM decreases, and TC3 increases. (2) As observed, a high inventory carrying rate, r, will bring about a greater a0 value at a close distance to the design target, m. The reason is that a high carrying rate, r, causes a greater AI value, which is increased radically as cycle time increases. One way to decrease the cycle time is to locate the initial setting close to design target, m. Similar explanations can be found for the decrease in Q as the r value increases. (3) An increase in the inventory carrying rate, r, causes a greater t value or smaller unit cost. Consequently, a smaller AI is required to compensate for other increased costs. (4) An increase in the inventory carrying rate, r, results in a reduction of Q , Tp, and T values. This can be explained by the fact that when the inventory carrying rate, r, increases, the AI value or average inventory carrying cost will significantly increase

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Table 5. The values of a0 , t , Q , T1 , TP , T, and TC3 vs. P. Deterioration rate a0 t Q TP T1 T TC3

P ¼ 1200

P ¼ 1500

P ¼ 1800

P ¼ 2100

29.9774 0.335040 1038.72 0.865600 0.147935 1.03872 $20,369.37

29.9813 0.343197 1030.12 0.686747 0.200576 1.03012 $20,527.59

29.9887 0.349260 987.778 0.548766 0.245275 0.987778 $20,696.59

29.9893 0.354862 965.032 0.459539 0.266130 0.965032 $20,871.88

Note: K ¼ 2000, b ¼ 70, W ¼ 0.05, r ¼ 0.1.

as products are being produced. Hence, we prefer to produce a smaller production quantity Q (with a shorter production period Tp and production cycle T ) to avoid a greater AI value resulting. (5) An increase in the inventory carrying rate, r, causes a greater T1 value. This can be explained by the fact that the increase in the carrying rate, r, also results in an increase in AI. Thus, we would prefer a greater T1 for greater backorder in order to avoid a greater AI resulting. Table 5 intends to show the influence on the values of a0 , t , Q , T1, TP, T*, and TC3 under various production rates P. The results are discussed as follows: (1) When production rate, P, increases, the following happens: AS increases, AI decreases, AL decreases, AB increases, AF decreases, AR decreases, AM decreases, and TC3 increases. (2) A greater production rate, P, will result in a greater a0 value which is close to design target, m. The reason for this closeness is that a greater production rate, P, causes a greater AI value at the beginning of the production cycle time. One way to decrease the AI value is to reduce the production cycle time. This can be achieved by locating the initial setting in the neighbourhood of design target, m. (3) An increase in the production rate, P, causes a greater t value, namely, a smaller unit cost. Consequently, a smaller AI is gained, which is insensitive to a greater P value. (4) An increase in the production rate, P, results in a reduction of Q , Tp, and T values. The explanation appears under point (2). To avoid the AI value increasing as the production rate, P, increases, we prefer to produce a smaller production quantity Q , with a shorter production period Tp and production cycle T. (5) An increase in the production rate, P, causes a greater T1 value. This can be explained by the fact that AI has increased, also due to the increased production rate, P. Thus, we prefer to have a greater T1 for greater backorder to avoid a greater AI resulting. Table 6 describes the influence on the values of a0 , t , Q , T1 , TP, T, and TC3 under the various coefficient, b. The results are discussed as follows: (1) When the coefficient, b, increases, the following happens: AS increases, AI decreases, AL increases, AB increases, AF increases, AR decreases, AM decreases, and TC3 increases.

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International Journal of Production Research Table 6. The values of a0 , t , Q , T1 , TP , T, and TC3 vs. coefficient b of CM(t). Deterioration rate a0 t Q TP T1 T TC3

b ¼ 50

b ¼ 60

b ¼ 70

b ¼ 80

29.9776 0.323141 1040.67 0.867225 0.147522 1.04067 $20,368.30

29.9775 0.329560 1039.70 0.866417 0.147933 1.03970 $20,368.81

29.9774 0.335040 1038.72 0.865600 0.147935 1.03872 $20,369.37

29.9772 0.339832 1031.24 0.859367 0.149530 1.03124 $20,369.71

Note: K ¼ 2000, r ¼ 0.1, W ¼ 0.05, P ¼ 1200.

Table 7. Comparison of results between proposed approach and production-inventory management approach. Proposed approach W 0.001 0.01 0.05 0.1

Production-inventory management approach

Deviation between two approaches

Q

TC3 (1)

Q0

TC03 (2)

DTC(2) – (1)

1041.46 1039.43 1038.72 1026.58

$20,369.14 $20,369.17 $20,369.37 $20,370.01

1549.19 1549.19 1549.19 1549.19

$20,712.22 $20,712.33 $20,714.61 $20,721.58

$343.08 $343.16 $345.24 $351.57

Note: t ¼ 0.179813, a0 ¼ 30.

(2) As observed, a greater b value will result in a smaller a0 value indicating a farther distance from the design target, m. The reason is that a greater b value increases the tolerance cost. Naturally, we would like a greater tolerance value t indicating a smaller tolerance cost to pay off the additional cost resulting from the greater b value. To obtain a greater tolerance value and avoid an increase in the failure cost, we would like to establish a0 at a distance farther from the design target, m. (3) An increase in the b value causes a greater t value, namely, a greater AL and a greater AF. Thus, we prefer to have a smaller Q to reduce the cumulative quality loss and failure cost during the production cycle time. Similar explanations can be found for preferring smaller Tp and T values. (4) An increase in the b value causes a greater T1 value. This can be explained by the fact that AI has increased because of the b value. Thus, we prefer to have a greater T1 for greater backorder to avoid a greater AI resulting. To continue verifying the necessity of integrating the production-inventory management approach with the process-quality and cost design approach, two comparisons have been made. These comparisons appear in Tables 7 and 8. Table 7 compares the proposed model with the production-inventory management approach. Table 8 compares the proposed model with the process-quality and cost design approach. It is apparent in Table 7 that the total cost is increased when the production-inventory management approach is the only approach taken. The outcomes show that the total cost, TC3 , derived

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Table 8. Comparison of results between proposed approach and process-quality design approach. Proposed approach W 0.001 0.01 0.05 0.1

Process-quality and cost design approach

Deviation between two approaches

Q

TC3 (1)

Q00

TC003 (2)

œTC(2)–(1)

1041.46 1039.43 1038.72 1026.58

$20,369.14 $20,369.17 $20,369.37 $20,370.01

709.375 697.787 683.828 647.871

$21,026.37 $21,026.48 $21,029.61 $21,031.80

$657.23 $657.31 $658.24 $666.79

from the proposed model is always lower than the total cost, TC03 , obtained by the conventional EPQ model under a deterioration process, where TC03 is found by substituting a0 ¼ m, t ¼ tL, and Q ¼ Q0 into the TC3 expression. The design target, m, is 30 mm and the tightest tolerance, tL, is 0.179813 mm in this example. Q0 is gained from the conventional EPQ model, which lacks consideration of process-quality design. The reason for using the inputs, a0 ¼ m and t ¼ tL, as the process setting for the production-inventory management system is that the process designer normally establishes the process mean at design target, m, for quality consideration and has the process tolerance at the lower process limit, tL, for conservative consideration. Obviously, because the initial setting a0 has been established at target, m, there is no compensation for the deterioration process, and the impact on the total cost becomes significant as the deterioration rate, W, value increases. This represents an extreme situation. The other extreme is shown in Table 8. Here, the process-quality design system is run alone. As we can see from this table, the difference between TC3 and TC003 increases as the deterioration rate, W, increases. TC003 is found by substituting a0 ¼ a000 , t ¼ t00 , and Q ¼ Q00 into the TC3 expression. Having Wp ¼ W=P, the values, a000 , t00 , and Q00 are found by optimising the process-quality design formulation in Equation (33): q0 2 Z mþd 0Z m 1 3 2 2 ZQ K ðX mÞ fðXÞdX þ K ðX mÞ fðXÞdX B md C 07 1 6 B Cdq 7 þ CM ðtÞ, TCq0 ¼ 6 Z md Z m1 @ A 5 4 Q 0 þ CF f ðXÞdX þ CF fðXÞdX 0

mþd

ð33Þ where Uðq0 Þ ¼ a0 þ G WP q0 . Clearly, a recommendation for the incorporation of these two approaches can be derived from Tables 7 and 8. If the production-inventory management approach and the process-quality and cost design approach work separately, the total cost of each approach must be higher than the total cost of the proposed approach. Thus, integrating these two approaches in the proposed model is necessary to truly achieve an economic production order quantity that also represents quality. In order to greatly improve a producer’s ability to meet market opportunities, lower development cycle time, reduce production costs, and deliver high-quality products, the integration of production planning and process design becomes necessary (Jeang and Chung 2008). As demonstrated by the examples in Tables 1–8, an economic and quality product can be achieved through the proposed model early in the preparation stage.

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6. Summary The proposed models considered minimising the total cost, which includes: the costs of setup, inventory holding, backorder, quality loss, and tolerance, by conducting concurrent optimisation of the decision variables, including: the initial setting, process tolerance, production period for backorder, and production order quantity. Computer software, Mathematica, was used to find the optimal values for decision variables of interest. One example has been introduced to demonstrate the optimal solutions that this model is capable of achieving, which will augment its application. The significance of the proposed model in comparison to former approaches becomes evident as an enhanced high-quality product with enhanced functionality is produced at a minimal total cost.

Acknowledgments This research was carried out in the Design, Quality, and Productivity Laboratory (DQPL) at the Department of Industrial Engineering and Systems Management at Feng Chia University, Taichung, Taiwan, Republic of China. I would like to thank my research assistant, Mr. Yen-Pin Pai, graduate student in the I.E. Department.

References Arora, J.S., 1989. Introduction to optimum design. New York: McGraw-Hill. Chase, K.W., et al., 1990. Least cost tolerance allocation for mechanical assemblies with automated process selection. Manufacturing Review, 3 (1), 49–59. Chen, C.H. and Lai, M.T., 2007. Economic manufacturing quantity, optimum process mean, and economic specification limits setting under the rectifying inspection plan. European Journal of Operational Research, 183 (1), 336–344. Chung, K.L. and Hou, K.L., 2003. An optimal production run time with imperfect production process and allowable shortages. Computer and Operation Research, 30 (4), 483–490. Drezner, Z. and Wesolowsky, G.O., 1989. Optimal control of a linear trend process with quadratic loss. IIE Transactions, 21 (1), 66–72. Ganeshan, R., Kulkarni, S., and Boone, T., 2001. Production economics and process quality: a Taguchi perspective. International Journal of Production Economics, 71 (3), 343–350. Hax, A.C. and Canadea, D., 1984. Production and inventory mangement. Englewood Cliffs, NJ: Prentice-Hall. Kim, C.H. and Hong, Y., 1999. An optimal production run length in deteriorating production processes. International Journal of Production Research, 58 (2), 183–189. Jeang, A., 1994. Tolerance design: choosing optimal tolerance specification in the design machined parts. Quality and Reliability Engineering International, 10 (1), 27–35. Jeang, A., 1998. Reliable tool replacement policy for quality and cost. European Journal of Operation Research, 108 (2), 334–344. Jeang, A., 2001. Combined parameter and tolerance design optimization with quality and cost reduction. International Journal of Production Research, 39 (5), 923–952. Jeang, A. and Chung, C.P., 2008. Process mean, process tolerance, and use time determination for product life application under deteriorating process. International Journal of Advanced Manufacturing Technology, 36 (1), 97–113. Jeang, A. and Yang, K., 1992. Optimal tool replacement with nondecreasing tool wear. International Journal of Production Research, 30 (2), 299–314. Phadke, M.S., 1989. Quality engineering using robust design. Englewood Cliffs, NJ: Prentice Hall.

