Development of an Optimal Decision Policy for MTS-MTO System. Reno, Nevada, U.S.A. April 29 to May 2, 2011

Transcription

1 Development of an Optimal Decision Policy for MTS-MTO System Feng-Yu Wang 1,2, Rajesh Piplani 2, Yan-Guan Lim Roland 1, Eng-Wah Lee 1 1 Singapore Institute of Manufacturing Technology, Singapore 2 School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore (fywang@simtech.a-star.edu.sg) POMS 22 nd Annual Conference Reno, Nevada, U.S.A. April 29 to May 2, 2011 Abstract: MTS-MTO system is suitable where standard modules are shared by various finished products through divergent finalization. The system can satisfy product variety by taking advantage of flexibility of MTO, but its performance is lowered when the capacity is constrained. One way of dealing with the problem without adversely affecting system service level is to limit the acceptance of customer orders. A model is developed using Markov Decision Process where semi-finished module inventory position and WIP in MTO stage are reviewed periodically. The optimal decision policy with regards to order acceptance is then found using dynamic programming. Based on the policy, a relationship between inventory policy, WIP, and the order acceptance decisions is then shown. This study takes MTS-MTO system research to a new stage by incorporating operational decisions, and explores optimal inventory policy in MTS-MTO system. Key words: MTS-MTO system, Inventory Position, WIP, Decision Policy, Markov Decision Process

2 I. INTRODUCTION MTS-MTO supply chain is suitable where common modules are shared by various finished products through divergent finalization. The MTS-MTO supply chain inherits two important characteristics: it can lower the cost by taking advantage of economies of scale during Make-To- Stock (MTS) stage for the production of standard modules, while also satisfying the requirement of high product variety by taking advantage of flexibility of Make-To-Order (MTO) stage s flexibility. According to the literature, when there is sufficient capacity, the MTS-MTO system performs better than pure MTS and MTO system. Cost saving in terms of inventory holding and backlogging is one of the key criteria for comparison. Aviv and Federgruen (2001) reported that compared with MTS, the new system can achieve total cost reduction by 16% per month without seasonality, and up to 51% with strong seasonality. The study by Atali and Ozer (2008) showed a similar result with cost benefits from 12.38% to 22.69%, depending on production constraint and fluctuating demand patterns. Based on a numerical study, Kaminsky and Kaya (2008) concluded that the cost of the new combined strategy is significantly lower than the cost of either pure strategy. Service level is another key criterion for comparison between MTS and MTO systems. Based on a case study, Olhager (1990) explained that the new MTS-MTO system can reduce the manufacturing and delivery lead times, meanwhile improve the delivery dependability. Harrison and Skipworth (2006) suggested that the new combination system as an alternative to make-to-order reduces order lead-times and increases delivery reliability but introduces demand amplification. The MTS-MTO supply chain performs well because of the special two-stage logistics arrangement: at the MTS stage, standard modules are produced based on forecast and stocked as intermediate inventory; when customer orders arrive, these modules are assigned to orders and flow through the second (MTO) stage to become finished products as per the order s requirements. The boundary between the two stages is called the decoupling point, which decouples the

3 manufacturing of common modules from the customization of finished products. The decoupling point is also known as Order Penetration Point (OPP), Customer Order Decoupling Point (CODP), or Customer Order Point (COP); it is defined as the location where a product becomes earmarked for a particular customer, showing how far customer order can penetrate upstream into the supply chain (Sharman 1984, Hoekstra et al 1992, Olhanger 2003). The MTS-MTO supply chain does have its limitations also its performance decreases when the capacity becomes constrained. The decrease is particularly significant when the utilization in MTO stage is high. Gupta & Benjaafar (2004) reported that when the utilization at MTS stage is high, the cost advantage of new system over MTS diminishes, and when the utilization at MTO stage is sufficiently high, the new combination system becomes inferior to MTS. A similar result is also observed by Aviv et al (2001), Kaminsky et al (2008), and Atali et al (2008). The objective of this paper, thus, is to tackle the issue of constrained capacity by developing an optimal order acceptance and release policy, and further understand the interrelationship between inventory, WIP in MTO stage and the order policy. The paper is organized as follows: section 2 provides the literature review and the research gaps identified; section 3 describes the stochastic decision problem and the model based on Markov Decision Process; section 4 reports an experiment and the analysis of results; finally, section 5 concludes the study and identifies future research direction. II. LITERATURE REVIEW The research into MTS-MTO supply chains is relatively new and not well advanced yet. Most literature seems to address two fundamental questions (Gupta et al 2004, Jewkes et al 2009): the location of the decoupling point, and the optimal module inventory to be held at the decoupling point.

