selection. The Material Handling Equipment Selection (MHES) provides .... can be helpful for understanding the effectiveness equationâeven though most plant.
PROJECT REPORT
Selection of Material Handling Equipment M.Sc. (Mechanical Engineering Design) 2004-2006
Submitted By:
Engr. Rafiullah Khan
Supervised By: Prof. Dr. Iftikhar Hussain
Department of Mechanical Engineering, NWFP University of Engineering and Technology Peshawar, Pakistan.
Abstract The purpose of this work is to develop a new methodology for automating the determination of a material handling system by combining knowledge based and optimization approaches. The proposed system extends previous concepts of minimization of operating cost by including the cost for reliability, performance and flexibility into total cost. Mathematical model of the cost for Availability, Reliability, Maintainability and capability of different MHE (Material Handling Equipment) is developed. These cost values are then added into total cost (investment, Operating) of the individual MHE accordingly. This overall cost is then minimized by HASSAN’S construction algorithm for selection of Material Handling Equipment. The initial short listings of equipments were performed by the knowledge based system for the available MHE. Suitable code was used to develop the system. The models was tested, verified and validated by using 25 case studies.
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ACKNOWLEDGEMENT
All submission and glory is for the creator of talent and not the owner of it. It is in the recognition of blessings that the Merciful Almighty Allah has bestowed upon me. I am thankful from the core of my heart to the man whose loving guidance and cooperation was what I needed during my project work. He is my teacher and supervisor Professor Dr. Iftikhar Hussain Department of Mechanical Engineering. I also feel myself duty-bound to thank all my colleagues for their all time friendly and ever ready-to- cooperative behavior. Last but not the least, I do not find words to thank my parents who suffered all the hardships to educate and train me throughout my life for a better future.
Rafiullah khan
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TABLE OF CONTENTS ABSTRACT ........................................................................................................... I ACKNOWLEDGEMENTS .......................................................................................... II TABLE OF CONTENTS ............................................................................................ III
CHAPTER 1 INTRODUCTION .............................................................................. 1 1.1 Overview of Material Handling Equipment........................................ 1 1.2 Organization of the work ... ..................................................................3
CHAPTER 2 LITERATURE SURVEY.............................................................. .... 4 2.1 Materials Handling System Selection ……………………………….6 2.2 Research direction ............................................................................... 7
CHAPTER 3 AVAILABILITY, RELIABILITY, MAINTAINABILITY AND CAPABILITY ..........................................................................................................8 3.1 Effectiveness .......................................................................................8 3.1.1 Availability .......................................................................... 10 3.1.2 Reliability ............................................................................ 12 3.1.3 Maintainability ..................................................................... 14 3.1.4 Capability...................................................................................15
CHAPTER 4 KNOWLEDGE BASE SELECTION, OBJECTIVE FUNCTION AND COST MODELS OF MHE.............................................................................................16 4.1 A knowledge base system ........................................................................16 4.1.1 Knowledge base ..........................................................................16 4.1.2 Rules development .....................................................................18. 4.1.3. Knowledge based system...........................................................20 4.2 Objective Function...................................................................................20 4.3. Constraints ..............................................................................................21 4.4. Cost Models of MHE ..............................................................................22
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CHAPTER 5 METHODOLOGY ...................................................................................27 5.1. Heuristic approach .................................................................................27 5.1.1. Algorithm for the solution of Problem.......................................27 5.1.2. PROBLEM.................................................................................30 CONCLUSION...............................................................................................................33 REFERENCES ...............................................................................................................34 APPEDIX A....................................................................................................................35 APPENDIX B .................................................................................................................41
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Chapter 1
Introduction Material handling and transfer is defined as the movement of physical objects such as raw materials, component parts, sub-assemblies, assemblies and finished goods along within the manufacturing environment from receiving through shipping. The purpose of moving the material should be to increase its value. However, the handling, transporting, housing and controlling of materials and goods adds nothing but cost to the system [Sims (1991)]. The material handling and transfer is thus regarded as a burden and therefore, often carried out as a final step after product, process and layout design have been completed. Material handling activities may cost as much as 55% of the total production cost in an average industry [Pan et al (1992), Welgama and Gibson (1995)]. An efficient Materials Handling System (MHS) greatly improves the competitiveness of a product through a reduction of handling cost. The fundamental principles of material handling include the use of ‘systems approach’ where the material handling requirements of the entire factory is considered, and simplification of moves through the reduction or elimination of un-necessary and combination of several moves. Traditionally, ‘experts’ who analyze a few alternatives from which a selection is made based on their experience in the application environment have determined MHS. Selection of suitable MHE requires a complete analysis of the material handling problem. The design of MHS includes the selection of material handling devices to transport material between facilities, which also impact on lead time, safety, work in process, queue length, inventory levels and over all operating efficiency of a facility. Thus the proper design of MHS is very important for both conventional and advanced manufacturing systems. 1.1. Overview of Material Handling Equipment: A great variety of material handling equipment is available commercially. Material handling equipment includes: (1) transport equipment, (2) storage equipment, (3)
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unitizing equipment, and (4) identification systems. In this work only transport system is focused. 1.1.1 Transport equipment: Material transport includes equipment that is used to move material inside a factory, warehouse, or other facility. This equipment can be divided into the following five categories. 1. Industrial trucks. Industrial trucks divide into two types: non powered and powered. Non powered trucks are platforms with wheels that are pushed by human workers to move materials. Powered industrial trucks are steered by human workers. They provide mechanized movement of materials. 2. Automated guided vehicles (AGVs). AGVs are battery powered, automatically steered vehicles that follow defined pathways in the floor. The pathways are unobtrusive. AGVs are used to move unit loads between load and unload stations in the facility. Routing variations are possible, meaning that different loads move between different stations. 3. Monorails and other rail guided vehicles. These are self-propelled vehicles that ride on a fixed rail system that is either on the floor or suspended from the ceiling. The vehicles operate independently and are usually driven by electric motors. 4. Conveyors. Conveyors constitute a large family of material transport equipment that is designed to move materials over a fixed path, generally in large quantities or volumes. Examples include roller, belt, and tow-line conveyors. Conveyors can be either powered or non powered. Powered conveyors are distinguished from other types of powered material transport equipment in that the mechanical drive system into the fixed path. Non powered conveyors are either activated either by human workers or by gravity. 5. Cranes and hoists. These are handling devices for lifting, lowering and transporting materials, often very heavy loads. Hoists accomplish vertical lifting; both manually operated and powered types are available. Cranes provide horizontal travel and generally include one or more hoists.