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Porteus, E.L., 1986. Optimal lot-sizing process quality improvement and setup cost reduction. Operations Research, 34 (1), 137–144. Rahin, M.A. and Al-Hajailan, W.I., 2006. An optimal production run for an imperfect production process with allowable shortages and time-varying fraction defective rate. International Journal of Advanced Manufacturing Technology, 27 (11), 1170–1177. Rampersad, H.K., 2001. Total quality management: an executive guide to continuous improvement. New York: Springer. Rosenblatt, M.J. and Lee, H.L., 1986. Economic production cycle with imperfect production process. IIE Transactions, 18 (1), 48–55. Salameh, M.K. and Jaber, M.Y., 2000. Economic order quantity model for items with imperfect quality. International Journal of Production Economics, 64 (1), 59–64. Silver, E.A. and Peterson, R., 1985. Decision systems for inventory management and production planning. New York: Wiley. Taguchi, G., Elsayed, E., and Hsiang, T., 1989. Quality engineering in production systems. Singapore: McGraw-Hill. Trott, M., 2006. The Mathematica guide book for numerics. New York: Springer ScienceþBusiness Media.

Appendix 1: Necessary optimality conditions for a0 , t , and Q TC1 is given in Equation (20) for Case 1. To find the optimal initial setting a0 , process tolerance t , and economic production order quantity Q , we take the first derivative of TC1 with respect to a0 , t, and Q, respectively and then equate it to zero. (1) The derivative of TC1 with respect to a0 : 9bcect ð2DP DQr þ PQrÞ þ 4DKC2P t ¼0 18P

ðA1:1Þ

(2) The derivative of TC1 with respect to t: @TC1 9bc ect ð2DP DQr þ PQrÞ þ 4Dt : ¼ 18P @t

ðA1:2Þ

(3) The derivative of TC1 with respect to Q: @TC1 ¼ GWQ3 ½3Craw ðP DÞ r þ 4GDKW @Q þ 3Q2 PðP DÞ ða0 Craw þ aPÞ r þ GDPðCraw þ 2Kða0 mÞÞW þ bP2 ðP DÞ rect 6DP3 S:

ðA1:3Þ

The optimal a0 , t , and Q can be obtained by setting Equations (A1.1), (A1.2), and (A1.3) to be equal to zero and solving these simultaneous equations. Regarding whether the found a0 , t , and Q are global optimal solutions, then Appendix 2 for verifying the sufficient optimality conditions intends to answer this question.

Appendix 2: Sufficient optimality conditions for a0 , t , and Q The Hessian matrix of Equation (20) for Case 1 is: @2 TC1 h11 h12 h13 @a20 2 TC1 H ¼ h21 h22 h23 ¼ @@[email protected] h31 h32 h33 @2 TC01 @[email protected]

@2 TC1 @a0 @t @2 TC1 @t2 @2 TC1 @[email protected]

@2 TC1 @a0 @Q @2 TC1 : @[email protected] @2 TC1 @Q2

ðA2:1Þ

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The sufficient optimality conditions for a0 , t , and Q are that M1, M2, and M3 are positive (Arora 1989). The proofs for positive M1, M2, and M3 are: 2DK : P

ðA2:2Þ

DKð4D þ 9bc2 ect ð2DP þ PQr DQrÞÞ 9P2

ðA2:3Þ

M1 ¼ Because of P4D: M2 ¼

M3 ¼

2 ðPDÞrw DKW2 DK 4D þ 9bc2 ect ð2DP þ PQr DQrÞ 2DS þ GCraw2P þ 2G 3P 3 3 Q3 9P2

4 0:

ðA2:4Þ

Clearly, M1, M2, and M3 are positive, the solutions of a0 , t , and Q must be the global optimal solutions (Arora 1989).

Appendix 3: Relevant properties for proposed model Case 1:

Backorder is not allowed for conforming product

Property 1:

The optimal initial value a0 for Case 1 is: 1 Q Craw 1 1 1 Qr : 1þ a0 ¼ m GW 2 P 2 D P 2K

ðA3:1Þ

Derivation: Let @TC1 [email protected] ¼ 0, thus, Equation (A1.1) is equal to zero. Craw ð2DP DQr þ PQrÞ þ 2DKð2a0 P 2mP þ GQWÞ 2P2

ðA3:2Þ

1 Q Craw 1 1 1 1þ Qr : a0 ¼ m GW 2K 2 P 2 D P

ðA3:3Þ

where Craw, K, r, D, G, W, P are known values. Equation (A3.3) tells us that the optimal initial setting a0 established at the beginning of cycle time, Q=D, is the deviation, 1 Q Craw 1 1 1 GW þ Qr , 1þ 2 P 2 D P 2K away from target value m. Property 2: When the tolerance cost function CM ðtÞ is a þ bect , the optimal tolerance value t for Case 1 is: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 18DPbc2 þ 9Ac 4DKC2P þ 324DPbcðAc2 þ DPbc3 Þ þ 81A2 c2 , ðA3:4Þ t ¼ 18DPbc3 þ 9Ac2 where A ¼ bcQrðP DÞ. Derivation: Let @TC1 [email protected] ¼ 0, thus, Equation (A1.2) is equal to zero. 9bcect ð2DP DQr þ PQrÞ þ 4DKC2P t ¼ 0: 18P

ðA3:5Þ

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A. Jeang The ect can also be expressed as: ect ¼ 1 ct þ

ðctÞ2 : 2

ðA3:6Þ

Solve Equation (A3.5) by having the expression ect be replaced with Equation (A3.6); then: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 18DPbc2 þ 9Ac 4DKC2P þ 324DPbcðAc2 þ DPbc3 Þ þ 81A2 c2 , ðA3:7Þ t ¼ 18DPbc3 þ 9Ac2 where A ¼ bcQrðP DÞ. Property 3:

The optimal production order quantity Q for Case 1 is: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 q 3 q q2 p3 q2 p3 a þ Q ¼ þ þ þ , 2 2 4 27 4 27 3

ðA3:8Þ

where: ETC1 ¼ 3GWCraw r ðP DÞ þ 4G2 DKW2 FTC1 ¼ 3 PðP DÞ ða0 Craw þ aPÞ r þ GDPðCraw þ 2Kða0 mÞÞW þ bP2 ðP DÞ rect GTC1 ¼ 6DP3 S 1 FTC1 2 p¼ 3 ETC1 GTC1 2 FTC1 3 q¼ þ ETC1 27 ETC1 FTC1 a¼ : ETC1 Derivation: Let @TC1 [email protected] ¼ 0 and rearrange, we have: Q3 3rGWCraw ðP DÞ þ 4G2 DKW2 þ Q2 3 PðP DÞ ða0 Craw þ aPÞ r þ GDPðCraw þ 2Kða0 mÞÞW þ bP2 ðP DÞ rect 6P3 SD ¼ 0

ðA3:9Þ

Let: ETC1 ¼ 3GWCraw r ðP DÞ þ 4G2 DKW2 FTC1 ¼ 3 PðP DÞ ða0 Craw þ aPÞ r þ GDPðCraw þ 2Kða0 mÞÞW þ bP2 ðP DÞ rect GTC1 ¼ 6DP3 S: Then: Q3 þ

FTC1 2 GTC1 Q þ ¼ 0, ETC1 ETC1

ðA3:10Þ

where the coefficients, FTC1 =ETC1 and GTC1 =ETC1 , are known values. By the Cardano formula, we have the corresponding root of Equation (A3.10), which is: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 q 3 q q2 p3 q2 p3 a Q ¼ þ þ þ : þ ðA3:11Þ 4 27 4 27 3 2 2

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29

Backorder is allowed for conforming product

Property 4:

The average inventory cost, AI, and the backorder cost, AB, are: AI ¼

rCU ðP DÞðTP T1 Þ2 2TP

ðA3:12Þ

T21 ðP DÞ : 2TP

ðA3:13Þ

AB ¼ Derivation:

By referring to Figure 3, the production periods, TP, are: TP ¼ Q=P and TP ¼ T1 þ T2. The average inventory I is: Imax ¼ ðP DÞT2 ¼ ðP DÞðTP T1 Þ Imax ¼ DT3 Imax ðP DÞðTP T1 Þ ¼ T3 ¼ D D 1

I¼2

ðT2 þ T3 ÞImax ðP DÞðTP T1 Þ2 ¼ : T 2TP

ðA3:14Þ

The average backorder B is: ðP DÞT1 ¼ DT4 ðP DÞT1 T4 ¼ D 1

B¼2

T1 ðP DÞT1 þ 12 T4 DT4 T21 ðP DÞ : ¼ 2TP T

ðA3:15Þ

The average inventory holding cost AI is: AI ¼ rCU I ¼

rCU ðP DÞðTP T1 Þ2 : 2TP

ðA3:16Þ

The average backordering cost AB is: AB ¼ B ¼ Property 5:

T21 ðP DÞ : 2TP

ðA3:17Þ

The optimal initial value a0 for Case 2 is:

Craw PrðQ PT1 Þ2 þ D Q2 r þ P2 rT21 2PQð1 þ rT1 Þ þ 4mPDKQ 2DKGQ2 W : a0 ¼ 4PDKQ ðA3:18Þ Derivation: Let @TC2 [email protected] ¼ 0, we have: Craw PrðQ PT1 Þ2 D Q2 r þ P2 rT21 2PQð1 þ rT1 Þ þ 2DKQð2a0 P 2mP þ GQWÞ ¼ 0: 2P2 Q ðA3:19Þ

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A. Jeang Then:

Craw PrðQ PT1 Þ2 þ D Q2 r þ P2 rT21 2PQð1 þ rT1 Þ þ 4mPDKQ 2DKGQ2 W ðA3:20Þ a0 ¼ 4PDKQ The optimal tolerance value t for Case 2 is:

Property 6:

t ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 4 bc2 ð2D þ ZÞ þ 4D bc2 ð2D þ ZÞ þ 4D 9P þ 9P 2b c ð2D þ ZÞ bc3 ð2D þ ZÞ

,

ðA3:21Þ

where Z¼

rðP DÞðQ PT1 Þ2 : PQ

Derivation: Let @TC2 [email protected] ¼ 0, we have: 2Dt 1 ðctÞ2 ðP DÞðQ PT1 Þ2 þ bc 1 ct þ 2D þ r ¼ 0: 9P 2 2 PQ

ðA3:22Þ

Rearrangement: rðP DÞðQ PT1 Þ2 bc 2D þ PQ rðP DÞðQ PT1 Þ2 4D bc2 2D þ þ t PQ 9P 1 3 rðP DÞðQ PT1 Þ2 bc 2D þ t2 ¼ 0: þ 2 PQ

ðA3:23Þ

Then:

t ¼ Property 7:

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 4 bc2 ð2D þ ZÞ þ 4D bc2 ð2D þ ZÞ þ 4D 9P þ 9P 2b c ð2D þ ZÞ bc3 ð2D þ ZÞ

:

ðA3:24Þ

The optimal production order quantity Q for Case 2 is: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 q 3 q q2 p3 q2 p3 a þ þ þ , Q ¼ þ 4 27 4 27 3 2 2

where: ETC2 ¼ 8G2 KTP W2 FTC2 ¼ 3GCraw P2 WrðT1 TP Þ2 þ 3GPW Craw rðT1 TP Þ2 2TP þ 4KTP ðm a0 Þ GTC2 ¼ 12P3 STP 1 FTC2 2 p¼ 3 ETC2 GTC2 2 FTC2 3 q¼ þ ETC2 27 ETC2 a¼

