Production Planning and Quality of Service Allocation ... - IEEE Xplore

Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009

FrAIn3.13

Production Planning and Quality of Service Allocation across the Supply Chain in a Dynamic Lead Time Model Michael C. Caramanis, Chang-Chen Wu, Ioannis Paschalidis F

Abstract—We propose a supply chain (SC) production planning methodology optimizing inventory costs under quality of service (QoS) guarantees. Non-linear Clearing Functions are estimated using explicit models of stochastic production dynamics. A minimum inventory cost production plan is obtained subject to desired QoS guarantees. The proposed methodology employs time scale driven decomposition of the original problem to (i) a large time scale (e.g. week) planning master problem and (ii) short time scale (e.g. hour) facility-specific sub-problems estimating WIP and IFI levels and associated sensitivity information with respect to tentative master problem decisions. Computational experience and Monte Carlo simulation verification of the accuracy of the proposed SC production planning methodology is presented to demonstrate its potential effectiveness relative to industry practice.

I. INTRODUCTION Advanced Planning Systems are designed to provide feasible and near optimal plans across the supply chain while taking into consideration demand planning and resource constraints. At the operational level, the goal of Supply Chain Management is to fill orders on time at minimum inventory and holding cost, while observing supply chain constraints, including material balances, lead times and customer QoS guarantees. In today's highly competitive marketplace, lowering the overall SC inventory while providing short lead times and high customer QoS provides a significant comparative advantage. However, given the fundamental tradeoff between inventory and QoS, effective optimization requires synergistic management of the distribution of inventory, supply reliability and production capacity across the SC. Yet, state of the art practice in SC production planning [1]-[12] models only SC facility specific WIP while either ignoring Inter-Facility-Inventory (IFI) hedging (also known as base stock) policies or modeling IFI QoS as a static/exogenously specified requirement, thus leading to substantially sub optimal production planning. Related research in dynamic LT modeling is presented in a recent literature survey [13]. Work reported in [36]-[39] has focused on queuing approximations used to set optimal inventory targets in a base stock supply chain. Although production facility lead times and SC with base stock policy have been modeled in the literature mentioned above, a tractable supply chain production scheduling approach that NSF Grant CMMI-0300359 is acknowledged for partial support of research reported here. All authors are with Boston University, College of Engineering, e-mail: [email protected], [email protected], [email protected]

978-1-4244-3872-3/09/$25.00 ©2009 IEEE

optimizes WIP and hedging policies across SC production facilities and IFI locations has not been investigated up to now. In this paper we introduce a SC production planning methodology that guarantees the desired QoS to final demand while co-optimizing SC facility and IFI location specific production schedules and IFI hedging policies respectively. More specifically, the proposed methodology uses analytic models of stochastic production dynamics to estimate the non-linear SC production facility lead times and IFI levels in each period of the planning horizon. The analytic estimation models point towards promising directions for tractable co-optimization of facility specific production schedule and IFI hedging policies. The tractability of our optimal scheduling and hedging policy methodology is critically dependent on sufficiently accurate, yet computationally efficient analytic approximations of the SC performance resulting from the production capacity schedule and the IFI hedging policy implemented with a stop-and-go type SC production facility operation protocol. The analytic approximations include SC decomposition [17], Large Deviations (LD) asymptotics [15][19], G/G/1/K approximations [21], [22], two-moment Inverse Gaussian distribution approximation, and Monte-Carlo simulation based calibration of describing functions. The modeling accuracy of production facility and IFI operational dynamics is sensitive upon the accuracy of the describing functions, which we calibrate carefully using off line Monte Carlo simulation to provide the requisite data. The contribution of the research reported in this paper rests equally on two pillars: (i) the SC performance estimation modeling, and (ii) the iterative master-problem sub-problem optimization algorithm that we employ. Both of these pillars are presented below. The rest of the paper proceeds with the introduction of SC scheduling problem overview in Section II. The proposed optimization methodology, the master problem, and the subproblems of probabilistic facility-specific lead time and IFI QoS constraints are described in Sections III. Computational experience from the optimization of a two facility SC production schedule is presented in Section IV. Optimal solutions are obtained and evaluated against Monte Carlo simulation employed to verify the accuracy of our SC decomposition and optimization methodology. Finally, comparison of our optimal production schedules to industry practice supports the potential benefits that are realizable by developing further the proposed methodology.