4 The first question is a strategic decision about supply chain structure as the location of decoupling point will form the basis for the entire logistic organization and for the planning and control of the goods flow (Hoekstra 1992). The second question, however, is related to operational decision. These two questions significantly affect a supply chain s performance in terms of cost and responsiveness. Currently, these two questions are addressed by three types of research: strategy-driven, competition-driven and optimization-driven decoupling point positioning. A. Strategy-driven decoupling point positioning The strategy-driven method is relies on the well known postponement strategy. The postponement principle was first defined by Bucklin in 1965 as a device for individual institutions to shift the risk of owning goods to others. Since then the postponement principle has been generalized as a strategy to delay the time of making decision to avoid the risk of uncertainty. The postponement strategy development is widely applied in distribution/logistics, manufacturing and, recently, supply chain contexts. With postponement strategy, a supply chain needs to identify a postponement point whereby some decisions or activities for differentiation are made beyond the postponement point only once that uncertainty is cleared. By exploring the literature, Yang and Burns (2003) found that postponement is closely linked with the decoupling point and the level of postponement can also be related to the decoupling point. They further indicated that postponement application may be a reasonable starting point for making a decision on the location of the decoupling point. According to the strategy-driven method, once the postponement point is identified, the decoupling point can be located accordingly. Pagh & Cooper (1998) directly mapped out decoupling points located at warehouse and production plants to four postponement strategies: full speculation, logistics postponement, manufacturing postponement, and full postponement strategies. van Hoek

5 (1997, 2001) mapped seven decoupling points to four postponement strategies, namely bundle manufacturing at production plant, uni-centric at central warehouse, and deferred assembly and deferred packaging at theatre warehouse. One of the advantages of the strategy-driven method is that the decoupling points are generic and applicable to all industries because of the generality of business strategies. The second advantage is that it is easy to understand because of the widespread adoption of postponement strategy in industries and availability of plenty of case studies. But, the decoupling point is coarsely mapped to a wide range of activities at production plants, which actually requires extra effort to identify the exact location. B. Competition-driven decoupling point positioning The second research stream positions decoupling point by shifting the current location of the decoupling point. Olhager (1992) highlighted that the shifting of decoupling point is mainly driven by the need of strengthening competitive priority. Basically, the shift is to relocate the decoupling point away from current location for achieving certain competitive advantages. The primary research methodology currently is based on case studies. van der Vlist & et al (1997) presented a case in truck manufacturing with three key sequential manufacturing stages: 1) parts manufacture, 2) component assembly at supplier, and 3) truck assembly at the manufacturer. It was observed that the decoupling point has been shifted from after truck assembly to the beginning of truck assembly and the costly components such as fuel pumps are now stocked at the beginning of truck assembly process. The authors argued (using intuition) that if the decoupling point could be shifted further upstream and into supplier s processes, the track assembler would be able to achieve virtually zero (fuel pump) inventory, provided the lead time was short enough. van der Vorst et al (2001) presented another case study in food (poultry) processing industry, and found (by experience) that it was feasible to shift the current decoupling point backwards between

6 the processes of packaging and labeling, to increase flexibility in dealing with unpredictable demand and short delivery time. These case studies provide a rationale to understand why the decoupling point should be shifted and how far it can be shifted. But the shift may not have a global view for supply chain performance improvement, and therefore may not be justifiable. In addition, this method is mainly supported by case studies; without mathematical modeling, the studies have not been able to provide generic insights about the shift. C. Optimization-based decoupling point positioning The third approach to decoupling point location and corresponding inventory level for semifinished modules uses global optimization, disregarding the location of the current decoupling point. The optimization procedure normally employs mathematical modeling and operation research (OR) techniques for pursuing the lowest total cost, subject to service constraints. Gupta and Benjaafar (2004) developed an optimization model for locating the decoupling point and semi-finished module inventory by taking into consideration the inventory holding cost and product and process re-design costs. The optimization is subject to an upper bound of the average lead time. Jewkes & Alfa (2009) addressed the same problems by considering four types of costs: order delay cost, semi-finished module inventory holding cost, disposition cost for wrong semifinished module stocked, and the storage capacity cost for semi-finished modules, subject to time constraint that order processing time is limited to 5%-20% of total order lead time. These mathematical models provide a means for exactly positioning decoupling point and semifinished module inventory level, and the intensive experimental data and analysis also gives insights about the linkage between the decoupling point location, costs and the service constraints mathematically. Thus, optimization-oriented research is most suitable for operational decision study in MTS-MTO system.