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Table 1 summary of features and applications of five categories of Material Handling equipment Material handling equipment Features Typical applications Industrial trucks, manual Low cost Moving light load in a factory Low rate of delivery/hr Industrial trucks, powered
Medium cost
Automated guided vehicles
High cost Battery powered Flexible routing
Monorails and other rail guided vehicles
High cost Flexible Routing
Conveyors, Powered
Movement of pallet loads moving pallet loads in factory Moving work in process along variable routes Moving single assemblies products or poallet loadds along variable routes
Great varietyof equipment In-floor, On the floor, Mechanical powered
Cranes and Hoists
Lift capacities more than 100 tones
Moving large, heavy items in factories etc
1.2. Organization of the work: In chapter 2 literatures about the selection of Material handling system is revised. In chapter 3 Reliability, Maintainability, Availability and Capability are described in detail to find suitable indices to be used in the effectiveness equation. Chapter 4 outlines the attempt to model the costs associated with MHE which are further use in the selection of suitable MHE. A new approach to model the MHE cost based on equipment reliability is developed. This is followed by the constraints and objective function of the MHE selection problem. Chapter 5 presents the Methodology used to solve MHE selection problem.
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Chapter 2
Literature Survey The research done by various researchers in the field of MHE selection is given in the following lines. 2.1. Materials Handling System Selection The literature covers optimization approach, expert systems (knowledge based) and hybrid systems for the selection of MHE. WEBSTER AND REED (1971) used optimization technique for finding a suitable minimum cost MHE for each move without initially being concerned about improving utilization, and subsequently combining several moves and assigning to some selected MHE in an attempt to improve utilization. HASSAN (1985) proposed construction algorithm which selects a minimum cost MHE from a candidate MHE set and assigns moves to it until its utilization reaches an acceptable level, the moves assigned to the equipment are assigned to some other equipment type. One advantage of this method over Webster’s procedure is that the method itself estimates the operating times and operating costs, however an operating cost per unit load distance per period is required for each item of equipment. Both procedures require the user to determine a feasible candidate MHE set for each move and t6he cost to performing each move by each MHE. FARBER AND FISHER (1985) have developed MATHES, a Material Handling Equipment Selection Expert System. To arrive at a decision, the values of four main parameters viz. path, volume of flow, size of load and distance between facilities are determined. Fisher et al. (1988) have improved MATHES by taking into account both technological and economic considerations using heuristic rules. The results indicate that due to technological considerations the conveyors should not be selected for material moves over a variable path, while an AGV is not selected for a low volume move. Even though an AGV is technically feasible for low volume moves, it can justify its high cost only for high volume moves.
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GABBERT AND BROWN (1988) have developed MAHDE (Material Handling Design), a hierarchical frame structured KB system. The MAHDE initially selects the equipment for an MHS design based on the physical capacities of the equipment size, payload and throughput. The equipment, which does not meet the initial parameters are removed from the subsequent searches to narrow down the search space. The MAHDE system combines formal and Expert System (ES) methodologies to address the complexity of the problem and is able to select an equipment type based on optimal cost, an availability measure, lead time, a feasibility measure and a security measure. HOSNI (1989) has presented an ES for material handling method and equipment selection. The Material Handling Equipment Selection (MHES) provides suggestion for an MHS configured to meet a particular purpose and limited by some constraints such as cost, area, material type, material weight and move characteristics and frequency. The MHES is basically based on the famous material handling equation: MATERIAL + MOVE = METHOD devised by Apple (1976). For the selection of an equipment type, a set of questions guide the user through the various frames leading to one or more equipments. NOBEL AND TANCHOCO (1993) have presented a framework for an MHS design justification. Design justification refers to a design procedure where the economic ramifications of design decisions are considered simultaneously with design development. The goal of design justification is to guide the designer to a design that is justifiable from both a performance and economic perspective. The MHS design justification framework consists of system designer, design interface, design inference model, model generator, rule base and database. The comparison between system alternatives is facilitated through graphs showing total system cost, total system flexibility or unit flexibility cost. RUBINOVITZ AND KARNI (1994) have presented a detailed description of the use of ES for the selection of material handling and transfer equipment type. The ES compares a set of attributes of the intended operating environment with a set of attributes of the Material Handling Transport (MHT) equipment. After comparison, the system selects the most appropriate equipment type and model. The MHT specification is created
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in the form of a questionnaire, listing the interface design attributes and their possible values. Integrating the ES into the design process is also achieved. PRASERT AND K.J.ROGERS (1994), focus on the activity-based costing approach to the equipment selection problem for manufacturing systems. This technique help decision-makers select the appropriate set of equipment or machines to be used in the system based on the objective to minimize total operating cost subject to the availability of machines in the system. WELGAMA AND GIBSON (1995) have developed a hybrid KB/optimization system for automated selection of MHS, through minimization of total cost and aisle space requirements. The model also allows the design considerations to be treated as parameters, determines the equipment’s design load capacity and selects a candidate set of equipment through a knowledge base, based around certain constraints. A mathematical model (which in fact is the extension of Hassan and Hogg (1985)) is developed with the objective function of minimizing the total cost. The constraints ensure that all the moves are assigned to the MHE and one move to assign to only one MHE type. The knowledge base of the hybrid methodology obtains a feasible set of MHE for each move, and then an optimization algorithm determines the optimum MHE for all moves using a system approach. RAMZAN YAMAN (1999) used a knowledge-based approach for selection of Material Handling Equipment and Material Handling System. This approach speed up the design process and to extends personal abilities. In this approach, MHS equipment selection is defined as a matching problem between product, process handling requirements and equipment specifications using rule sets. HUSSAIN et al. (2006) used a hybrid (production rules, fuzzy logic and analytical approaches). KB part selects MHE with certain confidence level and the analytical part calculates the cost factors of the selected MHE in detail and practical manner. Some of the cost factors (such as intangible) that are difficult to estimate are calculated using fuzzy logic. Once the adjusted costs of the selected MHE are calculated, then various moves between the departments are assigned to the most feasible (within the selected MHE) MHE based on minimum cost.