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FrAIn3.13 II. OVERVIEW OF THE SUPPLY CHAIN SCHEDULING PROBLEM We propose a supply chain production planning methodology that addresses inventory costs under QoS guarantees. Non-linear supply chain production facility WIP and IFI levels are estimated in each period of the planning horizon using explicit models of stochastic production dynamics corresponding to the operational short-time-scale of minutes to hours. Facility specific production targets and IFI hedging policies are optimized for each of the long-time-scale weekly periods constituting the multiple-week planning horizon. The optimization minimizes WIP and IFI costs subject to meeting desired QoS guarantees at the customer end. There are various modeling alternatives for evaluating the WIP and Lead Time (LT) of production facilities, including on the analytic side Open Queuing Networks (OQN) [23], [24] and Closed Queuing Networks (CQN) [25], or Monte Carlo Simulation. We selected the OQN modeling option because (i) it is sufficiently general in representing topology and stochasticity of workstations and material routing within each production facility, (ii) it can efficiently model the propagation of variance from input to output locations, a feature that is important in evaluating WIP as well as the QoS at the downstream IFI location, and finally (iii) it can handle a general release process into the production facility from the upstream IFI location. As for the modeling of IFI locations, we selected G/G/1/K queue and LD asymptotic approximations employing a fictitious single part aggregation of multiple part types [1], [2], [8]. Whereas the above choices are not uniquely preferable, we made them because they are reasonably accurate and flexible and also because they are sufficient for the purpose of providing proof of concept which is our major goal. If more accurate implementation is required more accurate but also more computationally onerous choices can be made. β1 (tk )

ε0 = 0

N

R1 (tk ) R1 (t k )

µ1,1 N

X 1 (t k )

Z1 (t k )

Define: Decision Variables X c (tk ) : Average throughput target of facility c during period t k Rc (tk ) : Average release target of I c during time period t k

ε c (tk ) : Average stock out probability for Ic during time period t k β c (tk ) : Average blocking probability for facility c during time period t k

Dependent Variables Qc (tk ) : WIP of facility c at the end of time period t k I c (tk ) : Inventory level of IFI Ic at the end of time period t k Qc (tk ) : Average WIP of facility c during time period t k I c (tk ) : Average level of Ic during time period t k Z (tk ) : Hedging point of Ic during time period t k Known Parameters D (tk ) : Average external demand arrival rate during time period t k CtD2 C (tk ) : SCV of external demand inter-arrival time

µc,m : Q

Production capacity of machine m in facility c

κ c , I κ c , I κ C + , I κ C − are cost coeffieients

Pre-determined external demand is met from the products available at the last IFI (or FGI) after facility C, and it is backordered if a product is not available. Backlog is not βC (tk )

N

R2 (t k ) R2 (tk )

µ2,1

µ 2,M N

X 2 (tk )

2

µC ,1 N

εC

IC

QC (tk )

Q2 (t k )

X 1( tk )

µ1,M 1

denoted by t k , kÛ {1, 2, . . . , T}. Performance evaluation takes place at a short time scale (minutes to hours) that characterizes the performance evaluation sub-problem layer dynamics and is called a time slot.

ε 1 (tk )

I1

Q1 (t k )

exogenously pre-determined QoS at the final IFI with a minimum SC WIP and IFI cost. Planning (i.e. production scheduling) and QoS hedging point policy parameters vary across the weekly long time scale defined as a period and

X C (t k )

µC , M

C

ZC ( tk )

D (tk )

X C (t k )

Fig. 1. A single part type SC with limited production capacity at each facility and limited hedging level at each IFI location Ic.