7 D. Research gap However, current optimization-based research into MTS-MTO system has its drawbacks. First of all, the combination of decoupling point location and semi-finished module inventory optimization gives the incorrect impression that both the decoupling point change and inventory level change must happen at the same time. In practice, inventory level is an operational decision; besides change of decoupling point location, it could be made based on other operational adjustments such as capacity change, service level change, etc. Secondly, because of the combination of two important decisions i.e. decoupling point location and semi-finished module inventory, optimization becomes more complex and can only be done at an aggregate level. Thirdly, the impact of WIP on system performance is ignored. As a result, current optimization-based research is not able to support operational decision-making in MTS-MTO system. III. PROBLEM STATEMENT AND SYSTEM MODELING This research considers a MTS-MTO supply chain where the decoupling point is predetermined. The MTO stage has finite capacity and is modeled as a single server. The server s processing time is an exponential variable (µ) for all types of finished products. Customer orders arrive according to Poisson distribution with arrival rate of λ i i=1,2,3,, as shown in Figure 1. The system service level is measured by the lead time (the time from receiving customer order to when the finished product is ready for delivery, denoted as LT) and a fraction indicating the percentage of customer orders that can be fulfilled within the lead time. A periodic-review inventory policy (s, S) is employed for the modules with a review period of T.

8 Figure 1: Overview of an MTS-MTO system Other assumptions for the MTS-MTO system include: One product family with one set of common modules caters for different end products. The module inventory is stocked at the decoupling point. A single per-specified service level is applicable to all customer orders. The lead time for module replenishment LT is fixed, and is longer than the review period T. The MTS stage has sufficient capacity to supply the module within the fixed lead time. The assumption eliminates the possibility that MTS stage becomes the system bottleneck. There will be only one replenishment order outstanding at any given moment. All accepted customer orders will immediately be converted into WIP and queue up in front of the server. A. Service constraints and order acceptance In MTS-MTO supply chains, order cannot be filled immediately and customers have to tolerate some delay before receiving the finished products. But, unlike traditional MTO system, the delay in MTS-MTO supply chain is not quoted by manufacturing company, instead, it is determined by the market (Andries et al 1995, Olhager 2003, Yang et al 2003). These service constraints should not be

9 compromised, and they will restrict the system behavior in the way that some controllable variables, e.g. module inventory and WIP in MTO, need to be managed to satisfy the service level. Since production capacity in MTO stage is finite, one way to meet the service level is to control WIP at a reasonable level, which depends on the number of customer orders that are accepted by the system. At the same time, if the system keeps sufficiently high module inventory, then by monitoring WIP variable alone, the decision of order acceptance can be made. This would, however, result in very high inventory holding cost. If the system keeps a low module inventory and allows certain inventory backlog and without jeopardizing system service level by taking advantage of queuing time in MTO stage, then the system can achieve a lower cost. This requires monitoring of both module inventory and WIP in MTO. B. Inventory related costs Inventory costs include inventory holding and backlog costs and are estimated based on the expected Inventory Level (IL). During an inventory replenishment period, we have: IL = IP D LT, (1) where IP is the Inventory Position at beginning of the inventory replenishment period, and D(LT ) is the demand during the period. Assuming it takes two review periods to replenish inventory of modules, then LT = 2T. At an arbitrary time point, denoted as time t, if IP is raised up to S (target inventory level), then the modules will arrive at time point (t+2t). Because the nearest next possible replenishment will arrive at (t+3t), the IL at (t+2t) is, therefore, critical in determining the holding cost and backlog cost incurred during the period (t+2t, t+3t) as shown in Figure 2. The interest of the study is to know