FTC2 : ETC2

ðA3:25Þ

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8 9 < 12P3 STP = 2 3GCraw P Q WrðT1 TP Þ þ 3GPQ2 W Craw rðT1 TP Þ 2TP þ 4KTP ðm a0 Þ : ; 8G2 KQ3 TP W2 ¼0 12P3 Q2 TP ðA3:26Þ 2

2

2

Rearrangement: Q3 ð8G2 KTP W2 Þ þ Q2 3GCraw P2 WrðT1 TP Þ2 þ 3GPW Craw ðrðT1 TP Þ2 2TP Þ þ 4KTP ðm a0 Þ 12P3 STP ¼ 0:

ðA3:27Þ

Having ETC2 , FTC2 , and GTC2 as defined in the above expressions, then Equation (A3.27) can be written as the following: FTC2 2 GTC2 Q þ Q3 þ ¼ 0, ðA3:28Þ ETC2 ETC2 where the coefficients, FTC2 =ETC2 and GTC2 =ETC2 , are known values. Similarly, by the Cardano formula, the root is: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 q 3 q q2 p3 q2 p3 a Q ¼ þ þ þ : þ 4 27 4 27 3 2 2 Property 8:

ðA3:29Þ

The optimal production period T1 for Case 2 is: T1

Qr 2bP2 þ ect ½2Pða0 Craw þ aPÞ þ GCraw QW : ¼ 2bP3 r þ Pect ½2Pða0 Craw r þ Par þ PÞ þ GQWCraw r

ðA3:30Þ

Derivation: Let @TC2 [email protected] ¼ 0, we have: i h 0 PþGQWÞ ðP DÞ T1 T2P rðTP T1 Þ CM ðtÞ þ Craw ð2a2P 2 TP

¼ 0:

ðA3:31Þ

Then: T1 ¼

Qr 2bP2 þ ect ½2Pða0 Craw þ aPÞ þ GCraw QW : 2bP3 r þ Pect ½2Pða0 Craw r þ Par þ PÞ þ GQWCraw r

ðA3:32Þ

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Economic production order quantity and quality Angus Jeang* Department of Industrial Engineering and Systems Management, Feng Chia University, P.O. Box 25-150, Taichung, Taiwan, ROC (Received 1 March 2009; final version received 10 December 2009) The purpose of this study is to integrate a conventional production-inventory management approach and a process-quality design approach so as to promote quality and reduce costs. Because the integration of these two approaches into a single system is conducted under the influence of a deterioration process, estimated costs of production must be modified. Only then can these various costs be presented as a total cost for the present integrated system. This total cost includes: a setup cost for production reordering and process resetting, quality-related costs stemming from the loss of product quality along with a tolerance-related production process cost and a failure cost for product defects, backorder costs for production shortage, carrying costs generated by the storing and handling of inventory, and material costs for determining the process mean. The Taguchi quality loss function is introduced to assess quality loss in the system. The decision variables include the initial setting (process mean), process tolerance, and production order quantity. These variables need to be determined simultaneously to minimise the average total cost for a cycle time. An example is presented to demonstrate the proposed approach. Keywords: initial setting; process tolerance; production order quantity; quality; cost; deterioration process; optimisation

1. Introduction One of the drawbacks of conventional inventory models is the idealistic assumption that produced parts are all of good quality with the production process being in perfect condition. The problem with this assumption is that it does not hold true in practical application. Quality value may fall outside product specification due to the stochastic nature of produced quality values, and some defective parts may be produced due to a deterioration process related to systemic causes. Another drawback to conventional models is that decision problems are solved by using economic criteria obtained within a company. For a life cycle application, the relevant measurement criteria in resolving the production ordering problem should not be limited to costs solely involved in the production process; rather, quality-related costs that accrue after the delivery of goods to customers should also be included. Production process related issues in inventory modelling require additional consideration as these issues affect the quality and cost of the produced product in general. Hence, inventory, production, and quality costs should be simultaneously minimised when determining the optimal levels of production

*Email: [email protected] ISSN 0020–7543 print/ISSN 1366–588X online 2010 Taylor & Francis DOI: 10.1080/00207540903555528 http://www.informaworld.com

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order quantity. The proposed model is designed to meet these requirements which have long been neglected. The conventional belief that products with quality values that meet specification requirements will not result in quality loss is simply untrue. Recent studies have shown that this narrow definition of quality greatly underestimates the costs of poor quality and leads to decisions that are less than optimal, except in particular local situations. It is now no longer appropriate to base product acceptance or rejection on specification requirements. The Taguchi quality loss function was introduced to assess the loss arising from the quality value deviating from its design target (Phadke 1989, Taguchi et al. 1989). This assessment can also evaluate quality-related costs that accrue after the delivery of goods to customers. Consequently, the minimisation of quality variability is now the main concern of process and quality engineers in terms of product and process design. There are two ways of reducing variability: one is to establish the process mean as near as possible to the design target, and the other is to tighten the tolerance value as small as possible within the limits of process capability. Because process mean and process tolerance are mutually interdependent, a concurrent determination of process mean and process tolerance in balancing tolerance cost and quality loss is essential for a process design to realise economic and quality achievement (Jeang 2001). Defective items resulting from imperfect quality production processes were originally considered by several authors (Porteus 1986, Rosenblatt and Lee 1986, Kim and Hong 1999, Salameh and Jaber 2000). Chung and Hou (2003) have generalised the work of Kim and Hong and have assumed that shortage was permitted. However, as mentioned in the work of Rahin and Al-Hajailan (2006), the common assumption of the above models is that a fixed percentage of defective items were assigned. Thus, Rahin and Al-Hajailan have further generalised the work of Chung and Hou by introducing a time varying fraction defective rate. However, none of these works considered quality-related costs after the delivery of goods to customers, which means they lack a life cycle view in regard to cost evaluation. To offset this drawback Ganeshan et al. (2001) have linked the lot-size problem with the Taguchi quality loss function in inventory modelling development. The purpose was to determine the optimal levels of inventory and production order quantity in order to minimise inventory and quality-related costs. Chen and Lai (2007) revisited this interaction between the economics of production with the Taguchi quality cost perspective in determining both process mean and product specifications. An ordinary assumption among Ganeshan et al., and Chen and Lai, is that Taguchi quality loss is independent of time spent and products produced. However, as time passes, the output values of quality characteristics may change during the production process as time passes. Consequently, quality loss should be expressed as a function of the time period in which a product is produced. Another assumption appearing in the work of Ganeshan et al., and Chen and Lai is that Taguchi quality loss is purely a function of process mean. The assumption is mistaken, as the formulation excluding the influence of process tolerance is insufficient in representing quality-related costs, particularly when deterioration phenomena exist in the production process (Jeang 2001, Jeang and Chung 2008). The proposed approach integrates a process-quality design approach with a conventional production-inventory management approach in order to view process mean and process tolerance concurrently in assessing quality loss using the Taguchi quality loss function. A process-quality design model was recently developed in which the resetting cost, Taguchi quality loss, and tolerance cost are considered as functions of decision variables

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such as initial setting (process mean), process tolerance, and cycle time under the deterioration process (Jeang and Chung 2008). A unit cost subject to deterioration process is approximated as a function of process mean and process tolerance. This cost affects the inventory carrying cost, which in turn influences production order quantity as determined in the production-inventory management system. The chain of influence extends even further as the production order quantity influences the initial setting (process mean) and process tolerance obtained in process design. The inter-influence becomes great when the production process is subject to the deterioration process. The proposed model aims to take these influences into account by integrating production-inventory management and process-quality design. Due to the deterioration process, economic production order quantity (EPQ) is a good starting-point in representing the production-inventory management system. Inventory-related costs include an ordering cost and a carrying cost. The processrelated costs include: a resetting cost, quality-related costs stemming from loss of quality, and a tolerance cost. These costs are represented in the proposed model. Quality loss is expressed by the Taguchi quality loss function, and the tolerance cost appears as a tolerance cost function. Because the costs associated with a production-inventory management approach and a process-quality design approach are characterised by periodical recurrence, a single setup with one common cycle time for production reordering and process resetting is assumed in the proposed model (Hax and Canadea 1984, Silver and Peterson 1985, Drezner and Wesolowsky 1989, Jeang and Yang 1992, Jeang 1998). Other common costs, including a backorder cost for shortage, a failure cost for defectives, and a raw material cost for process mean, are also included. Thus, the costs of production-inventory and process-quality design are combined as a total cost in the proposed model. The purpose of this model is to concurrently optimise the cycle time, initial setting (process mean), process tolerance, and production order quantity by integrating previous process-design models with the production-inventory EPQ model. To achieve this optimisation, a trade-off among the various costs pertaining to both inventory and process-design is presented via mathematical representation. The paper itself is divided into six sections. Section 1 provides an introduction to the current study. Section 2 defines the relevant assumptions and notation. Section 3 outlines the background related to this research. Section 4 describes the model development. Section 5 demonstrates the proposed model by an example. Finally, a summary is given in Section 6.

2. Assumptions and notation Assumptions: (1) Produced quality value may deteriorate during the production process. (2) The deterioration rate is a linear function of time. (3) After each cycle time, the process mean is reset to its initial value to compensate for process deterioration. (4) Common cycle time for production reordering and process resetting. (5) Setup costs include production reordering cost and process resetting cost. (6) Shortages are allowed for backorder in Cases 2 and 3. (7) The demand rate and production rate are constant and deterministic.

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(8) Unit cost does not remain fixed over the cycle time, which is the sum of material costs and process costs. Material cost is a function of process mean and process cost is tolerance cost. Notation: AC AB AF AI AL AM AR AS a0 a000 B CF CM(t) CP CR Craw CU D d F(Q/P) f(X ) f(s) G h I Imax K L(X ) L(Q/P) l(s) m P Q Q0 Q00 QL(s) q0 RC r S

consumer loss; average backorder cost per time; average failure cost per time; average inventory cost per time; average quality loss per time; average tolerance cost per time; average material cost per time; average setup cost per time; initial setting for process mean at the beginning of the cycle time; initial setting for process-quality design approach (see Equation (33)); average backorder during production period T1; associated failure cost when quality value X exceeds its limits m d; tolerance cost per unit which is a function of tolerance t; process capability index; total raw material cost for production order quantity Q; raw material cost; unit cost of a product; demand rate in units per time; design specification (design tolerance) of produced quality value X; cumulative failure cost during production period Q/P; normal probability density function of produced quality value X; failure cost at time s; þ1 or 1 which is dependent on the direction of process deterioration; inventory carrying cost per unit per time; average on-hand inventory level for a cycle time, T; maximum on-hand inventory level for a cycle time, T; quality loss coefficient, which is: AC =S2max ; quality loss of produced quality value X; cumulative quality loss for nonconforming product within production period Q/P; quality loss at time s when failure exists; design target of produced quality value X; production rate in units per time (P4D); backorder cost for shortage; production order quantity requiring determination; production order quantity for conventional EPQ model; production order quantity for process-quality design approach (see Equation (33)); expected quality loss at time s for conforming product; produced units ranging from zero to production order quantity Q; raw material cost per unit; inventory carrying rate, percent of dollar value per time; setup cost for each cycle time;

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Smax s T T1 T2 T3 T4 TP t tL t00 TC1 TC2 TC3 U U(s) U(q0 ) W Wp X X(s)

5

maximum deviation from design target; time ranging from 0 to cycle time Q/D; standard deviation of produced quality value X; cycle time, when backorder exists (Cases 2 and 3), T ¼ T1 þ T2 þ T3 þ T4; production period for producing backorder (see Figure 3); production period for cumulating inventory (see Figure 3); consumption period for consuming inventory (see Figure 3); consumption period for cumulating backorder (see Figure 3); production period, TP ¼ T1 þ T2 (see Figure 3); process tolerance for nominal-the-best type quality characteristic; lower limit of process capability range; process tolerance for process-quality design approach; average total cost per time for Case 1; average total cost per time for Case 2; average total cost per time for Case 3; process mean; process mean expressed as a function of time s under a deterioration process; process mean expressed as a function of produced units q0 under a deterioration process; deterioration rate per time; deterioration rate per unit, which is equal to W/P; quality value for produced product, which is a nominal-the-best type quality characteristic; quality value produced at time s, which is a nominal-the-best type quality characteristic.