To describe our SC model and establish notation, we consider the system depicted in Figure 1. Since our objective is to optimize the distribution of IFI levels across the SC, or equivalently the selection of IFI specific QoS levels, we consider a SC consisting of production facilities and IFI locations in tandem. In particular we consider C production facilities. Each facility is an OQN of machines with effective processing times modeled as a random variable with a probability distribution approximated by two moments. The objective of supply chain production scheduling is to meet an

allowed at intermediate IFI locations. The release process in front of each production facility follows a time process with the average release rate specified by the scheduling problem and a user specified probability distribution that represents the transportation process from the IFI to the next SC production facility. The release process in front of the first facility draws materials from an infinite source and delivers it at a rate with exogenously specified inter release times that are not affected by any form of internal starvation type disturbance. To reduce the complexity of SC modeling while providing

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FrAIn3.13 proof of concept in implementing SC optimization, we introduce a starvation and blocking hedging point policy implemented by stopping all activity at the facility, more specifically freezing it into inaction when the facility is starved or blocked. We call the resulting SC operation protocol the stop-and-go policy. While such a policy may introduce inaccuracies when large facilities are modeled – e.g. a fabrication facility with hundreds of stages – it is reasonable when the total processing time in each SC facility is small relative to the planning problem time scale as discussed in section III. Since IFI hedging points are finite, a production facility is idled when it is starved or blocked. For facility c to achieve a throughput target X c (tk ) during a planning time period t k , it must produce at a higher effective facility production intensity when it is neither starved nor blocked, so that the N X c (t k ) capacity lost during the idle time periods can be compensated and the requisite amount of material flow realized. This effective facility production intensity is quantified as: N

X c (tk ) =

X c (tk ) 1 − ε c −1 (tk ) − βc (tk )

(1)

In this equation, throughput target X c (tk ) is a value determine by the production plan. The stock out probability ε c −1 (tk ) is also a decision variable as it is equivalent to deciding on the QoS level (in fact QoS=1- ε c −1 (tk ) ) or the hedging level that is consistent with that QoS level. Due to the interaction of the facility idle probability and its effective production intensity that in turn affects the hedging level required to achieve a certain QoS level associated with a

throughput targets, (ii) IFI release rates, (iii) IFI stock out probabilities, and (iv) facility blocking probabilities. III. THE PROPOSED OPTIMIZATION METHODOLOGY A. Decomposition to Multiple Layers The non-linear facility WIP and QoS relations are modeled as convex constraints that can be approximated by outer linearization constraints. They capture short time scale SC stochastic dynamics and are amenable to analytic estimates of only single points on the constraint surface. To deal with these issues we propose a time scale driven decomposition of the original problem to (i) a multiple long time period planning master problem that determines average tentative production targets for each SC facility and inter-facility QoS levels and (ii) short time period sub-problems that determine facility-specific lead times and IFI levels which are consistent with the tentative master problem targets. The master problem layer uses as input facility specific performance and its sensitivity averaged over the length of each planning period (week). In order for the time scale decomposition to yield steady state estimates that are reasonable, the total processing time of each facility must be shorter than the planning period. This assumption holds under appropriate combinations of facility size and facility work station capacities. As pointed out correctly by Missbauer [40], steady state estimates of clearing functions – the essence of the methodology used here – are only appropriate when convergence to steady state, namely an adequate time scale separation, holds. Whereas the lead time at each facility is determined by estimating the facility’s WIP using a two moment OQN approximation, the QoS determination requires a model that D(tk )

n

X c (tk )

n

n

Rc (tk ) n ε c (tk )

n

β c (tk )

g c (i), ∇ n gc (i)

n hc (i), ∇ n hc (i) ∀c, tk

Fig. 2. The proposed optimization scheme with Master Problem (MP) and QoS-HC and Performance Evaluation Sub-problems.