10 what the costs i.e. inventory holding cost and backlog cost are incurred between (t+2t, t+3t) just before the next replenishment arrives. Figure 2: Module inventory variation over time According to (1), the IL at time (t+2t) equals to: IL = IP D 2T = S D 2T. (2) The probability of variable IL at (t+2t) can be derived based on the lead time demand probability density function; IL will consistently reduce during the period (t, t+2t). There are two possible situations for IL at (t+2t): a) it is greater than reorder point s, or b) it is lower than or equal to reorder point s. In the first case, no replenishment would arrive at (t+t) and (t+2t), the IP and IL at (t+2t) would identical. In the second case, the IP had been raised to order-up-to level S at time points (t+t) and/or (t+2t), as result at (t+2t) one IL is mapped to multiple possible IPs and vice versa. Let the demand during (t+2t, t+3t) be d, if E IL t+2t d 0, holding cost will be incurred. The holding cost during (t+2t, t+3t) can calculated from c h n = u h E IL t+2t d, (3) where u h represents the holding cost per unit.

11 If E IL t+2t d < 0, then backlog cost will be incurred during the period (t+2t, t+3t). However, in MTS-MTO system the backlog cost is not a linear function of backlog orders a small amount of backlogged orders will not affect the delivery, because there is time buffer in MTO queue; only relatively high backlogged orders indicate a substantial shortage of the inventory, and will cause delivery delay. Thus, the total backlog cost can be calculated as c b n = f b E IL t+2t + d. (4) where the cost function f b (x) is exponential. C. Service related costs Two types of service related costs are considered in this study: order rejection cost and late delivery cost. Let x d n denote the order acceptance decision at an arbitrary review period. The order rejection cost can be calculated as c r n = u r d (i x n ) Pr i, λ, (5) i=x n d +1 Where u r is the rejection cost per unit, Pr(i, ) is Poisson probability distribution function with order arrival rate of. The late delivery cost is closely related to average WIP in MTO stage. The average time WIP spends in the system has a Gamma distribution with parameters of (the rate of the single server) and k (the average WIP). The total expected late delivery, denoted as c l n, equals the sum of the late delivery probability of the average WIP as follows:

12 c n l = u l k Pr t, k, μ t > LT LT = u l 1 Pr t, k, μ t LT = u l (1 f t, k, μ dt k k 0, (6) where f(t) is Gamma distribution, u l is the late cost per unit. D. Objective function In this study, at any review period, four types of costs are taken into account: modules holding cost c h n, modules backlog cost c b n, customer order rejection cost c r n, and customer order late delivery cost c l n. All of costs are directly influenced by the decision of order acceptance: accepted orders will consume inventory and change into WIP level in MTO stage; rejected orders will incur rejection cost. The objective of the optimization, thus, is to find the optimal orders acceptance policy that minimizes the total cost incurred over the planning horizon. E. System model This is a typical stochastic decision problem where the dynamic decisions need to be determined at every review period based on the system state at the moment. The problem can be modeled using Markov Decision Process (MDP), whereby the system transitions from one state to another with a probability and corresponding costs. The probability of transition is influenced by the decisions made. The MDP is a 4-tuple {U, X, P, C } where U represents system state space; X represents decision (action) space; P represents one-step transition probability matrices indicating the probability from one state transitioning to another; and C represents the one-step cost matrices incurred when the system transitions from a state to another. Both transition probability matrices and cost matrices are defined for decisions in the decision space individually.

13 The solution to a MDP can be expressed as a policy, a series of decisions for all members in the state space. We consider stationary policy only, meaning the decision is made based only on Markov chain state regardless of time; in other words the policy is a function mapping states to decisions. Another output of MDP is a cumulative cost array which indicates the cumulative costs when the system starts initially from a different state and evolves following the optimal policy. State space and decision space MDP model requires the Markov property (or memory-less property). When applying the (s, S) inventory policy, the variable IP (Inventory Position) demonstrates the memory-less property, i.e. IP n+1 at review period t n+1 depends on IP n at last review period t n and the number of accepted customer orders x d only, irrespective of previous status and decisions, whereas, IL stochastic variable does not have the memory-less property. So, IP rather than IL, is modeled in MDP, and the values of IP fall in between the lowest reorder point s and the highest order-up-to level S. The values of the other stochastic variable WIP are related to the cap for WIP, denoted as WIP MAX. If average WIP is less than or equal to WIP MAX, the system is guaranteed to fulfill the service level (i.e. LT and ). However, WIP level may go above the cap, which is allowed. Hence, a new variable, denoted as WIPR, representing the room for the stochastic variable WIP to reach WIP MAX, is modeled. By combining the two variables IP and WIPR, the state space is formed based on stochastic variable U: U = u i = IP, WIPR i, i = 1,2,, M. (7) It is evident that the stochastic variable U has the Markov property, because at time period t n, the future values of the IP n+1 and WIPR n+1 only depend on the corresponding values at t n, not the past ones. So the stochastic process {u i } is a discrete Markov chain.