3. Quality loss, tolerance cost, variance estimation, and simultaneous determination of mean and tolerance values 3.1 Quality loss Traditionally, the view that no loss would occur as long as the product quality characteristics fell within the specifications range, was accepted. However, this view is no longer accepted, since a quality characteristic that falls within the specifications range can still lead to varying degrees of quality loss depending on its deviation from the design target (see Figure 1). The Taguchi’s nominal-the-best quality loss function is considered in this study. The loss function is (Phadke 1989, Taguchi et al. 1989): 0, if X ¼ m LðX Þ ¼ : ð1Þ K ðX mÞ2 , otherwise The expected quality loss of Equation (1) is (Phadke 1989, Taguchi et al. 1989): E½LðXÞ ¼ K ðU mÞ2 þ 2 :

ð2Þ

3.2 Tolerance cost Usually, a high tolerance cost is associated with a tight process tolerance, while a low tolerance cost results from a loose process tolerance. It is known that a tight process

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A. Jeang f(X) L(X)

K (X– m)2

σ

0

X m–d

m

U

m+d

Figure 1. Quality loss function L(X ) and probabilistic normal distribution f(X ).

tolerance results in a higher tolerance cost and a lower failure cost due to the need for additional manufacturing operations and expensive equipment and materials, whereas a loose process tolerance results in a lower tolerance cost and a higher failure cost (Jeang 1994). The tolerance cost can be formulated according to various function expressions. To evaluate the tolerance cost, this paper directly adopts the tolerance cost function as developed in the literature (Chase et al. 1990): CM ðtÞ ¼ a þ b expðc tÞ,

ð3Þ

where a, b, and c are the coefficients found by regressing the real data from the shop floor.

3.3 Variance estimation Process variance is a common feature in all manufacturing processes. However, in most cases, the process variance is unknown due to the lack of previous data, particularly for new products or new processes, unless previous data from similar processes is available. Therefore, a reasonable estimate is important to obtain. It is known that variance is a function of process tolerance. If a manufacturing process is stable after a period of production has been assumed, the product engineers or process engineers can then indirectly estimate the process variance through the following relation (Jeang 1994): t 2 : ð4Þ 2 ¼ 3CP CP is the process capability index which becomes a stable value when the production process is run for a period of time. In other words, the equality in Equation (4) becomes true. Hence, Equation (2) can be further expressed as: " # t 2 2 : ð5Þ E½LðXÞ ¼ K ðU mÞ þ 3CP

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This indicates that the expected quality loss is a function of controllable variables such as process means, U, and process tolerances, t.

3.4 Simultaneous determination of parameter and tolerance values The total quality-related cost incurred during the life cycle of a product is normally the sum of expected quality loss E[L(X )] and tolerance cost CM(t). The quality loss characterises the quality cost for customers after the production process, and the tolerance cost represents the quality cost for producers during the production process. To minimise the total cost, there must be a trade-off between quality loss E[L(X )] and tolerance cost CM(t). For example, a small t will result in a small quality loss and give rise to a greater tolerance cost; conversely, a great t will result in a great quality loss and a smaller tolerance cost. Again, when we intend to minimise the total cost, E[L(X )] þ CM(t), a small(great) quality loss coefficient, K, followed by a great(small) tolerance value results. The purpose of these examples is to emphasise that when an objective function contains decision variables, process mean U and process tolerance t, they have to be determined simultaneously for true optimisation. Another reason for simultaneous optimisation of the process mean and process tolerance is that they are interdependent by nature.

4. Proposed model development TQM programmes have increasingly focused their attention on the need for better quality, thereby increasing productivity and lowering cost (Rampersad 2001). With higher quality and a lower price, a product on the marketplace has an enhanced competitive position. The procedure for improving quality starts by taking into consideration the needs of customers. A focus on the customer is evident in the current trend toward customer-driven manufacturing and the incorporation of customer requirements in the production process, resulting in what is called ‘process-quality design’. However, all of this ignores the fact that production-inventory management is among the most important of related functions, since inventory requires a great deal of capital and affects the delivery of goods to customers. Additionally, quality level and manufacturing process cost are both closely related to the production order quantity, which is decided at the stage of production-inventory management. Thus, an exclusive process-quality design approach is not sufficient to guarantee an economic and quality product. Instead, a further incorporation is necessary. In a situation of a deterioration process various inventory levels reveal different collective quality levels for produced products. Hence, production inventory based solely on the conventional production-inventory management approach may be over-produced, due to a lack of consideration of varied quality costs at different points of time along with the production run length. For cyclical quality promotion and cost reduction, there is a motivation to extend the conventional production-inventory management approach by adding the process-quality design approach, which considers the varied quality-related costs as a function of time. One of the main objectives in the production process is to achieve greater accuracy with respect to economical products. In the production process, there are various sources of errors that result in poor, inaccurate and defective products, meaning products that deviate randomly and systematically from the design target. These deviations reflect inaccuracies resulting from the collective effects of various working conditions,

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including: material variations, inaccurate operations, environmental differences, and process deterioration. All these errors, individually and collectively, can eventually lead to quality loss. Product accuracy can be depicted by a set of controllable terms such as process mean and process tolerance. Thus, one way to achieve the accuracy of a product is to determine the best process mean and process tolerance for each operation in the production process. Fortunately, these findings are available before production processes are realised. While it is important to have the process setting and process tolerance established at the beginning of the production process, this cannot always guarantee a quality product since process deterioration or systematic error may occur during the production process (Jeang and Chung 2008). Owing to the deterioration process, a continuous change in the quality of a product may take place and the product may fail before the expiration of its intended life. This typically results in a great loss of customer satisfaction. For this reason, controllable variables affecting the quality values of a product should be incorporated into the process design. In this regard, process designers often have the process mean reset cyclically for an initial deviation from target value in order to compensate for process deterioration. In the long run, this reduces the production cost and results in a higher quality product. Production-inventory management is crucial, owing to the amount of capital it requires. For this reason, the costs arising from a production-inventory management approach should also be included, as previously mentioned. This study incorporates the classic EPQ model of a production-inventory management approach with the process-quality design approach to ensure that decisions made at the early production planning and process design stage are able to effectively reduce costs and enhance quality. Consequently, the chances of business success in a competitive environment are increased.

4.1 Having EPQ model contained with quality costs for process design under deterioration process Conventional inventory decision problems are solved by offsetting economic criteria, such as ordering cost and holding cost, to determine economical production quantity for a cycle time Q/D. However, under the deterioration process, the product quality level and process cost among Q may vary at different points of time along the cycle time Q/D. Because previous inventory models failed to consider these aspects of the deterioration process, the determined production order quantity generally exceeded the true order quantity. Other aspects ignored in conventional inventory decision problems are the variations in unit costs and quality levels resulting from process designs that differ from one another. These variations are a concern, since on-line quality improvement leading to process redesign may occur after production and inventory decisions have been made. It is known that variations in unit costs will result in different inventory holding costs and that differing quality levels will bring about dissimilar quality-related costs. These, in turn, will drive the determined production order quantity deviation even further from the true order quantity. Thus, the proposed model intends to integrate the conventional production ordering problem with a process-quality design problem for off-line application to avoid uneconomical and overproduced low-quality products. Considerable savings and quality improvement can be realised under process deterioration if the process mean, U, is reset to its initial value, a0, regularly with cycle

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time Q/D in the opposite direction of the deteriorating course to compensate for the amount of shift occurring during the production process. Each resetting involves an appreciable cost. There is the cost of process resetting itself, but also costs associated with lost production time. Thus, for the sake of quality concern and cost savings, there is a motivation to simultaneously conduct the resetting of the initial establishment for quality assurance and the ordering of the commencement preparation for economic production. A single cost is applied to collectively represent the resetting cost and the ordering cost in one setup. Other than the resetting and ordering costs involved at the beginning of the cycle time, the holding cost for inventory, quality related costs, and a tolerance cost should also be included. In this study, quality loss is the value determined by a loss function which characterises economic loss due to deviation from the target value. Tolerance cost is the production cost for quality products. When the quality loss function and tolerance cost function are integrated, the production and quality related costs can be truly reflected. Only then can producers offset these cost factors by determining the production quantity and the optimal process parameters concurrently. The optimal solutions presented in the proposed model stand for pre-eminent decisions pertaining to product cost and product quality that are in the best interests of both producers and customers.

4.2 Proposed model derivation Since process setting and process tolerance are established prior to the production process, the process mean will be close to the preselected process setting, whereas the process variance will be close to the estimated value of the process tolerance after the production process. To formulate the model, we assume that the product quality value is a random variable following a normal distribution with the process mean, U, and the process variance, 2 . Thus, the associated probability density function is: 1 ðX UÞ2 p ﬃﬃﬃﬃﬃ f ðX Þ ¼ : ð6Þ exp 2 2 2n However, with the possibility of process deterioration or systematic error being included in the production process, the process mean U in Equation (6) turns out to be time-dependent and is expressed as a function of time s: ð7Þ UðsÞ ¼ a0 þ G W s: a0 is the initial establishment, G is þ1 or 1, depending on the direction of process deterioration, and W is the process deterioration rate. This can be seen in Figure 2. The reason for having a0 established in a deviation from the target value is to provide an allowance, so that the cycle time can be extended for economic and quality considerations. For this reason also, the time element s must be included as an additional allowance, G W s, during the problem formulation. In the Sections 4.2.1 and 4.2.2, we develop a model that takes into account the costs pertaining to setup, inventory holding, and quality loss, as well as the related costs for process mean, process tolerance, and production order quantity under a deterioration process. Three cases are introduced: Case 1: Conforming product is produced with the assumption that backorder is not allowed. The corresponding total cost per time is TC1.

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m+d

U(s)

U(s2) U(s1) m a0

m–d

0

s

s2

s1

Figure 2. A time-dependent quality value for nominal-the-best case under deterioration process.

Case 2: Conforming product is produced with the assumption that backorder is allowed. The corresponding total cost per time is TC2. Case 3: Nonconforming product is produced with the assumption that backorder is allowed. The corresponding total cost per time is TC3.

4.2.1 Conforming product is produced for inventory management Case 1 and Case 2 are introduced in this section. Total raw material cost for quantity Q: Assume the raw material cost is in a linear function of process mean U(s), which varies in time because of the deterioration process. Then, the total raw material cost for production order quantity Q is: Z

Q=P

CR ¼

Craw UðsÞds, 0

where Q/P is the production period.