certain IFI level, there is a resulting tradeoff between IFI and WIP. As a result, the optimal (i.e., minimum inventory cost) SC production/hedging plan is underdetermined unless the blocking probability β c (tk ) is also treated as a decision variable. Decision Variables. The conclusion of the preceding discussion is that decision variables include: (i) facility

involves the effective capacity feeding an IFI location and the effective demand withdrawing material from that IFI location. Since the latter requires horizontal coordination across the SC, we call it the QoS-HC sub-problem. Figure 2 demonstrates the proposed master problem and sub-problems employed in an iterative feedback loop framework. We use superscript n on decision variables and corresponding facility-specific LT and QoS-HC constraint functions in the

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FrAIn3.13 proposed optimization diagram to represent the iteration index. The major tasks of the two layers can be categorized as follows: 1) Master Problem Layer: Responsible for incorporating outer linearization constraints provided iteratively by the sub-problems and solving the SC production planning problem (i.e. determining tentative production and release targets and blocking and starvation probabilities) via linear programming. 2) Sub-Problem Layer: Responsible for SC probabilistic modeling and sensitivity evaluation used to generate linearization of LT and QoS constraints in the vicinity of each tentative Master Problem solution. Using this time scale driven decomposition approach, the higher level master problem is used to set tentative facility throughput targets, IFI release targets, IFI stock out and facility blocking probabilities for each weekly period or long time scale in the planning horizon. For each master problem tentative solution, the sub-problems provide sensitivity information to the master problem in the form of support hyperplanes of the non-linear facility WIP and IFI level relationships tangent to the point of the tentative solution. The resulting iterative algorithm evolves along multiple pairs of master-sub problem solutions until the master problem representation of the non-linear relationships is sufficient to allow the master problem to converge to the optimal solution subject to a user specified tolerance. Generally speaking, the sub-problem layer of our framework includes:
Dynamic Lead Time Modeling: We use standard OQN results to estimate average facility WIP/lead times as a function of production targets, stock out and blocking probabilities, and input IFI release variation. These lead time point estimates, g (i) , and their sensitivity are used to construct constraints on weekly master problem decision variables.
Quality of Service Modeling: We introduce QoS HC constraints that bound the probability of stock out at IFI locations and satisfy an exogenously determined QoS at customer end. We model these QoS guarantees as constraints in our production scheduling framework. We determine point estimates on the surface of the QoS HC constraints which depend on tentative master problem decisions. These non-linear QoS-HC constraint surfaces are denoted by h (i) and their sensitivity are similarly used to construct constraints on weekly master problem decision variables. B. The Master Problem The master problem minimizes overall SC WIP and IFI holding cost over decision variables including IFI stock out probability ε c (tk ) , facility blocking β c (tk ) probability, facility throughput X c (tk ) and IFI release target Rc (tk ) . The optimal solution is obtained subject to the material balance constraints (C1), facility WIP and IFI averages (C2), the facility maximum utilization constraints (C3), the non-linear

facility-specific LT constraints (C4), and the non-linear QoS-HC constraints (C5) as shown below: T

min

ε c ( tk ), βc ( tk ), X c ( tk ), Rc ( t k )

 C −1

∑ ∑ ( tk



Q

κ cQc (tk ) + I κ c I c (tk ) ) + I κ C + I C + (tk ) + I κ C − I C − (tk )  

c

Subject to: The material balance constraints (C1) Qc (t k ) = Qc (tk −1 ) + Rc (tk ) − X c (t k ) ∀c, tk I c (tk ) = I c (t k −1 ) + X c (t k ) − Rc +1 (tk ) ∀c ≠ C , t k I C (tk ) = I C (tk −1 ) + X C (t k ) − D (tk ) ∀tk ∀tk I C (tk ) = I C + (t k ) − I C − (tk ) Facility WIP and IFI average (C2) Qc (tk ) = α Q Qc (tk −1 ) + (1 − α Q )Qc (tk ) ∀c, tk I c (tk ) = α I I c (tk −1 ) + (1 − α I ) I c (tk ) ∀c, tk where α Q and α I are user defined weighting coefficients The facility maximum utilization constraints (C3) X c (t k ) ≤ ηc µc* (1 − ε c −1 (tk ) − β c (tk )) ∀c, tk where µc* = min {µ c } is the bottleneck machine capacity in facility c and