14 The decisions are made at individual review periods for determining the cap of number of customer orders that can be accepted. Obviously, the lowest available value for the decision is zero, meaning no customer order can be accepted. One-step transition and cost matrices A one-step transition matrix and a cost matrix are required for each decision in the decision space. At any time t n, the system could be in any state u i U. x d X, after taking the decision x d, the system state would transition from state u i to u j U according to a one-step probability in the transition matrix. The sum of the transition probability for the state u i equals to one. Correspondingly, during the transition, the total cost incurred in the form of sum of inventory holding cost, backlogging cost, order rejection cost, and late delivery cost is specified in the cost matrix. IV. EXPERIMENT AND RESULT ANALYSIS A. The experiment The key parts of the experiment are the four elements of the MDP model, i.e. state space, action space, one-step transition matrices, and one-step cost matrices. The periodic reviewed inventory policy (s, S) is set as reorder point s = 14 and order-up-to level S = 28. So, the stochastic variable IP belongs to the set {15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28}. In MTO stage, since the service rate = 1/6, WIP MAX is derived according to the service level, i.e. LT= 2 and β = 90% as shown in Table 1. When WIP is smaller than or equal to eight, the system can fulfill the service level, i.e. more than 91% orders can be delivered within the lead time. When WIP increases to 10, β declines to 75.76%. In order to make the order acceptance decision more

15 flexible, the WIP MAX is set to 10, correspondingly, the random variable WIPR has a set of countable values: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Table 1: WIP level and corresponding probability of on-time delivery WIP β Based on the random variables IP and WIPR, state space U={(IP, WIPR)} is formed with total of 154 members as shown in Table 2. Table 2: MDP state space WIP > WIPR IP (15,0) (15,1) (15,2) (15,3) (15,4) (15,5) (15,6) (15,7) (15,8) (15,9) (15,10) 16 (16,0) (16,1) (16,2) (16,3) (16,4) (16,5) (16,6) (16,7) (16,8) (16,9) (16,10) 17 (17,0) (17,1) (17,2) (17,3) (17,4) (17,5) (17,6) (17,7) (17,8) (17,9) (17,10) 18 (18,0) (18,1) (18,2) (18,3) (18,4) (18,5) (18,6) (18,7) (18,8) (18,9) (18,10) 19 (19,0) (195,1) (19,2) (19,3) (19,4) (19,5) (19,6) (19,7) (19,8) (19,9) (19,10) 20 (20,0) (20,1) (20,2) (20,3) (20,4) (20,5) (20,6) (20,7) (20,8) (20,9) (20,10) 21 (21,0) (21,1) (21,2) (21,3) (21,4) (21,5) (21,6) (21,7) (21,8) (21,9) (21,10) 22 (22,0) (22,1) (22,2) (22,3) (22,4) (22,5) (22,6) (22,7) (22,8) (22,9) (22,10) 23 (23,0) (23,1) (23,2) (23,3) (23,4) (23,5) (23,6) (23,7) (23,8) (23,9) (23,10) 24 (24,0) (24,1) (24,2) (24,3) (24,4) (24,5) (24,6) (24,7) (24,8) (24,9) (24,10) 25 (25,0) (25,1) (25,2) (25,3) (25,4) (25,5) (25,6) (25,7) (25,8) (25,9) (25,10) 26 (26,0) (26,1) (26,2) (26,3) (26,4) (26,5) (26,6) (26,7) (26,8) (26,9) (26,10) 27 (27,0) (27,1) (27,2) (27,3) (27,4) (27,5) (27,6) (127,7) (27,8) (27,9) (27,10) 28 (28,0) (28,1) (28,2) (28,3) (28,4) (28,5) (28,6) (28,7) (28,8) (28,9) (28,10) In the experiment, the cap of customer orders that can be accepted is set to 10, so, the decision space is, X = {0,1,2,3,4,5,6,7,8,9,10}. Assuming customer order arrival rate = 8 and service rate = 1/6, the demands basically overwhelms the system capacity. The cumulative probability function for order acceptance is F (cap=10; =8) = Pr (n<=10 =8) = , meaning, on average, about 81.59% of the demand can be accepted when the cap is10. The cumulative probability function for order completion is F (cap=10; 1/ =6) = Pr (n<=10 1/ =6) = , meaning to complete the maximum