ð8Þ

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Raw material cost per unit: During production period Q/P, production quantity, Q, is produced. Thus, the raw material cost per unit is: Z Q=P

RC ¼ CR =Q ¼ Craw ða0 þ G W sÞds Q: ð9Þ 0

The reason for taking the average of production quantity Q is that under the deterioration process the product quality level varies at different points of time along the production period Q/P. Because Craw is independent of times S, Craw will be moved to the other side of the integral sign in the following model derivation. Unit cost CU: The unit cost consists of raw material cost per unit and production cost per unit. The latter cost is estimated from tolerance cost, CM(t): CU ¼ RC þ CM ðtÞ

Z Q=P ða0 þ G W sÞds Q þ ½a þ b expðc tÞ: ¼ Craw

ð10Þ

0

The average setup cost AS: AS ¼ S

D : Q

ð11Þ

The average material cost AR: Q D Z Q=P D ¼ Craw ða0 þ G W sÞds : Q 0

AR ¼ CR

ð12Þ

The average tolerance cost AM: The tolerance cost per unit, CM(t), is a þ b expðc tÞ. Then, the average tolerance cost is:

Q AM ¼ ½ða þ b expðc tÞÞ Q D ð13Þ ¼ ½a þ b expðc tÞ D: The average inventory cost AI (backorder does not exist): Q PD I ¼ 2 P Q PD AI ¼ h, 2 P

ð14Þ

ð15Þ

where h ¼ r CU , and r is the inventory carrying rate. That is,

Z Q=P Q PD AI ¼ ða0 þ G W sÞds Q þ ða þ b expðc tÞÞ r: ð16Þ Craw 2 P 0

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The average quality loss AL: Based on Equation (5), the expected quality loss for conforming product at time s is: " # t 2 2 QL ðsÞ ¼ K ðða0 þ G W sÞ mÞ þ : ð17Þ 3CP Thus, when the time, s, and production order quantity, Q, are connected in the formulation, the average quality loss per time for cycle time Q=D is: Z Q=P

Q QL ðsÞds AL ¼ D 0 ( ) Z Q=P D t 2 2 ¼ K ½ða0 þ G W sÞ m þ ds: ð18Þ Q 0 3CP The average total cost is the sum of the following: average setup per time, AS, average inventory cost per time, AI, average quality loss per time, AL, average raw material cost per time, AR, and average tolerance cost per time, AM, for the cycle time Q=D. Because the unit cost, RC þ CM(t), varies according to order quantity and process designs, the expressions, AR and AM, must be included in the total cost expression for the proposed model. The corresponding terms: AS, AI, AL, AR and AM can be derived from Equations (11), (16), (18), (12) and (13). To reflect the economic reasons a backorder is sometimes allowed. Thus, the total cost per time for a conforming product can be further divided into two cases: Case 1 – backorder is not allowed; Case 2 – backorder is allowed. The differences for these two cases come from the average inventory holding cost AI and the average backordering cost AB. Case 1: Backorder is not allowed for conforming product The total cost per time is: TC1 ¼ AS þ AI þ AL þ AR þ AM

ð19Þ

D Q

Z Q=P Q PD þ ða0 þ G W sÞds Q þ ða þ b expðc tÞÞ r Craw 2 P 0 " # Z Q=P D t 2 2 K ð a0 þ G W s m Þ þ þ ds Q 0 3CP Z Q=P D ða0 þ G W sÞds þ Craw Q 0

TC1 ¼ S

þ D ½a þ b expðc tÞ:

ð20Þ

Decision variables are a0 , t, and Q. The objective is to find the optimal values for decision variables, a0 , t , and Q which minimise TC1 as given in Equation (20). Obtaining optimal values of these decision variables is a typical problem in calculus. By taking the derivatives of TC1 and setting them equal to zero, the resulting expressions of a0 , t , and Q can be obtained.

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Details concerning the optimal a0 , t , and Q have been derived and are shown in Equations (A3.1), (A3.4), and (A3.8) in Appendix 3. The associated necessary and sufficient conditions are proved in Appendices 1 and 2, respectively. The optimal solutions can be found either by solving Equation (20) directly with the computer software, Mathematica (Trott 2006), or by taking the steps for solution algorithm here introduced. The steps in finding optimal solutions for a0 , t , and Q are: Step 1: Have an estimation a0 ¼ m and t ¼ tL be substituted into Equation (A3.8) to find Q initially, where m and tL are known values. Step 2:

Have the just found Q be plugged into Equations (A3.1) and (A3.4) to find a0 and t.

Step 3:

Similar to Step 1, let the found a0 and t be substituted into Equation (A3.8) to find the next Q.

Step 4: Go back to Step 2. Continue the loop until the values a0 , t, and Q are convergent to an acceptable level. These values will then be optimal solutions. It is economical to have process design with an initial establishment a0 and process tolerance t in producing quality products in order quantity Q over a production period of time Q =P. No further parts are established until the lot is depleted at time Q =D. This makes it possible to spread the setup cost of initial establishment and production preparation over a number of parts Q to offset the quality loss resulting from quality deviation and the holding cost which cumulates from inventory build-up. Consequently, the process parameters and production quantity for an economic and quality product are determined. By observing Equations (A3.1), (A3.4), and (A3.8) from Appendix 3, we can conclude: (1) The decision variables, a0 and t, which are process design related, are in a functional relationship to the production and inventory related decision variable, Q, and also to related conditions such as r, P and D. Additionally, these decision variables are influenced by the coefficient of quality loss function, K. (2) The decision variable, Q, which is production and inventory related, is in a functional relationship to the process design related decision variables, a0 and t, and to production and inventory related conditions such as r, P and D. Of course, the decision variable, Q, is also affected by the quality related costs, K value. Clearly, based on the evidence of relationship and dependence among these variables, the production-inventory management approach and the process-quality design approach must work together for optimisation of the global system. Case 2:

Backorder is allowed for conforming product

By referring to Equations (A3.12) and (A3.13) from Appendix 3, it is apparent that the average inventory holding cost AI and the average backordering cost AB can be expressed in the following. The average inventory cost AI (backorder exists): AI ¼

rCU ðP DÞðTP T1 Þ2 : 2TP

ð21Þ

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The average backordering cost AB: AB ¼

T12 ðP DÞ : 2TP

ð22Þ

As indicated in Figure 3, TP is the production period which is also equal to Q=P. Similarly, AS, AL, AR, and AM are from Equations (11), (18), (12) and (13). TC2 ¼ AS þ AI þ AB þ AL þ AR þ AM TC2 ¼ S

D Q

ðP DÞðTP T1 Þ2 þ

ð23Þ

h

. i R Q=P Craw 0 UðsÞds Q þ CM ðtÞ r 2TP

T21 ðP

DÞ 2TP ( ) Z Q=P D t 2 2 þ K ½ða0 þ G W sÞ m þ ds Q 0 3CP Z Q=P D ða0 þ G W sÞds þ Craw Q 0 þ

þ D CM ðtÞ:

ð24Þ

Decision variables are a0 , t, Q, and T1

Lot size

Inventory level

T

−D

P−D

0

Time s

T1

T2 TP

Figure 3. EPQ model with shortage.

T3

T4

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The evidence of relationship and dependence among variables in Case 1 also appears in Case 2. By referring to Equations (A3.18), (A3.21), (A3.25), and (A3.30), it is apparent that the production-inventory related decision variables, Q and T1, are correlated with process-quality decision variables, a0 and t. The process related decision variables, a0 and t, are linked with production-inventory related decision variables, Q and T1, and related conditions such as r, P and D. Obviously, both the production-inventory management approach and the process-quality design approach are allied even when a backorder exists. For this reason, it is necessary to integrate these two approaches to achieve product quality at an economical cost.

4.2.2 Nonconforming product is produced for inventory management In contrast to Cases 1 and 2, Case 3 does not assume a conforming product. Rather, a nonconforming product is assumed. Because of this relaxation of assumptions, the failure cost needs to be considered and a revision in terms of quality loss needs to be made. The expected quality loss AL and AF: Let the quality loss and failure cost at time s be: Zm Z mþd 2 l ðsÞ ¼ K ðX mÞ f ðXÞdX þ K ðX mÞ2 f ðXÞdX md

ð25Þ

m

Z

Z

md

1

f ðXÞdX þ CF

f ðsÞ ¼ CF

f ðXÞdX:

0

ð26Þ

mþd

The cumulative quality loss for imperfect product for production period Q=P is: Z Q=P l ðsÞds LðQ=PÞ ¼ 0 Z Q=P Z m Z mþd ¼ K ðX mÞ2 f ðXÞdX þ K ðX mÞ2 f ðXÞdX ds ð27Þ 0

md

Z

m

Q=P

FðQ=PÞ ¼

f ðsÞds Z Q=P Z ¼ CF 0

0

Z

md

1

f ðXÞdX þ CF 0

f ðXÞdX ds:

ð28Þ

mþd

The expected quality loss and expected failure cost for cycle time Q=D are: D LðQ=PÞ Q Z Q=P Z m Z mþd D 2 2 ¼ K ðX mÞ f ðXÞdX þ K ðX mÞ f ðXÞdX ds Q 0 md m

AL ¼

D FðQ=PÞ Q Z Q=P Z md Z1 D ¼ CF f ðXÞdX þ CF f ðXÞdX ds : Q 0 0 mþd

ð29Þ

AF ¼

ð30Þ

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Backorder is allowed for nonconforming product

Case 3:

With the exception of AL and AF, the remaining AS, AI, AB, AR, AM are the same as those in Case 2. Thus, by referring to Equations (11), (21), (22), (29), (30), (12), and (13), it is apparent that the total cost, TC3, with nonconforming product and backorder is: TC3 ¼ AS þ AI þ AB þ AL þ AF þ AR þ AM TC3 ¼ S

D Q

ðP DÞðTP T1 Þ2

h

Craw

R Q=P

þ þ þ

0

ða0 þ G W sÞds

.

ð31Þ

i Q þ CM ðtÞ r

2TP T21 ðP D Q

D þ Q

DÞ 2TP Z Q=P Z 0

Z

m

K ðX mÞ2 f ðXÞdX þ md

Q=P

Z CF

0

Z

md

Z

K ðX mÞ2 f ðXÞdX ds

m

1

f ðXÞdX þ CF 0

Z Q=P D þ Craw ða0 þ G W sÞds Q 0

mþd

f ðXÞdX ds

mþd

þ D CM ðtÞ:

ð32Þ

Decision variables are a0 , t, Q, and T1.

5. Illustrative example For the purpose of illustrating the proposed model and the themes arising from the preceding discussion, an example is provided that can be demonstrated. Let m ¼ 30 mm, K ¼ $2000/mm2, W ¼ 0.05 mm/month, G ¼ 1, CP ¼ 1, P ¼ 1200 units per month, D ¼ 1000 units per month, S ¼ $150 for each production run (setup cost), and r ¼ 10 percent of dollar value per unit time. Then, the coefficients of the tolerance function, a ¼ 20.00, b ¼ 70.00, c ¼ 30.00, and d ¼ 0.30 mm. Table 1 attempts to prove the necessity of integrating the production-inventory management approach with the process-quality design approach. The numerical analysis is

Table 1. Various process designs, quality levels, and quality costs vs. identical unit cost and production order quantity. i

U

1 2 3 4 5

29.70 29.85 30.00 30.15 30.30

t

Q

(U T )2 þ 2

Failure rate

Failure cost

0.220785 395.628 0.163595 0.1742 $2.6130 0.220396 395.628 0.095965 0.1735 $2.6025 0.219799 395.628 0.073266 0.1723 $2.5845 0.219596 395.628 0.095699 0.1719 $2.5785 0.219391 395.628 0.163130 0.1715 $2.5725 R md R1 R md R1 Note: failure rate ¼ 0 f ðxÞdx þ mþd f ðxÞdx and failure cost ¼ CF ð 0 f ðxÞdx þ mþd f ðxÞdxÞ.