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2.2. Research direction The expert system approach only selects an MHE on the basis of its attributes. It has nothing to do with the economics concerned certain MHE. It selects an MHE on the basis of feasibility of it for handling a particular type of material. There are two optimization procedures proposed in the literature. The basic concept behind the optimization method in [1] is finding a suitable minimum cost MHE for each move without initially being concerned about improving utilization, and subsequently combining several moves and assigning to some selected MHE in an attempt to improve utilization. The construction algorithm proposed by HASSAN on the other hand, selects a minimum cost MHE from a candidate MHE set and assigns moves to it until its utilization reaches an acceptable predetermined level. The algorithm proposed considers equipment types one at a time. Moves are then assigned then to selected equipment. If the utilization of equipment is less than an acceptable level, the moves assigned to the equipment are assigned to some other equipment type. One advantage of this method over WEBSTER is that the method itself estimates the operating times and operating costs. However operating cost per unit load distance per period is required for each item of equipment, both procedures requires the user to determine a feasible candidate MHE set for each move and cost of performing each move by each MHE. Cost models used by both algorithms are too simplistic to be useful in practice. So there is a need to have a cost model for the MHE which is more realistic and incorporate the realistic factors. Then this cost model is incorporated into the optimization technique and then integrating this with the expert system approach. The effectiveness of an MHE is to be included in the cost. Effectiveness is the multiplication of Availability, Reliability, Maintainability and Capability. These factors are discussed in detail in the next chapter.
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Chapter 3
Availability, Reliability, Maintainability and Capability Availability, reliability, maintainability, and capability are components of the effectiveness equation. The effectiveness equation is a figure of merit which is helpful for deciding which component(s) detract from performance measures. For many equipments and machine tools the reliability component is the largest detractor from better performance. 3.1. Effectiveness. Effectiveness is defined by an equation as a figure-of-merit judging the opportunity for producing the intended results.
The effectiveness equation is
described in different formats (Blanchard 1995, Kececioglu 1995, Landers 1996, Pecht 1995, Raheja 1991). Each effectiveness element varies as a probability. Since components of the effectiveness equation have different forms, it varies from one writer to the next. Definitions of the effectiveness equation, and its components, generate many technical arguments. The major (and unarguable economic issue) is finding a system effectiveness value which gives lowest long term cost of ownership using life cycle costs, (LCC) (Barringer 1996a and 1997) for the value received: System effectiveness = Effectiveness/LCC Cost is a measure of resource usage. Lower cost is generally better than higher costs. Cost estimates never includes all possible elements, but hopefully includes the most important elements. Effectiveness is a measure of value received. Clements (1991) describes effectiveness as telling how well the product/process satisfies end user demands. Higher effectiveness is generally better than lower effectiveness. Effectiveness varies from 0 to 1 and rarely includes all value elements as many are too difficult to quantify. One form is described by Berger (1993):
Effectiveness = availability * reliability * maintainability * capability In plain English, the effectiveness equation is the product of: --the chance the equipment or system will be available to perform its duty, --it will operate for a given time without failure, --it is repaired without excessive lost maintenance time and --it can perform its intended production activity according to the standard. 8
Each element of the effectiveness equation requires a firm datum which changes with name plate ratings for a true value that lies between 0 and 1. Berger’s effectiveness equation (availability * reliability * maintainability * capability) is argued by some as flawed because it contains availability and components of availability (reliability and maintainability). For any index to be successful, it must be understandable and creditable by the people who will use it. Most people understand availability and can quantify it. Few can quantify reliability or maintainability in terms everyone can understand. The effectiveness equation is simply a relative index for measuring “how things are doing”. The importance of quantifying elements of the effectiveness equation (and their associated costs) is to find areas for improvement. For example, if availability is 98%, reliability is 70%, maintainability is 70%, and capability is 65%, the opportunity for improving capability is usually much greater than for improving availability. Table 1 contains a simple data set used to illustrate how some “—abilities” are calculated.
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Events are put into categories of up time and down time for a system. Because the data lacks specific failure details, the up time intervals are often considered as generic age-tofailure data. Likewise, the specific maintenance details are often considered as generic repair times. Add more details to the reports to increase their usefulness. This limited data can be helpful for understanding the effectiveness equation—even though most plant level people do not acknowledge the have adequate data for analysis (Barringer 1995). 3.1.1. Availability deals with the duration of up-time for operations and is a measure of how often the system is alive and well. It is often expressed as (up-time)/(up-time + downtime) with many different variants. Up-time and downtime refer to dichotomized conditions. Up time refers to a capability to perform the task and downtime refers to not being able to perform the task, i.e., uptime
not downtime. Also availability may be the
product of many different terms such as:
A = Ahardware * Asoftware * Ahumans * Ainterfaces * Aprocess and similar configurations. Availability issues deal with at least three main factors (Davidson 1988) for: 1) increasing time to failure, 2) decreasing downtime due to repairs or scheduled maintenance, and 3) accomplishing items 1 and 2 in a cost effective manner. As availability grows, the capacity for making money increases because the equipment is in service a larger percent of time. Three frequently used availability terms (Ireson 1996) are explained below. Inherent availability, as seen by maintenance personnel, (excludes preventive
maintenance outages, supply delays, and administrative delays) is defined as:
Ai = MTBF/(MTBF + MTTR) Achieved availability, as seen by the maintenance department, (includes both corrective
and preventive maintenance but does not include supply delays and administrative delays) is defined as:
Aa = MTBM/(MTBM + MAMT)
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Where MTBM is mean time between corrective and preventive maintenance actions and MAMT is the mean active maintenance time. Operational availability, as seen by the user, is defined as:
Ao = MTBM/ (MTBM + MDT) Where MDT is mean down time. A few key words describing availability in quantitative words are: on-line time, stream factor time, lack of downtime, and a host of local operating terms including a minimum value for operational availability. An example of 98% availability for a continuous process says to expect up-time of 0.98*8760 = 8584.8 hr/yr and downtime of 0.02*8760 = 175.2 hrs/yr as availability + unavailability = 1. Now, using the data set provided above in Table 1, the dichotomized availability is 98.6% based on up time = 8205.3 hours and downtime = 112.5 hours. Of course the dichotomized view of availability is simplistic and provides worst case availability numbers. Not all equipment in a train provides binary results of only up or only down sometimes it’s partially up or partially down. Clearly the issue is correctly defining failure. In the practical world, complexities exist in the definitions for when only some of the equipment is available in a train, and the net availability is less than the ideal availability i.e., a cutback in output occurs because of equipment failure which decreases the idealized output from say 95% to a lower value such as say 87% when failures are correctly defined. A key measure is defining the cutback (and thus loss of availability from a dichotomized viewpoint) when the cutback declines to a level causing financial losses— this is the economic standard for failure. In short, the area under the availability curve can be summed to calculate a practical level of availability and generate higher values for availability than when only dichotomized values are used. Lack of availability is a problem related to primarily to failures of equipment. But the root cause of the failure may lie in different areas than initially expected. Often deterioration, leading to economic failure, causes conflicts in the definitions of reliability, maintainability, and capability— real life issues are rarely simple and independent.
