optimizing joint maintenance and stock provisioning policy for a multi ...

Journal of the Chinese Institute of Industrial Engineers, Vol. 23, No. 4, pp. 289-302 (2006)

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OPTIMIZING JOINT MAINTENANCE AND STOCK PROVISIONING POLICY FOR A MULTI-ECHELON SPARE PART LOGISTICS NETWORK Mu-Chen Chen* Department of Business Management National Taipei University of Technology No. 1, Section 3, Chung-Hsiao E. Road, Taipei 106, Taiwan, ROC Chih-Ming Hsu Department of Business Administration Minghsin University of Science and Technology Shih-Wei Chen Data Processing Division, Taipei National Tax Administration Ministry of Finance

ABSTRACT Spare part stock management attempts to ensure that the failed equipment items can be replaced immediately to maintain a sufficient productivity level. In maintenance, the inventory policy determination for spare parts is an important issue. The age-based preventive replacement policy may seek the least total cost for spare part replacement. Considering the criticality of equipment where it is installed, demand for a certain spare part can be categorized into critical and non-critical. The stock level for critical demand can be set to reserve certain spare parts in stock for critical equipment users. This stock policy is named the inventory rationing policy. The stock policy has a significant impact on production system performance, particularly when a machine breakdown causes a large amount of production lost. This study optimizes the joint age-based preventive replacement and inventory rationing policy in a multi-echelon spare part logistics network. In this study, there are equipment users, spare part distribution center and spare part suppliers in the multi-echelon system. Inventory policy optimization for the spare part logistics network is extremely complicated. This paper adopts the scatter search based simulation-optimization method to obtain the optimal configurations with respect to maximizing the total profit of the spare parts logistics network. An implementation example of a spare part logistics system is used to demonstrate the advantages of employing the joint policy. Keywords: spare parts, preventive replacement policy, inventory rationing policy, simulation-optimization, scatter search.

1. INTRODUCTION To reduce the impact of machine breakdowns on production, equipment users have adequate machine maintenance policies for spare part stock management. The major objective of the spare part stock management is to ensure that the failed equipment items can be replaced immediately to maintain sufficient productivity. The cost of equipment maintenance is a major portion of the production costs in manufacturing systems. Preventive maintenance is adopted to reduce the maintenance and machine breakdown costs [3]. Additionally, preventive replacement can be performed during periods when the *

Corresponding author: [email protected]

equipment is not in operation to reduce equipment downtime [23]. Demand for certain spare parts can be categorized into critical (higher priority demand class) and non-critical (lower priority demand class) [6]. Different demand classes may not require the same spare part service levels. The stock level for critical demand can be set to reserve certain spare part stock for critical equipment users. The inventory rationing policy is used when the demand is categorized into different priority classes [21]. Dekker et al. [7] developed an inventory model in which spare parts are classified into critical and non-critical demand. The remaining stock is only supplied for critical demand when the inventory drops to the critical level SC . Dekker et al. [6] analyzed a spare part inventory model with several de

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Journal of the Chinese Institute of Industrial Engineers, Vol. 23, No. 4 (2006)

mand classes based on criticality. Two mathematical models were constructed. The first problem considered in determining the critical-level, lot-for-lot policy that can minimize the holding cost subject to the service level constraints. The other problem involves determining a policy that minimizes the inventory cost without considering the service levels. The total stockout cost depends on the total amount of unfulfilled items. Melchiors et al. [21] also considered the two classes of demands for the same spare part. However, Melchiors et al. applied an ( s , Q ) inventory model. For equipment users, the stock policy objective under an age-based preventive replacement policy seeks the least replacement cost [2], inventory cost [8], or minimizes the total replacement cost [16]. In the studies by Kabir and Al-Olayan [16] and Kabir and Farrash [17], a spare part stock policy involving the same spare parts installed in several identical pieces of equipment. They combined the age-replacement policy with the ( s , S ) inventory policy and introduced a decision variable that represented the preventive replacement age. If the spare part is available, the item is replaced immediately if either the part fails or the preventive replacement age has arrived. Sarker and Haque [23] adopted the same inventory policy used in [16, 17] to discuss the class B inventory items. However, the same items on several identical pieces of equipment were replaced in the entire equipment group after certain time intervals. This is called block replacement. Yoo et al. [25] also discussed the block replacement policy problem. They assumed that the initial spare part stock level is S and the failed item is replaced with a new one from the spare parts stock. A mathematical model was formulated and resolved using a recursive relational algorithm to obtain the optimal replacement time and initial spare part stock level that minimized the total inventory cost. The more details of inventory management issues of spare parts were discussed in the survey by Kennedy et al. [19]. Heuristic solution methods were developed to resolve the complicated spare part inventory models due to their stochastic nature. Dekker et al. [7] used enumeration to resolve the constructed mathematical model with the objective of minimizing the total inventory cost by determining two decision variables, the critical level and the maximum inventory level. A heuristic approach was utilized by Dekker et al. [6] to determine the optimal decision variables in the inventory policy. By resolving the mathematical inventory model for spare parts, Melchiors et al. [21] used enumeration and bounding optimization procedures with the objective of minimizing the average inventory cost. In the previous studies, simulation technique