ηc is the user specified facility/machine utilization The non-linear facility-specific LT constraints (C4) Qc (tk ) ≥ gc (µ c , C2c (tk ), CtR2 c (tk ), X c (tk ), ε c −1 (tk ), β c (tk ), Πc ) ∀c, tk The non-linear QoS-HC constraints (C5) I c (t k ) ≥ hc (µ c , Cc2 , CtR2 c (tk ), CtR2 c +1 (tk ), X c (tk ), X c +1 (tk ),.. .. Rc +1 (tk ), ε c −1 (t k ), ε c (t k ), β c (tk ), β c +1 (tk ), Π c ) I C (tk ) ≥ hC (µ C , CC2 , CtR2 C (tk ), CtD2 C (tk ), X C (t k ), DC (tk ),.. .. ε C −1 (tk ), ε C (tk ), βC (tk ), Π C )

∀c ≠ C , tk ∀tk

In these constraint functions, we use Π C to denote the SC operation policy, µ c = [ µc ,1 , ..., µc , M ] and Cc2 = [Cc2,1 , ..., Cc2, M ] c

c

to denote the vector form of facility c machine capacities and processing time Square Coefficients of Variation (SCVs), and CtR2 (tk ) to denote the SCV of IFI Ic inter-release time process. c

Because constraints (C4) and (C5) are non-linear and not analytically representable, we adopt the successive piece-wise linear approximation proposed by Caramanis and Anli [10]. As a result, the master problem layer accumulates information about the non-linear relationships through the feedback it receives from the sensitivity and performance evaluation of the sub-problem layer. C. The Sub-Problem Layer 1) The Facility-specific LT Sub-Problem A facility LT constraint is essentially the required average WIP a production facility has to maintain during a period of time in order to achieve a throughput target set by the master problem. This facility WIP requirement is a function of decision variables (e.g. facility throughput target and idle probabilities), the probability distribution of machine processing times, and the SC operation protocol shown in (C4). Based on the analytic OQN stochastic analysis and effective facility production intensity of equation (1), a numerical example for this minimum average facility WIP requirement of a single machine with exponentially distributed processing time is shown in Figure 3.

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FrAIn3.13 SCV of service time process

In this example, we assume the machine capacity µc ,1 equals 50 parts/hour and the facility throughput target H X c (tk ) (where

H

X c (t k ) = X c (t k ) H and H is the number of working

1.24

hours per week) equals 30 parts/hour. The resulting constraint surface is a function of facility upstream stock out probability and downstream blocking probability. The relationship is highly non-linear especially when either the stock out or blocking probability increases to the point that the corresponding effective facility production intensity gets close to the bottleneck capacity.

1.22 1.2

C

C2tS (t k)

1.18 1.16 1.14 1.12 1.1 1.08

0.1

0.2

0.08 0.06

0.15 0.04

0.1 0.02

LT constraint surface (the required WIP)

0.05

εC-1(t k)

β C(tk)

Fig. 4: The SCV of effective service time process. 6

Average FGI level

5

Qc (tk)

4 3 2

40

1

35 30

0 0.2

IC(tk)

0.15 0.15

0.1 0.1 0.05

βc (tk)

0

0

25 20

0.05

15 εc-1(tk)

10 0.02

0.05

Fig. 3: Minimum average facility WIP requirement.