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based on the conventional EPQ model with the above conditions except that the conditions of the deterioration process and quality loss are absent. By assuming a mixture of U and t for having a fixed and equal unit cost, $120, it is possible that various combinations of process mean, U, and process tolerance, t, representing alternative process designs, may result in an identical production order quantity, Q. However, aside from the systematic error resulting from process deterioration, associated quality measurements such as process variability, failure rate, and failure cost for each alternative are quite different. As a result, the possibility of process design for quality enhancement may be lost. When the assumption of a fixed unit cost is relaxed, quality improvement, along with a cost reduction in the determination of production quantity, becomes possible. Thus, a variable unit cost that takes into consideration alternative processes is absolutely necessary for model development. Because Case 3 is the most generalised case, we will focus on it for the purpose of numerical analysis. To prove the convexity of the total cost TC3 with respect to decision variables, a0 , t, Q, and T1, let us consider the following drawings. For example, when a0 is fixed at 29.9774 mm, the relationship between TC3, Q, and t gives a convex shape which is shown in Figure 4. Similarly, the convex shapes for TC3 versus any two combinations of a0 , t, Q, and T1 still hold. However, only partial combinations are shown in Figures 5 and 6 for the purpose of demonstration. Clearly, we can conclude that TC3 must be a convex set with global optimal solutions existing. The computer software, Mathematica, has been used to solve the proposed model (Trott 2006). With the conditions given in the beginning of this section, the optimal solutions are: initial setting a0 ¼ 29.9774 mm, process tolerance t ¼ 0.335040 mm, T1 ¼ 0.147935, production order quantity Q ¼ 1038.72 units, and total cost TC3 ¼ $20369.37/month.

Figure 4. TC3 versus Q and t. Note: a0 ¼ 29.9774 mm, T1 ¼ 0.147935.

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Figure 5. TC3 versus Q and a0 . Note: t ¼ 0.335040 mm, T1 ¼ 0.147935.

Figure 6. TC3 versus Q and T1. Note: a0 ¼ 29.9774, t ¼ 0.335040 mm.

A. Jeang

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International Journal of Production Research $ 800

TC3

AI + AB + AL + AR + AF

600

400

200

0

1000

Q*

2000

3000

4000

AS Q

= 1038.72

Figure 7. The partial components of TC3 such as AS and AI þ AB þ AL þ AR þ AF versus production order quantity Q.

Figure 7 is a plot of TC3 versus Q, with two components of TC3 shown separately. The first component is AS. The second component is AI þ AB þ AL þ AR þ AF. The tolerance cost CM(t), could be a þ b expðc tÞ. The reason for AM being excluded is that AM is independent of order quantity, Q. As shown in Figure 7, when Q increases, the component, AS, decreases; at the same time, the component, AI þ AB þ AL þ AR þ AF, increases; thus, one decreases while the other increases. This is precisely the trade-off between first and second components that was mentioned previously. In the case that quality-related costs AL or AF are absent, the new shape of the second component becomes flatter than the original shape. Consequently, the new Q turns out to be a greater value than the original Q. This result coincides with the observation that the conventional inventory model generally overproduces as a result of failing to take into consideration quality related costs. In fact, quality related costs are connected not only to the order quantity being produced, but also to process parameters being established. This connection becomes even tighter under a deterioration process. Thus, as mentioned in the preceding discussion on model development, it is essential to integrate production inventory management with a process quality design approach in order to truly optimise the production order quantity and process parameters for quality and cost. To confirm this thesis, a sensitivity analysis of the proposed model TC3 was carried out to study the effects of variables associated with a production inventory management approach and a process-quality design approach. These variables include: the deterioration rate, W, quality loss coefficients, K, the coefficient, b of CM(t), production rate, P, and inventory carrying rate, r. For various deterioration rate W values, the influence on the values of a0 , t , Q , T1 , TP , T, and TC3 are displayed in Table 2. From this table, the following results were obtained: (1) When the deterioration rate, W, increases, the following happens: AS increases, AI decreases, AL increases, AB increases, AF increases, AR decreases, AM decreases, and TC3 increases.

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A. Jeang Table 2. The values of a0 , t , Q , T1 , TP, T, and TC3 vs. W. Deterioration rate a0 t Q TP T1 T TC3

W ¼ 0.001

W ¼ 0.01

W ¼ 0.05

W ¼ 0.1

29.9986 29.9947 29.9774 29.9558 0.334671 0.334684 0.335040 0.33618 1041.46 1039.43 1038.72 1026.58 0.867883 0.866192 0.865600 0.855483 0.147357 0.147785 0.147935 0.150533 1.04146 1.03943 1.03872 1.02658 $20,369.14 $20,369.17 $20,369.37 $20,370.01

Note: K ¼ 2000, b ¼ 70, r ¼ 0.1, P ¼ 1200.

Table 3. The values of a0 , t , Q , T1 , TP , T, and TC3 vs. K. Deterioration rate a0 t Q TP T1 T TC3

K ¼ 1000

K ¼ 1500

K ¼ 2000

K ¼ 2500

29.9763 0.359010 1040.14 0.866783 0.147630 1.04014 $20,359.01

29.9769 0.345079 1039.94 0.866617 0.147674 1.03994 $20,364.27

29.9774 0.335040 1038.72 0.865600 0.147935 1.03872 $20,369.37

29.9776 0.327273 1028.50 0.857083 0.150123 1.02850 $20,374.12

Note: W ¼ 0.05, b ¼ 70, r ¼ 0.1, P ¼ 1200.

(2) The reason that a high deterioration rate W will end up with a small a0 at a farther distance from target, m, can be explained by the fact that a high deterioration rate W will increase the probability of the product quality value becoming poorer early on in the cycle time, while the cumulative quality loss will increase significantly as products are being produced. Therefore, a0 should be strategically located so as to ensure that the product quality value remains at an acceptable level for the remainder of the cycle time. (3) An increase in the deterioration rate, W, causes a greater t value, namely, a smaller unit cost. Consequently, a smaller AI is needed to compensate for other increased costs. (4) An increase in the deterioration rate, W, results in a reduction of Q , Tp, and T values. This may be explained as follows: when the deterioration rate increases, it is possible that the cumulative quality loss will significantly increase as products are being produced. Hence, we prefer to have a smaller production quantity to ensure that the produced product performs its desired function. Here again, smaller Tp, and T values will result. (5) An increase in the deterioration rate W causes a greater T1 value. This can be explained by the fact that AI is reduced because of shortage being held. Similarly, the influence on the values of a0 , t , Q , T1 , TP, T, and TC3 under various quality loss coefficients, K, are displayed in Table 3. The following results were

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International Journal of Production Research Table 4. The values of a0 , t , Q , T1 , TP , T, and TC3 vs. r. Deterioration rate a0 t Q TP T1 T TC3

r ¼ 0.1

r ¼ 0.2

r ¼ 0.3

r ¼ 0.4

29.9774 0.335040 1038.72 0.865600 0.147935 1.03872 $20,369.37

29.9803 0.335127 892.699 0.743916 0.182359 0.892699 $20,510.52

29.9812 0.335284 836.852 0.697377 0.197175 0.836852 $20,642.74

29.9817 0.335456 805.348 0.671123 0.205804 0.805348 $20,771.80

Note: K ¼ 2000, b ¼ 70, W ¼ 0.05, P ¼ 1200.

obtained: (1) When the quality loss coefficient, K, increases, the following happens: AS increases, AI decreases, AL increases, AB increases, AF decreases, AR increases, AM increases, and TC3 increases. (2) An increase in quality loss coefficient, K, results in a reduction of the Q value. This may be explained as follows: when K increases, the cumulative quality loss will increase considerably as units are produced. To trim this loss, we prefer frequent resetting of the production process at a quality level. Similar explanations can be found for smaller T and TP values. (3) A greater K value causes a smaller process tolerance value t . This result can be explained by the fact that a greater K value will drive a greater value of AL. Naturally, a smaller process tolerance t value is preferred. (4) A greater K value causes a greater a0 value. The reason is that the closeness of the process mean to the design target, m, reduces the cumulative quality loss for greater K values. (5) An increase in quality loss coefficients, K, causes a greater T1 value. This is because a higher shortage occurs under greater T1 values, allowing AI to be reduced significantly. The influence on the values of a0 , t , Q , T1 , TP, T, and TC3 under various inventory carrying rates, r, is listed in Table 4. The following results were obtained: (1) When the inventory carrying rate, r, increases, the following happens: AS increases, AI increases, AL decreases, AB increases, AF decreases, AR decreases, AM decreases, and TC3 increases. (2) As observed, a high inventory carrying rate, r, will bring about a greater a0 value at a close distance to the design target, m. The reason is that a high carrying rate, r, causes a greater AI value, which is increased radically as cycle time increases. One way to decrease the cycle time is to locate the initial setting close to design target, m. Similar explanations can be found for the decrease in Q as the r value increases. (3) An increase in the inventory carrying rate, r, causes a greater t value or smaller unit cost. Consequently, a smaller AI is required to compensate for other increased costs. (4) An increase in the inventory carrying rate, r, results in a reduction of Q , Tp, and T values. This can be explained by the fact that when the inventory carrying rate, r, increases, the AI value or average inventory carrying cost will significantly increase

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Table 5. The values of a0 , t , Q , T1 , TP , T, and TC3 vs. P. Deterioration rate a0 t Q TP T1 T TC3

P ¼ 1200

P ¼ 1500

P ¼ 1800

P ¼ 2100

29.9774 0.335040 1038.72 0.865600 0.147935 1.03872 $20,369.37

29.9813 0.343197 1030.12 0.686747 0.200576 1.03012 $20,527.59

29.9887 0.349260 987.778 0.548766 0.245275 0.987778 $20,696.59

29.9893 0.354862 965.032 0.459539 0.266130 0.965032 $20,871.88

Note: K ¼ 2000, b ¼ 70, W ¼ 0.05, r ¼ 0.1.

as products are being produced. Hence, we prefer to produce a smaller production quantity Q (with a shorter production period Tp and production cycle T ) to avoid a greater AI value resulting. (5) An increase in the inventory carrying rate, r, causes a greater T1 value. This can be explained by the fact that the increase in the carrying rate, r, also results in an increase in AI. Thus, we would prefer a greater T1 for greater backorder in order to avoid a greater AI resulting. Table 5 intends to show the influence on the values of a0 , t , Q , T1, TP, T*, and TC3 under various production rates P. The results are discussed as follows: (1) When production rate, P, increases, the following happens: AS increases, AI decreases, AL decreases, AB increases, AF decreases, AR decreases, AM decreases, and TC3 increases. (2) A greater production rate, P, will result in a greater a0 value which is close to design target, m. The reason for this closeness is that a greater production rate, P, causes a greater AI value at the beginning of the production cycle time. One way to decrease the AI value is to reduce the production cycle time. This can be achieved by locating the initial setting in the neighbourhood of design target, m. (3) An increase in the production rate, P, causes a greater t value, namely, a smaller unit cost. Consequently, a smaller AI is gained, which is insensitive to a greater P value. (4) An increase in the production rate, P, results in a reduction of Q , Tp, and T values. The explanation appears under point (2). To avoid the AI value increasing as the production rate, P, increases, we prefer to produce a smaller production quantity Q , with a shorter production period Tp and production cycle T. (5) An increase in the production rate, P, causes a greater T1 value. This can be explained by the fact that AI has increased, also due to the increased production rate, P. Thus, we prefer to have a greater T1 for greater backorder to avoid a greater AI resulting. Table 6 describes the influence on the values of a0 , t , Q , T1 , TP, T, and TC3 under the various coefficient, b. The results are discussed as follows: (1) When the coefficient, b, increases, the following happens: AS increases, AI decreases, AL increases, AB increases, AF increases, AR decreases, AM decreases, and TC3 increases.