16 of 10 new incoming orders will take 95.74% of the total capacity, without considering the existing WIP, if any. One-step transition matrices A transition matrix is defined for one decision, specifying the probabilities for a state transitioning to any of the 154 states in the state space. But not all transitions are possible. For example, for states whose IP = 24, i.e. the states of (24,0), (24,1), (24, 2), (24,3), (24,4), (24,5), (24,6), (24,7), (24, 8), (24,9), (24,10), if the decision of order acceptance cap x d = 2, then they can only transition to those states whose IP = 22, 23 and 24. The probability for transitioning to other remaining states in the state space is infinitely high. Table 3 and Table 4 show the transition probability for states whose IP = 24 to those states whose IP = 22, 23. Table 3: Transition probability matrix for states IP = 24 to 22 u i \u j (22,0) (22,1) (22,2) (22,3) (22,4) (22,5) (22,6) (22,7) (22,8) (22,9) (22,10) (24,0) (24,1) (24,2) (24,3) Inf (24,4) Inf Inf (24,5) Inf Inf Inf (24,6) Inf Inf Inf Inf (24,7) Inf Inf Inf Inf Inf (24,8) Inf Inf Inf Inf Inf Inf (24,9) Inf Inf Inf Inf Inf Inf Inf (24,10) Inf Inf Inf Inf Inf Inf Inf Inf One-step cost matrices Inventory related cost calculation is based on estimated IL at the end of each review period. One IP can map to multiple possible ILs. For instance when IP = 24, there are multiple ILs, each having their own probability as shown in Table 5.

17 Table 4: Transition matrix for states from IP = 24 to 23 u i \u j (23,0) (23,1) (23,2) (23,3) (23,4) (23,5) (23,6) (23,7) (23,8) (23,9) (23,10) (24,0) (24,1) (24,2) Inf (24,3) Inf Inf (24,4) Inf Inf Inf (24,5) Inf Inf Inf Inf (24,6) Inf Inf Inf Inf Inf (24,7) Inf Inf Inf Inf Inf Inf (24,8) Inf Inf Inf Inf Inf Inf Inf (24,9) Inf Inf Inf Inf Inf Inf Inf Inf (24,10) Inf Inf Inf Inf Inf Inf Inf Inf Inf IP IL Table 5: Inventory holding and backlog costs (IP = 24) Prob of IL Order received Inv holding Holding cost Backlog cost Proportional holding cost Proportional backlog cost total: In Table 5, Inv holding = IL orders received. If Inv holding >= 0, the Holding cost = Inv holding * 2; if Inv holding < 0 then Backlog cost is calculated based on exponential function Y = x The proportional holding cost and proportional backlog cost are the costs after

18 factoring in the probability of IL. The summaries of the proportional holding cost and proportional backlog cost are the final estimated inventory costs. The order rejection cost is calculated based on the assumptions of order arrival rate λ = 8 and rejection cost per unit u r = 2. The rejection cost decisions in the decision space are listed in Table 6. Late delivery cost is calculated based on WIPR and orders received. An example of the late cost is shown in Finally, the four costs are summed up for each state in the state space and form the one-step cost matrix for decisions in decision space. The cost matrices and transition matrices are the inputs to the dynamic programming model for solving the MDP model. Table 7 for the case where order received = 2. Some of the values are infinite as the scenarios are impossible, for example, it is impossible to transition states from WIPR=4 to WIPR= 1 when only two customer orders are accepted. Table 6: Order rejection costs Decision cap rejected # of orders rejection cost

19 Finally, the four costs are summed up for each state in the state space and form the one-step cost matrix for decisions in decision space. The cost matrices and transition matrices are the inputs to the dynamic programming model for solving the MDP model. Table 7: Late delivery cost calculation for order accepted = 2 WIPR n \WIPR n Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf B. Results analysis The optimization of the experiment is done using dynamic programming algorithm, more precisely, the discounted value-iteration algorithm which actually derives the optimal decision policy through backward induction. The algorithm stops after a total of 47 iterations,the optimal decision policy is found and corresponding cumulative costs are calculated for each state following the optimal policy. For the sake of analysis, the optimal policy and costs are reformed and each is put into two 14 x 11 matrices, where the rows represent IPs and the columns represent WIRPs as shown in Table 8 and Table 10.