0.04

0.1

0.06 0.15

2) The QoS-HC Sub-Problem A QoS-HC constraint is essentially the required average IFI level the SC has to maintain at each IFI location during a period of time in order to meet the demand and satisfy the material/product stock out probability. This IFI level requirement is a function of decision variables (e.g. facility throughput target, IFI release rate, and idle probabilities), the probability distribution of machine processing times, and the SC operation protocol as shown in (C5). We estimate the QoS at some IFI location by aggregating effective upstream supply and effective downstream demand as represented by its mean and SCV. To this end, we use Monte-Carlo-Simulation based calibration of SCV describing functions and employ the inverse Gaussian (IG) two-moment approximation method to approximate unknown effective demand and service process at each IFI location. Analytically obtained results are then applied to calculate required IFI hedging points and corresponding average IFI levels that satisfy the tentative IFI QoS. Since backlog is only allowed at the final IFI location or Finished Good Inventory (FGI), it is necessary for us to perform the probabilistic modeling and sensitivity evaluation of each intermediate IFI using a G/M/1/K or G/G/1/K approximation. For the FGI location, however, we can and do use LD asymptotics. For illustration purposes, the numerical result shown in Figure 4 demonstrates the SCV describing function of effective FGI service time process for an all-exponential two facility SC. This SCV describing function is obtained through

0.08 0.2

0.1

β C(tk)

εC-1(t k)

Fig. 5: Minimum average IFI level requirement.

off line Monte-Carlo-simulation based data gathering that is then used to calibrate the describing function’s parameters. In Figure 4 the throughput target X C (tk ) is fixed and the effective SCV is shown as a function of FGI blocking probability βC (tk ) and upstream IFI stock out probability ε C −1 (tk ) . Meanwhile, under our stop-and-go SC operation protocol, the effective service capacity equals: X C (tk ) (1 − ε C −1 (tk ) ) M

X C (tk ) =

1 − ε C −1 (tk ) − βC (tk )

(2)

At this point, we can use the first two moments to approximate the effective FGI service time process using the Inverse Gaussian distribution. Given that the probability distribution of customer demand arrival is known (e.g. demand arrival rate D (tk ) and SCV CtD2 (tk ) are known) at the FGI location, the C

single stage SC decomposition and LD asymptotics results allow us to calculate the required hedging point and its corresponding average FGI level that guarantees the FGI stock out probability to not exceed the pre- determined level ε C (tk ) . For a pre-determined stock out probability of 0.05, the result of minimum average FGI level is shown in Figure 5. We observe a slight increase in the minimum average FGI level as upstream stock out probability decreases. The reason is that, when βC (tk ) is fixed, the effective service capacity of (2) increases while upstream stock out probability increases.

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FrAIn3.13 As other experiments of fixed service to demand ratio demonstrate, this implies that a slight increase in the effective FGI service capacity is more important than the SCV adjustment when upstream stock out probability varies. This result also demonstrates that increasing the facility blocking probability increases the facility’s capacity when it is not blocked which in turn decreases the IFI cost in order to achieve the final customer QoS guarantee. Based on the aggregation of the SC behavior upstream and downstream from an IFI which we call single stage SC decomposition, IFI hedging points and average IFI levels can be obtained using the analytic G/M/1/K result or the G/G/1/K approximation. In addition to the two-moment representation of the effective IFI service process, the effective demand process at an intermediate IFI must also be represented using the IG two-moment approximation. Similar to the effective service process, Monte-Carlo-simulation calibrated describing functions/approximation multipliers are also obtained for the IFI location and the effective demand arrival rate is calculated as: M

Rc (tk ) =

Rc (tk ) (1 − βc (tk ) ) 1 − ε c −1 (tk ) − βc (tk )

(3)