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International Journal of Production Research Table 6. The values of a0 , t , Q , T1 , TP , T, and TC3 vs. coefficient b of CM(t). Deterioration rate a0 t Q TP T1 T TC3

b ¼ 50

b ¼ 60

b ¼ 70

b ¼ 80

29.9776 0.323141 1040.67 0.867225 0.147522 1.04067 $20,368.30

29.9775 0.329560 1039.70 0.866417 0.147933 1.03970 $20,368.81

29.9774 0.335040 1038.72 0.865600 0.147935 1.03872 $20,369.37

29.9772 0.339832 1031.24 0.859367 0.149530 1.03124 $20,369.71

Note: K ¼ 2000, r ¼ 0.1, W ¼ 0.05, P ¼ 1200.

Table 7. Comparison of results between proposed approach and production-inventory management approach. Proposed approach W 0.001 0.01 0.05 0.1

Production-inventory management approach

Deviation between two approaches

Q

TC3 (1)

Q0

TC03 (2)

DTC(2) – (1)

1041.46 1039.43 1038.72 1026.58

$20,369.14 $20,369.17 $20,369.37 $20,370.01

1549.19 1549.19 1549.19 1549.19

$20,712.22 $20,712.33 $20,714.61 $20,721.58

$343.08 $343.16 $345.24 $351.57

Note: t ¼ 0.179813, a0 ¼ 30.

(2) As observed, a greater b value will result in a smaller a0 value indicating a farther distance from the design target, m. The reason is that a greater b value increases the tolerance cost. Naturally, we would like a greater tolerance value t indicating a smaller tolerance cost to pay off the additional cost resulting from the greater b value. To obtain a greater tolerance value and avoid an increase in the failure cost, we would like to establish a0 at a distance farther from the design target, m. (3) An increase in the b value causes a greater t value, namely, a greater AL and a greater AF. Thus, we prefer to have a smaller Q to reduce the cumulative quality loss and failure cost during the production cycle time. Similar explanations can be found for preferring smaller Tp and T values. (4) An increase in the b value causes a greater T1 value. This can be explained by the fact that AI has increased because of the b value. Thus, we prefer to have a greater T1 for greater backorder to avoid a greater AI resulting. To continue verifying the necessity of integrating the production-inventory management approach with the process-quality and cost design approach, two comparisons have been made. These comparisons appear in Tables 7 and 8. Table 7 compares the proposed model with the production-inventory management approach. Table 8 compares the proposed model with the process-quality and cost design approach. It is apparent in Table 7 that the total cost is increased when the production-inventory management approach is the only approach taken. The outcomes show that the total cost, TC3 , derived

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Table 8. Comparison of results between proposed approach and process-quality design approach. Proposed approach W 0.001 0.01 0.05 0.1

Process-quality and cost design approach

Deviation between two approaches

Q

TC3 (1)

Q00

TC003 (2)

œTC(2)–(1)

1041.46 1039.43 1038.72 1026.58

$20,369.14 $20,369.17 $20,369.37 $20,370.01

709.375 697.787 683.828 647.871

$21,026.37 $21,026.48 $21,029.61 $21,031.80

$657.23 $657.31 $658.24 $666.79

from the proposed model is always lower than the total cost, TC03 , obtained by the conventional EPQ model under a deterioration process, where TC03 is found by substituting a0 ¼ m, t ¼ tL, and Q ¼ Q0 into the TC3 expression. The design target, m, is 30 mm and the tightest tolerance, tL, is 0.179813 mm in this example. Q0 is gained from the conventional EPQ model, which lacks consideration of process-quality design. The reason for using the inputs, a0 ¼ m and t ¼ tL, as the process setting for the production-inventory management system is that the process designer normally establishes the process mean at design target, m, for quality consideration and has the process tolerance at the lower process limit, tL, for conservative consideration. Obviously, because the initial setting a0 has been established at target, m, there is no compensation for the deterioration process, and the impact on the total cost becomes significant as the deterioration rate, W, value increases. This represents an extreme situation. The other extreme is shown in Table 8. Here, the process-quality design system is run alone. As we can see from this table, the difference between TC3 and TC003 increases as the deterioration rate, W, increases. TC003 is found by substituting a0 ¼ a000 , t ¼ t00 , and Q ¼ Q00 into the TC3 expression. Having Wp ¼ W=P, the values, a000 , t00 , and Q00 are found by optimising the process-quality design formulation in Equation (33): q0 2 Z mþd 0Z m 1 3 2 2 ZQ K ðX mÞ fðXÞdX þ K ðX mÞ fðXÞdX B md C 07 1 6 B Cdq 7 þ CM ðtÞ, TCq0 ¼ 6 Z md Z m1 @ A 5 4 Q 0 þ CF f ðXÞdX þ CF fðXÞdX 0

mþd

ð33Þ where Uðq0 Þ ¼ a0 þ G WP q0 . Clearly, a recommendation for the incorporation of these two approaches can be derived from Tables 7 and 8. If the production-inventory management approach and the process-quality and cost design approach work separately, the total cost of each approach must be higher than the total cost of the proposed approach. Thus, integrating these two approaches in the proposed model is necessary to truly achieve an economic production order quantity that also represents quality. In order to greatly improve a producer’s ability to meet market opportunities, lower development cycle time, reduce production costs, and deliver high-quality products, the integration of production planning and process design becomes necessary (Jeang and Chung 2008). As demonstrated by the examples in Tables 1–8, an economic and quality product can be achieved through the proposed model early in the preparation stage.

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6. Summary The proposed models considered minimising the total cost, which includes: the costs of setup, inventory holding, backorder, quality loss, and tolerance, by conducting concurrent optimisation of the decision variables, including: the initial setting, process tolerance, production period for backorder, and production order quantity. Computer software, Mathematica, was used to find the optimal values for decision variables of interest. One example has been introduced to demonstrate the optimal solutions that this model is capable of achieving, which will augment its application. The significance of the proposed model in comparison to former approaches becomes evident as an enhanced high-quality product with enhanced functionality is produced at a minimal total cost.

Acknowledgments This research was carried out in the Design, Quality, and Productivity Laboratory (DQPL) at the Department of Industrial Engineering and Systems Management at Feng Chia University, Taichung, Taiwan, Republic of China. I would like to thank my research assistant, Mr. Yen-Pin Pai, graduate student in the I.E. Department.

References Arora, J.S., 1989. Introduction to optimum design. New York: McGraw-Hill. Chase, K.W., et al., 1990. Least cost tolerance allocation for mechanical assemblies with automated process selection. Manufacturing Review, 3 (1), 49–59. Chen, C.H. and Lai, M.T., 2007. Economic manufacturing quantity, optimum process mean, and economic specification limits setting under the rectifying inspection plan. European Journal of Operational Research, 183 (1), 336–344. Chung, K.L. and Hou, K.L., 2003. An optimal production run time with imperfect production process and allowable shortages. Computer and Operation Research, 30 (4), 483–490. Drezner, Z. and Wesolowsky, G.O., 1989. Optimal control of a linear trend process with quadratic loss. IIE Transactions, 21 (1), 66–72. Ganeshan, R., Kulkarni, S., and Boone, T., 2001. Production economics and process quality: a Taguchi perspective. International Journal of Production Economics, 71 (3), 343–350. Hax, A.C. and Canadea, D., 1984. Production and inventory mangement. Englewood Cliffs, NJ: Prentice-Hall. Kim, C.H. and Hong, Y., 1999. An optimal production run length in deteriorating production processes. International Journal of Production Research, 58 (2), 183–189. Jeang, A., 1994. Tolerance design: choosing optimal tolerance specification in the design machined parts. Quality and Reliability Engineering International, 10 (1), 27–35. Jeang, A., 1998. Reliable tool replacement policy for quality and cost. European Journal of Operation Research, 108 (2), 334–344. Jeang, A., 2001. Combined parameter and tolerance design optimization with quality and cost reduction. International Journal of Production Research, 39 (5), 923–952. Jeang, A. and Chung, C.P., 2008. Process mean, process tolerance, and use time determination for product life application under deteriorating process. International Journal of Advanced Manufacturing Technology, 36 (1), 97–113. Jeang, A. and Yang, K., 1992. Optimal tool replacement with nondecreasing tool wear. International Journal of Production Research, 30 (2), 299–314. Phadke, M.S., 1989. Quality engineering using robust design. Englewood Cliffs, NJ: Prentice Hall.

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Porteus, E.L., 1986. Optimal lot-sizing process quality improvement and setup cost reduction. Operations Research, 34 (1), 137–144. Rahin, M.A. and Al-Hajailan, W.I., 2006. An optimal production run for an imperfect production process with allowable shortages and time-varying fraction defective rate. International Journal of Advanced Manufacturing Technology, 27 (11), 1170–1177. Rampersad, H.K., 2001. Total quality management: an executive guide to continuous improvement. New York: Springer. Rosenblatt, M.J. and Lee, H.L., 1986. Economic production cycle with imperfect production process. IIE Transactions, 18 (1), 48–55. Salameh, M.K. and Jaber, M.Y., 2000. Economic order quantity model for items with imperfect quality. International Journal of Production Economics, 64 (1), 59–64. Silver, E.A. and Peterson, R., 1985. Decision systems for inventory management and production planning. New York: Wiley. Taguchi, G., Elsayed, E., and Hsiang, T., 1989. Quality engineering in production systems. Singapore: McGraw-Hill. Trott, M., 2006. The Mathematica guide book for numerics. New York: Springer ScienceþBusiness Media.

Appendix 1: Necessary optimality conditions for a0 , t , and Q TC1 is given in Equation (20) for Case 1. To find the optimal initial setting a0 , process tolerance t , and economic production order quantity Q , we take the first derivative of TC1 with respect to a0 , t, and Q, respectively and then equate it to zero. (1) The derivative of TC1 with respect to a0 : 9bcect ð2DP DQr þ PQrÞ þ 4DKC2P t ¼0 18P

ðA1:1Þ

(2) The derivative of TC1 with respect to t: @TC1 9bc ect ð2DP DQr þ PQrÞ þ 4Dt : ¼ 18P @t

ðA1:2Þ

(3) The derivative of TC1 with respect to Q: @TC1 ¼ GWQ3 ½3Craw ðP DÞ r þ 4GDKW @Q þ 3Q2 PðP DÞ ða0 Craw þ aPÞ r þ GDPðCraw þ 2Kða0 mÞÞW þ bP2 ðP DÞ rect 6DP3 S:

ðA1:3Þ

The optimal a0 , t , and Q can be obtained by setting Equations (A1.1), (A1.2), and (A1.3) to be equal to zero and solving these simultaneous equations. Regarding whether the found a0 , t , and Q are global optimal solutions, then Appendix 2 for verifying the sufficient optimality conditions intends to answer this question.