20 Table 8: Optimal policy WIPR IP Table 8 clearly shows that about 86% of decisions above the dotted line do not exceed WIPR. However, as IP reaches higher level, starting from IP = 24, and WIPR <= 6, system starts to take more orders according to the policy and the cap quantity exceeds WIPR. This happens to 14% of total states that are below the dotted line, meaning that for these cases the average WIP could exceed WIP MAX and cause late delivery. From IP perspective, when IP is small, WIPR has less influence on decisions. For example, when IP = 15, the decisions of acceptance cap remain as 1 across all levels of WIPR. However, when quantity of IP is high, IP >= 20, the decisions on order acceptance cap get close to and even exceed WIPR. In summary, when the value of IP is low, IP confines the decisions of order acceptance to avoid backlog cost which is exponential. Under such situation, WIPR has a little role to play. When IP is high, to avoid high inventory holding cost, the influence of WIPR on the decision increases. So, the combination of IP and WIPR, rather than individual values, affects the decision of order acceptance.

21 Table 9: The maximum average WIP WIPR IP The maximum average WIP for each state in the state space is further considered in the analysis. The maximum average WIP for each state can be calculated based as, Max Ave WIP i = 1 2 WIPMAX WIPR i + Decision i, (8) where i = 1, 2, 154; Decision i represents the decision for state u i U. About 84% of maximum average WIPs in Table 9 are from 3.5 and 5.5 and they fall between the two dotted lines. While, the highest average WIP is 6 for the states which have high IPs and high WIP (low WIPR) and the lower average WIP from 0.5 to 3 comes from the states that have lower IP and lower WIP (high WIPR). The average WIP does not exceed the WIP MAX, implying that under all circumstances the optimal policy can guarantee the service level, i.e. LT = 2 and β = 90%.

22 Table 10: The cumulative costs as per the optimal policy WIPR IP Table 11: Statistics of cumulative costs statistics value Max Min Mean Standard Deviation 6.38 Table 10 shows the cumulative costs for all the states, indicating the cumulative cost when the system initially starts from any one of the states. The statistics of the 154 cumulative costs, as shown in Table 11, suggest that whatever state the system might start from, the difference in cumulative cost is very small following the optimal policy. V. CONCLUSION AND FUTURE RESEARCH It is well recognized in literature that the MTS-MTO supply chain in general outperforms pure

23 MTS and MTO systems, but the value of new system declines as the production capacity becomes constrained, which remains as an outstanding issue for MTS-MTO system. In this study, an experiment was conducted using Markov Decision Process where module inventory position and WIP in MTO are reviewed periodically. The quantified optimal decision policy with regards to order acceptance was found using dynamic programming. Following the policy, an interrelationship between inventory policy, WIP, and the order acceptance decisions is shown. The preliminary analysis answers some of the research questions raised in the study. First of all, the outstanding issue in MTS-MTO system can be solved at operational level by properly controlling both IP and WIP in MTO stage by limiting order acceptance. Secondly, both IP and WIP (WIPR) play roles in influencing the decision making in terms of customer order acceptance. When IP is low, it confines the decisions and WIPR has a little role to play. When IP is high, the influence of WIPR on the decision increases obviously. Thirdly, the optimal decision policy can guarantee the stability of the system and meet the service level under any circumstance. Finally, if the system follows the optimal policy, it can achieve the optimal cumulative cost consistently, regardless of which state it may start from. This study extends MTS-MTO system research to a new stage by taking into account operational decisions. In future research, the assumption of fixed lead time for replenishing the semi-finished module will be relaxed; and a wider range of bands for the reorder-point s and order-up-to level S will be evaluated in order to validate the interrelationship between the inventory policies and order acceptance decision policies identified in the study. The optimization procedure basically tests the baseline for the lowest system cost and reveals the interrelationship between decisions and system variables. In this study, a period-reviewed (s, S) policy is followed; in future, other types of inventory policies, such as (r, Q), will be considered for evaluation and comparison. The MTS-MTO system performance comparison will also be done between periodic-review and continuous-review decision policies.

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