Referring to Figures 3 and 5, it is interesting to note that whereas IFI level (I) increase with quality of service while decreasing with blocking, WIP (Q) has the opposite behavior (note that the positive direction in the axes of figures 3 and 5 are reversed). The optimization algorithm captures this difference and selects blocking and starvation probabilities that achieve the optimal tradeoff among WIP and IFI levels. D. Complexity and Scalability Considering the number of master problem constraint equations as a measure of computational effort, a SC with C facilities and T planning time periods, requires at the first iteration (2C+1)T material balance constraints, 2CT facility WIP and IFI average constraints, and CT facility utilization constraints. Despite these (5C+1)T initial linear constraints, there are 2CT constraints that will be appended to the optimization problem after each iteration until the problem converges. Overall (without considering the general boundary and positivity constraints), if a problem is convex and takes n iteration to converge, the total number of constraints will reach (5C+1+2nC)T. When mild non-convexities are present, Caramanis and Anli [9] have suggested a two-phase outer linearization approach to obtain a feasible local optimal solution. This is achieved by dropping hyperplanes generated in earlier iterations, keeping only a subset of constraints generated during a fixed number of the most recent iterations. The implementation of our analytic approximations of IFI performance evaluation relies on the use of SCV describing functions and G/G/1/K approximation multipliers to achieve an efficient SC decomposition. The complexity and scalability of the optimization problem depends on the calibrations of these describing functions/approximation multipliers. For a SC with C production facilities/IFI, there are C-1 approximation multipliers of stock out probability and C-1 approximation multipliers of average IFI levels that must be

calibrated for the G/G/1/K approximation at intermediate IFI locations and one SCV describing function of FGI effective service time process required at the final demand location. Because the external demand arrival process is assumed to be known, there are (C-1) SCV describing functions of IFI effective demand arrival process that must be calibrated. So, overall there will be 3(C-1)+1 SCV describing functions required if the SC has C production facilities. The total number of describing functions can be reduced to 2(C-1)+1 if the average IFI level adjustments can be ignored in the G/G/1/K approximation. Although complex, computational experience presented below proves that our SC decomposition approach utilizing SCV describing functions and G/G/1/K approximations is practical and the optimization methodology scalable to industry size problems with multiple production facilities. IV. COMPUTATIONAL EXPERIENCE To verify the modeling accuracy of our proposed optimization framework, analytic inventory and QoS and blocking probability estimates obtained from the proposed optimization algorithm were compared to Monte Carlo simulation based estimates. Although longer planning horizon production scheduling problems can be solved, the examples presented here consist of three time periods. For illustration purposes, we conducted experiments on a two production facility SC. In particular, there are two types of SC cases that we consider: (i) all-exponential SC with single machine facilities, (ii) multiple machine facility SC with non-exponential production times and IFI release processes. The cost coefficients are listed in Table I and the demand scenario is shown in Table II. In both cases, the probability distribution of the external demand inter-arrival times is assumed exponential and hence CtD2 (tk ) = 1 . C

Q

c=1 c=2

κc

10 20

I

κc ( I κ 2 15 25

+

)

I

κ2

−

N/A 1000

Table I: Cost coefficients.

H

D2 (tk )

tk = 1

2

3

14

15

13

Table II: Demand scenario.

c=1 c=2

CtR2 c

µc ,1

Cc2,1

1 1

20 20

1 1

Table III: Case 1 IFI Release and production machine parameters.

Machine capacities and processing time SCVs and IFI inter-release time SCVs for the two SC cases are listed in Table III and Table IV. Each machine is allowed to operate at a maximum utilization of 0.95. The initial facility WIP and IFI

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FrAIn3.13

c=1 c=2

CtR2 c

µc ,1

µc ,2

Cc2,1

Cc2,2

1 0.5

20 20

20 20

1 0.7

1 0.9

Table IV: Case 2 IFI Release and production machine parameters.