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3.1.2. Reliability deals with reducing the frequency of failures over a time interval and is a measure of the probability for failure-free operation during a given interval, i.e., it is a measure of success for a failure free operation. It is often expressed as
R(t) = exp(-t/MTBF) = exp(-λt) Where λ is constant failure rate, and MTBF is mean time, between failures. MTBF measures the time between system failures and is easier to understand than a probability number. For exponentially distributed failure modes, MTBF is a basic figure-of-merit for reliability (failure rate, λ, is the reciprocal of MTBF). For a given mission time, to achieve high reliability, a long MTBF is required. Also reliability may be the product of many different reliability terms such as
R = Rutilities * Rfeed-plant * Rprocessing * Rpackaging * Rshipping and similar configurations. To the user of a product, reliability is measured by a long, failure free, operation. Long periods of failure free interruptions results in increased productive capability while requiring fewer spare parts and less manpower for maintenance activities which results in lower costs. To the supplier of a product, reliability is measured by completing a failure free warranty period under specified operating conditions with few failures during the design life of the product. Improving reliability occurs at an increased capital cost but brings with it the expectation for improving availability, decreasing downtime and smaller maintenance costs, improved secondary failure costs, and results in better chances for making money because the equipment is free from failures for longer periods of time. While general calculations of reliability pertain to constant failure rates, detailed calculations of reliability are based on consideration of the failure mode which may be infant mortality (decreasing failure rates with time), chance failure (constant failure rates with time), or wear-out (increasing failure rates with time). 12
A few key words describing reliability in quantitative words are: mean times to failure, mean time between failures, mean time between/before maintenance actions, mean time between/before repairs, mean life of units in counting units such as hours or cycles, failure rates, and the maximum number of failures in a specified time interval. An example of a mission time of one year with equipment which has a 30 year mean time to failure gives a reliability of 96.72% which is the probability of successfully competing the one year time interval without failure. The probability for failure is 3.278% as reliability + unreliability = 1. For reliability issues, defining the mission time is very important to get valid answers. Notice from the example that high reliability for mission times of one year or more require high inherent reliability (i.e., large mean times to failure)—often the inherent reliability is not achieved due to operating errors and maintenance errors. The data in Table 1 shows the mean time between maintenance actions is 683.8 hours. Calculate the system reliability using the exponential distributions described above and a mission time of one year. The system has a reliability of exp(-8760/683.8) = 0.00027%. The reliability value is the probability of completing the one year mission without failure. In short, the system is highly unreliable (for a one year mission time) and maintenance actions are in high demand as the system is expected to have 8760/683.8=12.8 maintenance actions per year! So how can high availability be achieved with systems requiring many maintenance actions? The maintenance actions must be performed very quickly to minimize outages!!!!! This leads to pressures for establishing world class maintenance operations. A better way to solve the problem is to reduce the number of failures—thus demands for world class maintenance operations is avoided and costs are decreased—particularly when life cycle costs drive the actions. Remember failures carry hidden costs resulting from the hidden factories associated with production losses for disposal of scrap and the slow output incurred while reestablishing steady state conditions—the lost time may be 1.5 to 5 times the obvious lost time costs. The real issue for studying reliability is driven by a simple concept called money—particularly when the cost of unreliability (Barringer 1996c) is identified and used for motivating trade-off studies.
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High reliability (few failures) and high maintainability (predictable maintenance times) tend toward highly effective systems. 3.1.3. Maintainability deals with duration of maintenance outages or how long it takes to achieve (ease and speed) the maintenance actions compared to a datum. The datum includes maintenance (all actions necessary for retaining an item in, or restoring an item to, a specified, good condition) is performed by personnel having specified skill levels, using prescribed procedures and resources, at each prescribed level of maintenance. Maintainability characteristics are usually determined by equipment design which set maintenance procedures and determine the length of repair times. The key figure of merit for maintainability is often the mean time to repair (MTTR) and a limit for the maximum repair time. Qualitatively it refers to the ease with which hardware or software is restored to a functioning state. Quantitatively it has probabilities and is measured based on the total down time for maintenance including all time for: diagnosis, trouble shooting, teardown, removal/replacement, active repair time, verification testing that the repair is adequate, delays for logistic movements, and administrative maintenance delays. It is often expressed as
M(t) = 1- exp(-t/MTTR) = 1 - exp(-µt) Where µ is constant maintenance rate, and MTTR is mean time to repair. MTTR is an arithmetic average of how fast the system is repaired and is easier to visualize than the probability value. Note the simple, easy to use criteria shown above, is frequently expressed in exponential repair times. A better and more accurate formula requires use of a different equation for the very cumbersome log-normal distributions of repair times describing maintenance times which are skewed to the right. The maintainability issue is to achieve short repair times for keeping availability high so that downtime of productive equipment is minimized for cost control when availability is critical. An example of a stated maintainability goal is a 90% probability that maintenance repair times will be completed in 8 hours or less with a maximum repair time of 24 hours. This requires a system MTTR of 3.48 hours. Also the cap of 24 hours (99.9% of repairs will
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be accomplished in this time, or less) requires control of three main items of downtime: 1) active repair time (a function of design, training, and skill of maintenance personnel), 2) logistic time (time lost for supplying the replacement parts), and 3) administrative time (A function of the operational structure of the organization). The probability for not meeting the specified 8 hour repair interval in this example is 10% based on a MTTR of 3.48 hours as Maintainability + unmaintainability = 1. Data in Table 1 shows mean down time due to maintenance actions is 9.4 hours. Calculate the system maintainability using the exponential distributions and an allowed repair time of 10 hours. The system has a maintainability of 1-exp(-10/9.4) = 65.5%. The maintainability value is the probability of completing the repairs in the allowed interval of 10 hours. In short, the system has a modest maintainability value (for the allowed repair interval of 10 hours)! High availability (high up-time), high reliability (few failures) and high maintainability (predictable and short maintenance times) tend toward highly effective systems if capability is also maintained a high levels.
3.1.4. Capability deals with productive output compared to inherent productive output which is a measure of how well the production activity is performed compared to the datum. This index measures the systems capability to perform the intended function on a system basis. Often the term is the synonymous with productivity which is the product of efficiency multiplied by utilization. Efficiency measures the productive work output versus the work input. Utilization is the ratio of time spent on productive efforts to the total time consumed. For example, suppose efficiency is 80% because of wasted labor/scrap generated, and utilization is 82.19% because the operation is operated 300 days per year out of 365 days. The capability is 0.8*0.8219 = 65.75%. These numbers are frequently generated by accounting departments for production departments as a key index of how they are doing. Thus these calculations need few explanations. As we have defined the factors of effectiveness equation in detail. Now we are able to incorporate these factors in the cost model of MHE, which is discussed in the next chapter.