Appendix 2: Sufficient optimality conditions for a0 , t , and Q The Hessian matrix of Equation (20) for Case 1 is: @2 TC1 h11 h12 h13 @a20 2 TC1 H ¼ h21 h22 h23 ¼ @@[email protected] h31 h32 h33 @2 TC01 @[email protected]

@2 TC1 @a0 @t @2 TC1 @t2 @2 TC1 @[email protected]

@2 TC1 @a0 @Q @2 TC1 : @[email protected] @2 TC1 @Q2

ðA2:1Þ

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The sufficient optimality conditions for a0 , t , and Q are that M1, M2, and M3 are positive (Arora 1989). The proofs for positive M1, M2, and M3 are: 2DK : P

ðA2:2Þ

DKð4D þ 9bc2 ect ð2DP þ PQr DQrÞÞ 9P2

ðA2:3Þ

M1 ¼ Because of P4D: M2 ¼

M3 ¼

2 ðPDÞrw DKW2 DK 4D þ 9bc2 ect ð2DP þ PQr DQrÞ 2DS þ GCraw2P þ 2G 3P 3 3 Q3 9P2

4 0:

ðA2:4Þ

Clearly, M1, M2, and M3 are positive, the solutions of a0 , t , and Q must be the global optimal solutions (Arora 1989).

Appendix 3: Relevant properties for proposed model Case 1:

Backorder is not allowed for conforming product

Property 1:

The optimal initial value a0 for Case 1 is: 1 Q Craw 1 1 1 Qr : 1þ a0 ¼ m GW 2 P 2 D P 2K

ðA3:1Þ

Derivation: Let @TC1 [email protected] ¼ 0, thus, Equation (A1.1) is equal to zero. Craw ð2DP DQr þ PQrÞ þ 2DKð2a0 P 2mP þ GQWÞ 2P2

ðA3:2Þ

1 Q Craw 1 1 1 1þ Qr : a0 ¼ m GW 2K 2 P 2 D P

ðA3:3Þ

where Craw, K, r, D, G, W, P are known values. Equation (A3.3) tells us that the optimal initial setting a0 established at the beginning of cycle time, Q=D, is the deviation, 1 Q Craw 1 1 1 GW þ Qr , 1þ 2 P 2 D P 2K away from target value m. Property 2: When the tolerance cost function CM ðtÞ is a þ bect , the optimal tolerance value t for Case 1 is: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 18DPbc2 þ 9Ac 4DKC2P þ 324DPbcðAc2 þ DPbc3 Þ þ 81A2 c2 , ðA3:4Þ t ¼ 18DPbc3 þ 9Ac2 where A ¼ bcQrðP DÞ. Derivation: Let @TC1 [email protected] ¼ 0, thus, Equation (A1.2) is equal to zero. 9bcect ð2DP DQr þ PQrÞ þ 4DKC2P t ¼ 0: 18P

ðA3:5Þ

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A. Jeang The ect can also be expressed as: ect ¼ 1 ct þ

ðctÞ2 : 2

ðA3:6Þ

Solve Equation (A3.5) by having the expression ect be replaced with Equation (A3.6); then: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 18DPbc2 þ 9Ac 4DKC2P þ 324DPbcðAc2 þ DPbc3 Þ þ 81A2 c2 , ðA3:7Þ t ¼ 18DPbc3 þ 9Ac2 where A ¼ bcQrðP DÞ. Property 3:

The optimal production order quantity Q for Case 1 is: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 q 3 q q2 p3 q2 p3 a þ Q ¼ þ þ þ , 2 2 4 27 4 27 3

ðA3:8Þ

where: ETC1 ¼ 3GWCraw r ðP DÞ þ 4G2 DKW2 FTC1 ¼ 3 PðP DÞ ða0 Craw þ aPÞ r þ GDPðCraw þ 2Kða0 mÞÞW þ bP2 ðP DÞ rect GTC1 ¼ 6DP3 S 1 FTC1 2 p¼ 3 ETC1 GTC1 2 FTC1 3 q¼ þ ETC1 27 ETC1 FTC1 a¼ : ETC1 Derivation: Let @TC1 [email protected] ¼ 0 and rearrange, we have: Q3 3rGWCraw ðP DÞ þ 4G2 DKW2 þ Q2 3 PðP DÞ ða0 Craw þ aPÞ r þ GDPðCraw þ 2Kða0 mÞÞW þ bP2 ðP DÞ rect 6P3 SD ¼ 0

ðA3:9Þ

Let: ETC1 ¼ 3GWCraw r ðP DÞ þ 4G2 DKW2 FTC1 ¼ 3 PðP DÞ ða0 Craw þ aPÞ r þ GDPðCraw þ 2Kða0 mÞÞW þ bP2 ðP DÞ rect GTC1 ¼ 6DP3 S: Then: Q3 þ

FTC1 2 GTC1 Q þ ¼ 0, ETC1 ETC1

ðA3:10Þ

where the coefficients, FTC1 =ETC1 and GTC1 =ETC1 , are known values. By the Cardano formula, we have the corresponding root of Equation (A3.10), which is: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 q 3 q q2 p3 q2 p3 a Q ¼ þ þ þ : þ ðA3:11Þ 4 27 4 27 3 2 2

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International Journal of Production Research Case 2:

29

Backorder is allowed for conforming product

Property 4:

The average inventory cost, AI, and the backorder cost, AB, are: AI ¼

rCU ðP DÞðTP T1 Þ2 2TP

ðA3:12Þ

T21 ðP DÞ : 2TP

ðA3:13Þ

AB ¼ Derivation:

By referring to Figure 3, the production periods, TP, are: TP ¼ Q=P and TP ¼ T1 þ T2. The average inventory I is: Imax ¼ ðP DÞT2 ¼ ðP DÞðTP T1 Þ Imax ¼ DT3 Imax ðP DÞðTP T1 Þ ¼ T3 ¼ D D 1

I¼2

ðT2 þ T3 ÞImax ðP DÞðTP T1 Þ2 ¼ : T 2TP

ðA3:14Þ

The average backorder B is: ðP DÞT1 ¼ DT4 ðP DÞT1 T4 ¼ D 1

B¼2

T1 ðP DÞT1 þ 12 T4 DT4 T21 ðP DÞ : ¼ 2TP T

ðA3:15Þ

The average inventory holding cost AI is: AI ¼ rCU I ¼

rCU ðP DÞðTP T1 Þ2 : 2TP

ðA3:16Þ

The average backordering cost AB is: AB ¼ B ¼ Property 5:

T21 ðP DÞ : 2TP

ðA3:17Þ

The optimal initial value a0 for Case 2 is:

Craw PrðQ PT1 Þ2 þ D Q2 r þ P2 rT21 2PQð1 þ rT1 Þ þ 4mPDKQ 2DKGQ2 W : a0 ¼ 4PDKQ ðA3:18Þ Derivation: Let @TC2 [email protected] ¼ 0, we have: Craw PrðQ PT1 Þ2 D Q2 r þ P2 rT21 2PQð1 þ rT1 Þ þ 2DKQð2a0 P 2mP þ GQWÞ ¼ 0: 2P2 Q ðA3:19Þ

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A. Jeang Then:

Craw PrðQ PT1 Þ2 þ D Q2 r þ P2 rT21 2PQð1 þ rT1 Þ þ 4mPDKQ 2DKGQ2 W ðA3:20Þ a0 ¼ 4PDKQ The optimal tolerance value t for Case 2 is:

Property 6:

t ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 4 bc2 ð2D þ ZÞ þ 4D bc2 ð2D þ ZÞ þ 4D 9P þ 9P 2b c ð2D þ ZÞ bc3 ð2D þ ZÞ

,

ðA3:21Þ

where Z¼

rðP DÞðQ PT1 Þ2 : PQ

Derivation: Let @TC2 [email protected] ¼ 0, we have: 2Dt 1 ðctÞ2 ðP DÞðQ PT1 Þ2 þ bc 1 ct þ 2D þ r ¼ 0: 9P 2 2 PQ

ðA3:22Þ

Rearrangement: rðP DÞðQ PT1 Þ2 bc 2D þ PQ rðP DÞðQ PT1 Þ2 4D bc2 2D þ þ t PQ 9P 1 3 rðP DÞðQ PT1 Þ2 bc 2D þ t2 ¼ 0: þ 2 PQ

ðA3:23Þ

Then:

t ¼ Property 7:

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 4 bc2 ð2D þ ZÞ þ 4D bc2 ð2D þ ZÞ þ 4D 9P þ 9P 2b c ð2D þ ZÞ bc3 ð2D þ ZÞ

:

ðA3:24Þ

The optimal production order quantity Q for Case 2 is: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 q 3 q q2 p3 q2 p3 a þ þ þ , Q ¼ þ 4 27 4 27 3 2 2

where: ETC2 ¼ 8G2 KTP W2 FTC2 ¼ 3GCraw P2 WrðT1 TP Þ2 þ 3GPW Craw rðT1 TP Þ2 2TP þ 4KTP ðm a0 Þ GTC2 ¼ 12P3 STP 1 FTC2 2 p¼ 3 ETC2 GTC2 2 FTC2 3 q¼ þ ETC2 27 ETC2 a¼

FTC2 : ETC2

ðA3:25Þ

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International Journal of Production Research Derivation: Let @TC2 [email protected] ¼ 0, we have:

8 9 < 12P3 STP = 2 3GCraw P Q WrðT1 TP Þ þ 3GPQ2 W Craw rðT1 TP Þ 2TP þ 4KTP ðm a0 Þ : ; 8G2 KQ3 TP W2 ¼0 12P3 Q2 TP ðA3:26Þ 2

2

2

Rearrangement: Q3 ð8G2 KTP W2 Þ þ Q2 3GCraw P2 WrðT1 TP Þ2 þ 3GPW Craw ðrðT1 TP Þ2 2TP Þ þ 4KTP ðm a0 Þ 12P3 STP ¼ 0:

ðA3:27Þ

Having ETC2 , FTC2 , and GTC2 as defined in the above expressions, then Equation (A3.27) can be written as the following: FTC2 2 GTC2 Q þ Q3 þ ¼ 0, ðA3:28Þ ETC2 ETC2 where the coefficients, FTC2 =ETC2 and GTC2 =ETC2 , are known values. Similarly, by the Cardano formula, the root is: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 q 3 q q2 p3 q2 p3 a Q ¼ þ þ þ : þ 4 27 4 27 3 2 2 Property 8:

ðA3:29Þ

The optimal production period T1 for Case 2 is: T1

Qr 2bP2 þ ect ½2Pða0 Craw þ aPÞ þ GCraw QW : ¼ 2bP3 r þ Pect ½2Pða0 Craw r þ Par þ PÞ þ GQWCraw r

ðA3:30Þ

Derivation: Let @TC2 [email protected] ¼ 0, we have: i h 0 PþGQWÞ ðP DÞ T1 T2P rðTP T1 Þ CM ðtÞ þ Craw ð2a2P 2 TP

¼ 0:

ðA3:31Þ

Then: T1 ¼

Qr 2bP2 þ ect ½2Pða0 Craw þ aPÞ þ GCraw QW : 2bP3 r þ Pect ½2Pða0 Craw r þ Par þ PÞ þ GQWCraw r

ðA3:32Þ