Case 1. Optimal Solution (cost) Case 1. Monte Carlo Simulation (%) Case 1. Industry Practice (%) Utilization 0.9, Z=[12, 25] Case 1. Industry Practice (%) Utilization 0.95, Z=[12, 19] Case 2. Optimal Solution (cost) Case 2. Monte Carlo Simulation (%) Case 2. Industry Practice (%) Utilization 0.9, Z=[11, 23] Case 2. Industry Practice (%) Utilization 0.95, Z=[11, 17]

Q1

Q2

I1

I2

TC

169.0296

468.0246

358.9257

1015.5

2011.5

99.11%

95.85%

106.50%

102.13%

101.19%

157.26%

114.48%

102.28%

145.46%

131.54%

311.08%

221.85%

107.12%

111.24%

153.03%

276.5704

547.0201

394.0168

1027.5

2245.1

100.16%

99.19%

100.34%

103.05%

101.28%

193.86%

135.80%

86.90%

132.51%

132.86%

384.02%

277.39%

90.67%

96.99%

175.19%

Table V: Numerical comparison results

levels were set to Q1 (t0 ) = 5 , Q2 (t0 ) = 5 , I1 (t0 ) = 7 , and I 2 (t0 ) = 10 . The optimal facility WIP and IFI inventory costs for each SC case are shown in Table V. In each example, we compared the optimal costs with the corresponding SC simulation results by simulating SC performance with the optimal solution of production and QoS schedule. Based on the averaged values of 1000 replications, the simulation results are presented in relative percentages in Table V. The discrepancies between optimal and simulated total costs (TCs) are well below 1.5%. Although not perfectly matched, the simulated SC inventory levels are reasonably close to the analytical estimates reported with the optimal solution. In addition, the simulation estimates of average FGI stock out probabilities for each time period in all three cases range from 0.048 to 0.052, hence are quite close to the desired customer stock out probability set at 0.05. The corresponding standard errors for the simulated FGI stock out probabilities are measured at roughly around 0.002. More complex but also more realistic cases reported in [41] have demonstrated the need to explicitly account for transient behavior that is prevalent when targets change significantly from one planning period to the next. Transient behavior has been effectively modeled by using appropriate values of α in the WIP and IFI average equations. In addition to demonstrating the accuracy of the proposed methodology, in Table V we compare the optimal SC inventory costs with simulated industry practice production scheduling costs. We assume that industry practice employs a non-adaptive constant facility/machine utilization and IFI hedging point values which were selected to meet customer QoS requirements under the worst case scenario. In each SC

case, industry practice results corresponding to facility utilization levels of 0.9 and 0.95 and the corresponding worst case compliant hedging points are reported and compared. Whereas the facility utilization levels and hedging points used in the industry practice examples are not uniquely preferable and some may argue that the use of lower facility utilization levels might produce better result, we chose them because (i) they are reasonably close to their corresponding SC optimal solutions and (ii) under our stop-and-go SC operation protocol the corresponding effective FGI service capacity of lower facility utilization cases may result in unrealistically high FGI hedging points (or more unstable SC) and hence inventory higher costs. Apparently, by comparing the results of facility utilization of 0.9 and 0.95 in each case, although high facility utilization can help to greatly reduce the average FGI and stock out levels, it also results in higher than necessary facility WIP and overall SC inventory cost. The sub optimality is accentuated when production facilities are modeled to contain multiple machines. In each case, industry practice is 30% more costly when SC is operated under a 0.9 facility utilization level. It should be noted, however, that when higher than required QoS is observed, no benefit or cost reduction is imputed. V. CONCLUSION We introduced a supply chain production planning methodology that guarantees the desired QoS to final demand while co-optimizing SC facility and IFI location specific production schedules and hedging policies respectively. In addition to the provisioning of customer QoS, the methodology achieves effective IFI QoS and inventory

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FrAIn3.13 allocation across the supply chain. The contribution of the research rests equally on: (i) our utilization of efficient and accurate SC analytic horizontal coordination and performance evaluation sub-problems, and (ii) a tractable and effective iterative master-problem sub-problem optimization algorithm that we developed. Reported computational experience and Monte-Carlo-simulation verification, albeit on small SC examples, provide proof of the concept that the stochastic inventory and material flow dynamics can be modeled effectively and exceed the performance of industry practice methodologies. Larger SC examples reported in [41] indicate the potential of the methodology to scale to real-size SC planning problems.

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