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Chapter 4
Knowledge base selection, objective function and Cost Model of MHE In this chapter first we introduce the knowledge base approach for the selection of Material handling equipment, and then the objective function of MHE will be discussed. After that, mathematical models for the cost of Material handling will be developed. 4.1 A knowledge base system A materials handling expert should analyze every move i and the capabilities of every MHE j. this involves analyzing the feasibility requirements. In recent years, a tendency exists to implement on expert systems approach to determine the feasibility of MHE for a particular move. In this chapter, a knowledge based system is developed to obtain a feasible set of MHE for each move, and then an optimization algorithm is used to determine the optimum MHE for all moves using a system approach. 4.1.1. Knowledge base The knowledge base consists of facts and rules that are used to obtain a feasible set of MHE types for each individual move. 4.1.1.1 Facts. These are the data values relevant to materials associated with moves, MHE data, location details of machines (source and destination of moves), and available time. The knowledge representation of facts is made in terms of lists. The following illustrates the knowledge representation. (i) The material associated with a move is represented as follows: Mat_data(F1i,F2i,Fi,[material type, nature, unit load, li, wi]) Here F1i=source associated with the move i F2i=destination associated with the move i Li= length of the unit load associated with the move i wi= width of the unit load associated with the move i Fi=the flow volume of move i.
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Material type and nature are considered because they are important in selecting a suitable MHE. Material type can be an ‘individual item’, ‘packaged’, or ‘bulk’. Material nature can be “fragile”, or “bulky”. (ii) MHE data are represented as follows. Equip(Rnj,eq.name,[Cj1,Cj2,Cj3],[special features], Vj, Cjp) Rnj=reference number for the MHE j. Eq.name=name of the MHE, e.g. tow tractor, AGV, bridge crane etc) Cj1, Cj2, Cj3=cost coefficients described before Special features=special features attached to the MHE, e.g. for a fork lift type 1 is “IC cushion type”: internal combustion engine with cushion tyres. Vj=speed of the MHE j Cjp=upper limit of the load carrying capacity of MHE j. Since in practice a wide range of load carrying capacities is available for a particular MHE type the upper limit of each type is considered here, as procedure will determine the appropriate “design and carrying capacity” for the optimum MHE. This information is useful to obtain a complete specification of the optimum set of MHE. The other facts such as available time are represented similarly in the knowledge base. 4.1.1.2. Rules. Rules are developed for obtaining a feasible set of MHE, calculating costs, and for combining moves which are parts of the optimization algorithm. The rules for obtaining a feasible set of MHE are developed using the material handling equipment selection guide. An example of these rules is: (R.1) IF material type is not “bulk” and Material nature is not “fragile” and Load < 100kg
and
Frequency is not “low” THEN
roller conveyor is feasible.
Also rules are developed to check the feasibility based on the unit load of material and the equipment capacity, and for checking feasibility of overhead cranes, the equipment cost calculations described before are also implemented in the knowledge base as rules. Since the material flow is in numerical form and the frequencies used in the above rules
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are in qualitative form, rules based on a volume matrix are used to convert flow into frequency. 4.1.2 Rules development General guidelines for the analysis and selection of MHE [Apple (1976)] are used to develop production rules in the knowledge base of this methodology. Although the chart [Apple (1976] cannot accurately depict relationships between a huge number of equipment and a large number of factors, but still it can serve as a guide to general types of equipment, since it again depicts the thinking process involved in the selection problem. Production rules for 20 different MHE are developed. Although the guidelines do not report information about the selection of Automated Guided Vehicle (AGV), and since AGVs play an important role in the modern day material handling activities, therefore, these factors are also taken into consideration based on the characteristics of the AGVs. These factors (attributes) are grouped into three major phases, material, move and method. Although no such “crutch” can compare with personal knowledge, nor can it indicate logical adaptations, which may make an equipment type applicable, but it might be helpful in pointing out possibilities with which the analyst may not be familiar. Forward chaining depth first inference strategy is used to execute rules [Kamran and Mark (1988)]. Figure 1 below shows this strategy.
A C
B C
D
E
F
Figure 1. Tree of possible paths for the search problem The depth-first-search burrows in to a tree looking for the goal state. By convention, the leftmost alternative below the current node is chosen as the next node to move to. 18
Thus, depth first search begins by examining the left most branches in Figure 1 (A-B-C). Since a terminal node is encountered without reaching the goal, the search method then moves back up the tree to the next untried path. In this Figure, it moves back up one node to B and going through the entire sequence (A-B-C-B-D-F). Since simple yes or no to a production rule is inadequate and that the real world is characterized by uncertainty (Zadeh, 1965), therefore, fuzzy logic approach has been used to handle uncertainty in the selection of MHE. An illustration of a rule using certainty factors is given below in Figure 2. (cf = 0.8)
(cf = 0.7)
(cf = 0.6)
(cf = 0.8)
(cf = 0.5)
Rule1
Rule2
Rule3
Rule4
Rule5
AND RI (cf)
Rule R(cf) = 0.7
Conclusion
(cf)
Figure 2. Rules illustration with certainty factors RI(cf) = min (0.8, 0.7, 0.6, 0.8, 0.5) = 0.5 Cf = RI(cf)*R(cf) = 0.5*0.7 = 0.35 RIk(cf) = min {Pi(cf)} if all Pi(cf) ≥ δ (0.2, assumed) Or RIk(cf) = 0, if Pi(cf) < δ for any i Where RIk(cf) is the composite rule input or premise confidence factor of rule k Pi(cf) is the confidence factor for premise clause i δ is confidence factor threshold level cfk = RIk(cf)*Rk(cf) Where cf is the output confidence factor of rule k Rk(cf) is the confidence factor of rule k
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4.1.3. Knowledge based system. The knowledge base described above is used initially to obtain a feasible set of MHE types, for each individual move, for further consideration in the optimization algorithm. This is carried out as follows: Consider a move, and test the feasibility of using each MHE type for the selected move using the knowledge base. All feasible MHE types for the move concerned form a set of feasible MHE types for further consideration. The process is repeated for all the moves. During the optimization process, the feasibility is maintained by referring to the knowledge base whenever a change is considered, in order to optimize the total system cost, to the initially selected MHE for a given move. 4.2 Objective Function The objective is to select a MHS such that total material handling cost is minimized. The total cost includes the increased capital cost due to effectiveness and operating cost of the MHE. Thus the objective function becomes:
N
Minimize z = ∑{ λj(Caj+CjO) }
(4.1)
j =1
N
Subject to
∑ aij * xij = 1
for i=1,2,…,m
(4.2)
for all I,j
(4.3)
∑ tij * xij ≤ µij*Ta
for j=1,2,….,N
(4.4)
xij ≤ aij
for all I,j
(4.5)
CjO= ∑ wij * xij
for j=1,2,…N
(4.6)
Wij=Cj3*tij*xij
for all i,j
(4.7)
for j=1,2,…N
(4.8)
for all i,j
(4.9)
j =1
xij ≤ λj m
i =1
m
i =1
λj={0,1}, xij={0,1}
µij ≥ 0
Where Tij=total operating cost of equipment type j required for move i Wij=operation cost equipment type j for move i
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aij=1 if equipment type j can be used for move I,0 otherwise λj=1 if MHE j is choosen,0 otherwise µj=number of units of MHE j required Ta=available time xij=1 if move I is assigned to j,0 otherwise Caj=adjusted increased cost of MHE j Cj3= operating cost of MHE j The above formulation is an extension of Hassan model. The objective function represents the minimization of the total cost (adjusted capital cost, operating cost). The constraints ensure that all the moves are assigned to MHE and one move to only one MHE type. 4.3. Constraints: The constraints required to be satisfied when searching for an optimum MHS are on feasibility, utilization and other system requirements. 4.3.1 Feasibility constraints. (a) Feasibility based on the material type, nature and flow volume: the MHE selected should be capable of handling the material in the technological sense. (b) Feasibility based on the unit load of the move and capability of MHE: the load carrying capacity of the MHE should be more or equal to the unit load associated with the move concerned. (c) Crane feasibility: bridge cranes and gantry cranes operate on rails. They can not be used for moves which extend beyond the span of these rails. 4.3.2 Utilization: The utilization of the selected MHE for all moves assigned to it should not exceed an acceptable limit. This limit should be decided, considering allowances required for operator changes, meal breaks if any, maintenance shutdown etc. 4.3 .3 other system constraints: (a) All moves should be assigned to material handling equipment. (b) One move should be assigned to only one equipment type. Although in practice on occasions, a move may be handled by more than one MHE type, this is not a very attractive option for management due to the complexities involved. For this reason and
21
for simplicity of analysis, a move is assigned to only one equipment type. However, one equipment type can handle many moves subject to feasibility and utilization limits. 4.4. Cost Models of MHE As the primary objective of any MHS selection problem is to minimize handling costs. For this purpose an accurate model of costs and easy to estimate cost coefficients should be used. Material handling cost consists of MHE investment (capital) cost and MHE operating cost. The cost considerations provided in Webster [1] and Hassan is too simplistic to be useful in most practical applications. They do not consider the costs of Reliability, Maintainability, Availability and Capability of an MHE. This might lead to a situation that an MHE would be selected which can not be efficient to perform its expected duty. The procedure developed here has the capability to incorporate the above mentioned factors in the model for cost of an MHS. 4.4.1 Investment/capital cost of MHE. Investment cost of MHE should be discounted to represent annual investment cost. This investment cost depends on many factors. The investment cost of variable path equipment j (e.g.AGVs, fork-lifts), Cj, which is assumed to be linearly proportionate to the lifting capacity, is given by,
Cj=Cj1 + Cj2 +Cjp
(1)
Where Cj1=a fixed cost Cj2=cost per unit load capacity Cjp=load carrying capacity. Of the fixed path equipment types, the investment cost of a bridge and gantry crane is proportionate not only to the load carrying capacity, but also to the span. Hence the investment cost of bridge/gantry crane j is modeled as:
22
CjI=Cj1+Cj2+*Cjp*S.
(2)
The investment cost of conveyers is mainly proportionate to the width of conveyors and distance associated with the move. It is assumed here that the coefficient Cj1 considers the effect of load. It is reasonable to approximate the width of a conveyor to be equal to the width of the unit load associated with the move concerned. Hence, the investment cost of conveyor j used for move i, Cji
Cj1=Cj1+Cj2*Wi*di
(3)
Where Di=distance associated with move i Wi =width of the unit load associated with move i. 4.4.2 Effect of Reliability, Maintainability, Availability and Capability on the capital/investment cost of MHE. In the previous section we modeled the cost of various MHE. During modeling process we ignored that up to what extent these equipments are reliable, maintainable, and available for service and what is the capability of a specific MHE. Before we introduce these factors in to the cost model we first define these factors briefly. These are discussed in detail in the previous chapter. Reliability is a measure of the probability of a system machine or process for failurefree operation during a given interval. Maintainability deals with the duration of maintenance outages or how long it takes to achieve the maintenance actions compared to a datum. Availability deals with the duration of uptime for operations and is a measure of how often the system is alive and well. Capability deals with the productive output compared to inherent productive output which is a measure of how well the production activity is performed compared to the datum.
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Let us define the Effectiveness. Effectiveness is a figure-of merit judging the opportunity of equipment for producing intended results. Effectiveness is a measure of value received. Clements (1991) describes effectiveness as telling how well the product/process satisfies end user demands. Higher effectiveness is generally better than lower effectiveness. Effectiveness varies from 0 to 1 and rarely includes all value elements as many are too difficult to quantify. One form is described by Berger (1993): Effectiveness = availability * reliability * maintainability * capability
(4)
In plain English, the effectiveness equation is the product of: --the chance the equipment or system will be available to perform its duty, --it will operate for a given time without failure, --it is repaired without excessive lost maintenance time and --it can perform its intended production activity according to the standard. Each element of the effectiveness equation requires a firm datum which changes with name plate ratings for a true value that lies between 0 and 1. If the effectiveness of an equipment/system is low, we have to pay for it. A more effective equipment is more cost competent than an equipment having less effectiveness. Thus an increased capital cost of less effective equipment. Let us denote effectiveness by Є, then the increased adjusted capital cost CDaj of a material handling equipment can be given by, Caj=Co + [1- Є]* Co
(5)
Where Co= original capital cost of the MHE In an ideal case the Effectiveness Є (all the four Reliability, maintainability, availability and capability in equation 4 are1) is considered 1, then Caj=Co, i.e. the increased capital is equal to the original capital cost.
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4.4.3. Operating cost of MHE. The operating costs include fuel, electricity, and cost of operators, costs of maintenance and cost of spare parts. Although modeling these factors is extremely difficult, it is very reasonable to consider that the operating cost is linearly proportional to the operating time (time of use). Thus operating time of MHE (j) required for move i (except for a tow tractor) is given by tij=2*di*Fi / Vj
(6)
Where Di=distance associated with move i Fi=flow volume (in unit load) in the move i Vj=speed of travel of MHE j. Here rectilinear distances are used. Although, the loading and unloading times are not included explicitly, the speed Vj can be adjusted to reflect the loading and unloading time. Also the MHE is assumed to be returning empty to the base; hence the multiplication factor is applied. Operating time for a tow tractor (j) required for move i: tij= [2*Fi / (Cjp/Li)]*di/Vj
(7)
Where Li= unit load associated with move i. The operating time of conveyors depend on the frequency of flows. If the frequency is too low ( i.e. if the interarrival time of material is more than the transfer time), a conveyor can be operated intermittently. Otherwise the conveyors are operated throughout available working time. Let the annual working is denoted by Ta Then, operating time of conveyor j required for move I is given by: tij = Fi*di / Vj
if Ta / Fi > di / Vj
tij = Ta
otherwise
(8)
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Let Cj3 be the operating cost of MHE j per unit operating time. Then operating cost of a MHE j is given by CjO=Cj3*tij.
(9)
Until now we have developed the procedures for finding the factors required for the selection of MHE. Now we are able to develop the methodology and integrating the concepts developed. The Methodology developed is discussed in next chapter.
26
Chapter 5
Methodology The methodology used for the selection of MHE is given in the following lines. 1. The Expert System (ES) selects a set of feasible MHEs from a pool of MHEs using knowledge base. ES uses questionnaires to acquire input data regarding material which have to be moved, moves, and attributes of MHEs. Then decides which type of MHEs is feasible to handle the moves. Thus short listing of MHEs is performed at first stage. 2. Cost models for the short listed MHEs are developed. Effectiveness equation factors (Reliability, Availability, Maintainability, and Capability) corresponding each MHE is calculated. Then these factors are used in the cost models. 3. Operating Cost Data corresponding to each MHE j for performing move I is collected. Total cost of material handling for performing a move i is calculated. 4. A move i is assigned to an MHE on the principle of minimizing the handling cost. Thus an optimum set of MHE is selected. 5.1. Heuristic approach After short listing of the MHEs by Expert system we have a set of MHEs to which the moves will be assigned. Since the problem can not be solved optimally, a heuristic approach has to be employed. In the following lines general steps for the solution of the MHE selection problem are given. 5.1.1. Algorithm for the solution of Problem: The algorithm considers the equipment types one at a time. Moves are assigned to a unit of the selected equipment until it is fully utilized or no other move can be assigned. A selection of the second unit or another type is then made, and the moves are assigned until that second unit or type is also fully utilized or no further assignment is possible. The algorithm terminates when all moves are assigned. Both equipment selection and move assignment are performed in a manner that helps in cost minimization. The steps of the algorithm are as follows.
27
1. for each equipment type, calculate the number of units that would be needed if the equipment performs all the moves as follows: Let
Yi=
∑ hij / Hi j
If the division is an exact integer, then λi=Yi If the division is not an exact integer, then λi=[Y]+1 Where the quantity in the brackets is integer portion of Y i. Usually, Hi is a set equal to 1, and hij is expressed as a fraction of Hi. 2. calculate the total cost of material handling for each equipment type as Zi=λiKi+ ∑ Wij jeEi
Where Ei is the vector of moves that can be performed by equipment type I, and the number of these moves is qi. 3. calculate the average cost for each equipment type per move as Zi(bar)=Zi/qi 4. Select the equipment having the smallest Zi(bar) first. Resolve ties by selecting the equipment with the smallest Zi. If the ties persist, resolve them by selecting in order of ascending λiKi. 5. For the selected equipment type, arrange the moves that can be performed by it in increasing order of equipment cost. 6. Assign the moves to the selected equipment starting with the move having the smallest operating cost. After each assignment, check to see whether the sum of hij is equal to Hi or within a tolerance Ei of it. If the sum of hij is equal to Hi, go to the next step; otherwise, check either of the following two cases: a. If the moves are the only remaining moves, or they can not be assigned to another piece of equipment, leave the assignment as it is. b. If the sum of hij is greater than Hi (or a multiple of Hi depending on the number of units required of the equipment so far), check the difference between the least integer multiple of H(making it greater than the sum of
28
hi) and the sum of hij, if the difference, which represents idle time, is less than or equal to E2 (a specified acceptable idle time), leave the assignment as it is. If the difference is larger than E2, remove moves from the equipment, starting with the last assigned move, until the acceptable utilization level is achieved. 7. Delete the moves assigned from consideration for the remaining moves, calculate a new value for Zi as before, and repeat the steps until all the moves are assigned. The process of MHE selection is shown in the flow chart below.
29
A sample problem is solved to illustrate the methodology which is given in detail next. 5.1.2. PROBLEM: Suppose we have 10 types of equipments which are short listed by ES to four. The data of Moves is given below
Equipment 1 Move 1 2 3 4 5 6 7 8 9 10 total
operating cost 400 600 400 500 100 200 0 0 0 0 2200
Operating time 0.4 0.6 0.5 0.4 0.3 0.7 0 0 0 0 2.9
Equipment 2 operating cost 0 400 500 400 300 900 400 100 200 100 3300
Operating time 0 1 1 1 1 1 1 1 1 1 9
Equipment 3 operating cost 200 900 900 800 0 0 400 300 800 0 4300
Operating time 0.5 0.8 0.9 0.9 0 0 0.7 0.7 0.7 0 5.2
Equipment 4 operating cost 0 0 0 0 0 0 200 200 900 500 1800
Operating time 0 0 0 0 0 0 0.4 0.4 0.3 0.2 1.3
The capital cost for each MHE is given below: Capital cost of MHE: Equipment Capital cost 1 2 3 4
5000 2777.8 3000 4000
30
The following table illustrates the basic calculations of step 1-3. Equipment Type
No.of possible mves
1 2 3 4
6 9 7 4
No. of unit of Eq 3 9 6 2
capital cost
total operating cost
Total cost
Zi
15000 24999.93 18000 8000
2200 3300 4300 1800
17200 28299.93 22300 9800
2866.667 3144.437 3185.714 2450
The smallest Zi is that of equipment 4; hence it is selected first, and the moves are arranged according to their operating costs as in table under: Move 7 8 10 9
W4j 200 200 500 900
h4j 0.4 0.4 0.2 0.3
∑h4j 0.4 0.8 1
After assigning moves 7,8 10, the ∑h4j=1; therefore this iteration is terminated. And one unit of 4 is fully utilized. For the next iteration the cost and the move time data are shown in the table, the moves already assigned are not included in the table.
Move 1 2 3 4 5 6 9 total
Equipment 1 operating Operating cost time 400 0.4 600 0.6 400 0.5 500 0.4 100 0.3 200 0.7 0 0 2200 2.9
Equipment 2 operating Operating cost time 0 0 400 1 500 1 400 1 300 1 900 1 200 1 2700 6
Equipment 3 operating Operating cost time 200 0.5 900 0.8 900 0.9 800 0.9 0 0 0 0 800 0.7 3600 3.8
Equipment 4 operating Operating cost time 0 0 0 0 0 0 0 0 0 0 0 0 900 0.3 900 0.3
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The calculation for Zi based on the data in the previous table is shown in the next table. Equipment Type
No. of possible moves
No unit of Eq
capital cost
total operating Cost
Total cost
Zi
1 2 3 4
6 6 4 1
3 6 4 1
15000 16666.62 12000 4000
2200 2700 3600 900
17200 19366.62 15600 4900
2866.667 3227.77 3900 4900
Equipment 1 has the smallest Zi; therefore it is the next to be selected. The ranked moves and their parameters, for a selected unit of equipment 1 second Iteration. Move 5 6 1 3 4 2
W1j 100 200 400 400 500 600
h1j 0.3 0.7 0.4 0.5 0.4 0.6
∑h1j 0.3 1
After assigning moves 5, 6 the ∑h1j=1; therefore this iteration is terminated. And one unit of 1 is fully utilized. If we continue in the same manner, the final assignment of the moves to the candidate equipment is the following: Move
1
2
3
4
5
6
7
8
9
10
Equipment
1
1
1
1
1
1
4
4
2
4
5.2. Development of the Matlab Program: In order to make the process of MHE selection automatic the algorithm is written in a Matlab program. This program is given in appendix A. 5.3. Testing and verification of the Methodology: Twenty five problems are solved using the proposed methodology. Results of the problems are verified by solving them manually and found them correct when checked by Matlab code. The problems tested and verified are given in Appendix B.
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Conclusion Following conclusions are made from the research work. •
Literature of MHE selection is reviewed. Both Expert system and optimization approaches developed by various researchers is discussed.
•
The previous mathematical models for the cost of material handling are improved by including the cost for the performance and effectiveness.
•
A new methodology is developed which combines the optimization approaches with the Expert system approach.
•
A detailed example is solved step by step in order to illustrate the developed methodology.
•
Methodology has been tested and verified by solving 25 problems manually and then using Matlab code.
•
The developed methodology can guide people in industries in more accurate way to arrive at a proper selection of MHE.
Future work: Short listing of the MHE can be done using fuzzy logic and then incorporating a mathematical model for cost of MHE to assign moves to the selected MHE with least cost.
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References 1. WEBSTER, D. B., and REED, R., 1971, A Material handling selection model. AIIE, 3, 13-21. 2. HASSAN, M. M. D., and HOGG, G. L., 1985, A construction algorithm for the selection and assignment of materials handling equipment, International Journal of Production Research, 23, 381-392. 3. FISHER, E. L., FARBER, J. B., AND KAY, M. G., 1988, MATHES: Material handling equipment selection, Engineering Costs and Production Economics, 14, 297-310. 4. HOSNI, Y. A., 1989, Inference engine for material handling selection, Computers and Industrial Engineering, 17(1-4), 79-84. 5. NOBEL, J. S., and TANCHOCO, J. M. A., 1993, A frame work for material handling system design justification, International Journal of Production Research, 31(1), 81-106. 6. WELGAMA, P. S., and GIBSON, P. R., 1995, A hybrid knowledge based/optimization system for automated selection of materials handling system, Computers and Industrial Engineering, 28(2), 205-217. 7. PAUL BARRINGER H., 1997, Availability, Reliability, Maintainability, and Capability, Barringer & Associates, Inc. Humble, TX. 8. HUSSAIN, I., 2002, Hybrid Approach to the Selection of Material Handling Equipment.
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APPEDIX A In this Appendix the code developed in MATLAB 6.5 is given. Here some instructions are given to efficiently use the program. 1. First of all write the moves in the first column of an excel file with name rda. In the same excel file write the operating cost and operating time corresponding to each move for each equipment. 2. Save the excel file as a text file rda.txt. 3.
In the MATLAB m-file moves2 write capital costs, for the each equipment, in the bracket in the line 7 in the name capcost.
4. Now the m-file is ready to be executed. 5. Run the file and see the results in the command line editor. 6. In the command line editor the 1st column is the moves and 2nd column gives the equipment to which the move in the 1st column is assigned.
35
clear all %for simplicity always save the text file from xcel %as rda.txt load rda.txt load capcost1.txt % in the following line write the capital % cost of each equipment in the bracket giving on espace %capcost is the capital cost capcost=capcost1(7,:); %capcost=[5000 2777.77 3000 4000]; % %nmove is the number of total moves and % is the the number of rows in rda.txt. nmove=length(rda(:,1)); % teq is the number of equipment and is the half % of the number of colums of rda.txt teq=length(rda(1,:))/2; mmm=[0]; mm1=[0]; % the maximum number of iteration will not be
36
% more than the No.of total moves for i=1:nmove jj=0; %for each equipment finding total %possible moves,total operating cost,total operating %time and no of equipment units in each iterations of %assignment of a move or moves
for j=1:teq %a is matrix of nonzero indices %possible moves by each equipment a=find(rda(:,2*j)>0); jj=jj+1; %tpm is total possible moves by equipment jj Tpm(jj)=length(a); %tocast is the operating cost and totime is the %operating time and noeq is the No of units %of equipment jj tocast(jj)=sum(rda(a,2*j)); totime(jj)=sum(rda(a,2*j+1)); noeq(jj)=ceil(totime(jj)); end
37
% tcapcost is the capital cost of noeq %No. of equipment jj and total cost is the %sum of operating and capital cost if eq jj tcapcost=noeq.*capcost; totalcost=tcapcost+tocast; dum=find(Tpm == 0); Tpm(dum)=1e-10; %finding average cost of equipment jj zbar=totalcost./Tpm; clear Tpm tocast totime noeq dum=find(zbar > 0); %finding minimum zbar value zbar_m=min(zbar(dum)); %eq_min is the equipment having minimum average cost eq_min=find(zbar==zbar_m); %a is the index matrix of equipment selected a=find(rda(:,2*eq_min)>0); b=[1 2*eq_min 2*eq_min+1]; %w_eq is the matrix of operating cost and time %of the selected equipment w_eq=rda(a,b); w_eq=sortrows(w_eq,2);
38
%sorting operating cost and calculating commulative %operating time of selected equipment h4j=cumsum(w_eq(:,3)); %a is the indices of moves possible to selected equipment %within its utilization df are the moves can be performed by %selected equipment a=find(h4j