was frequently used to model the complex spare part inventory policy. Kabir and Farrash [17] built a simulation model based on the SLAM II network and described a simulation procedure to select an inventory policy. Kabir and Al-Olayan [16] combined the age replacement with a continuous review ( s , S ) policy, and developed a simulation model to find the optimal decision variables in the joint policy by minimizing the costs. It is essential to carry a spare part inventory to ensure that failed equipment items can be immediately replaced. However, equipment users do not want to hold excessive amounts of stock because inventory is expensive and can become obsolete as equipment models change [14]. Distribution centers were adopted to aggregate small amounts of required stock for an array of equipment users. Hopp et al. [14] discussed a two-echelon spare part stock and distribution system that consists of a central distribution center and several regional facilities. Hopp et al. derived a mathematical model to minimize the total inventory investment in the system subject to a constraint on order frequency at the distribution center, ensuring that the average total delay observed by customers at the facilities was below a certain level. A heuristic was then developed to determine the optimal order quantity setting, the reorder point at the distribution center, and the base stock level at the facility. Previous studies regarding the spare part inventory management typically focused on local inventory of a single facility and little on the entire supply chain [15]. In most previous literature, suppliers directly provide spare parts to equipment users. Under such a system, equipment users must spend much time acquiring spare parts. The replenishment time is even longer if suppliers do not have enough stock. Hence, a distribution center between suppliers and users to form a multi-echelon spare part logistics network is desirable. Therefore, a distribution center can distribute spare parts for multiple suppliers to various equipment users, thus shortening the replenishment lead time, increasing the inventory turnover rate and decreasing the stockout level with a much lower inventory level. However, the cost or profit occurs only at the distribution center or equipment user and not along the entire spare part logistics network. The lost sale cost at the distribution center depends on customers. The higher priority demand class can obtain a higher service level. Suppliers are responsible for spare part production to meet the demand at the distribution center. This study proposes a multi-echelon spare part model that involves suppliers, distribution centers and equipment users. Both the age-based preventive replacement policy and priority demand classes are considered for equipment users. The inventory policy of this multi-echelon

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Chen: Optimizing Joint Maintenance and Stock Provisioning Policy system is very complicated. There are many decision variables included in the joint policy. It is difficult to build the mathematical model for this type of system. Therefore, we adopted the simulation-optimization method to produce the approximate results for maximizing the total profit of the spare part logistics network. This multi-echelon system inventory policy is very complicated. There are many decision variables included in the joint policy. A simulation-optimization methodology based on a meta-heuristic, Scatter Search [11, 12], is applied to determine the near optimal decision variable values with the objective of maximizing the total logistics network profit. Spare parts management along the entire spare part logistics network, maintenance and inventory policies are considered simultaneously. The impacts of different joint maintenance and stock provisioning policies for the spare part logistics network are also analyzed.

2. THE MULTI-ECHELON SPARE PART SYSTEM 2.1 Architecture of spare part logistics network This study explores the influence on profit along the spare part logistics network, consisting of equipment users, a distribution center and suppliers as shown in Figure 1. In this multi-echelon system, equipment users adopt a joint policy combining the age replacement and inventory rationing policies based on the importance of the equipment combined with a continuous review ( s , S ) policy. The spare part distribution center adopts the inventory rationing policy based on the equipment importance combined with the continuous review ( s , S ) policy. The spare part suppliers adopt the continuous review ( s , S ) policy. The total profit in a spare part logistics network can be obtained by considering the total revenue and cost arisen at the equipment users, spare part distribution center and suppliers. The total profit at equipment users, Pu , can be calculated by

Pu = Vu − Cu

(1)

where Vu is the total revenue of equipment at the equipment users ($), and Cu is the total cost at the equipment users ($). It includes stockout cost, holding cost, ordering cost, purchasing cost and replacement cost. For the spare part distribution center, the total profit at spare part distribution center can be found as

Pd = Vd − C d

(2)

Figure 1. The schematic architecture of spare part logistics network. where Vd is the total revenue at the distribution center ($), and C d is the total cost at the distribution center ($). It is the sum of tardiness cost, holding cost, ordering cost, and purchasing cost. For the spare part suppliers, the total profit at spare part suppliers is similarly obtained as

Ps = Vs − C s

(3)

where Vs is the total revenue at suppliers ($), and

C s is the total cost at suppliers ($). It is the sum of tardiness cost, holding cost and production cost. Finally, the total profit in the multi-echelon spare part system can then be calculated as

Psc = ∑ Pc + ∑ Pd + ∑ Ps

(4)

2.2 Spare part stocking management 2.2.1 Stocking management at equipment users An item is installed on each equipment, and the total number of demands varies according to the total number of equipment used by users. The ( s , S ) spare part stocking policy, that might be integrated with the inventory rationing and age-based preventive replacement policies, is applied to each user. Assuming that a certain type of item is installed on several machines, the equipment item with the higher priority is replaced when either the preventive replacement age has arrived or the part has failed. For equipment with lower priority the item can be replaced only when the number of this item exceeds the critical stock level even though the preventive replacement age has arrived or the part has failed. This is because spare parts below the critical stock level must be reserved for equipment with higher priority. The users must order spare parts from the

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distribution center and the lead time for an order will be prolonged by a certain period if the delivery is delayed. The costs relating to spare parts; holding cost, ordering cost, preventive replacement cost and failure replacement cost are charged to the equipment users. The failure replacement cost is much higher than the preventive replacement cost. The stockout cost relating to the failed item takes place when the spare part is out of stock. The stockout for equipment with a higher priority is more serious than that for parts with lower priority.

2.2.2 Stocking management at distribution center The distribution center supplies spare parts to important equipment users, i.e., users that demand a great quantity of spare parts, with higher priority. The ( s , S ) inventory model is utilized at the suppliers and distribution center. The distribution center and equipment users with different priorities have their own ordering costs and quantity. An order must be placed with the suppliers if the spare parts are out of stock and there is no undelivered order at the distribution center. Moreover, the distribution center sets up ( m − 1 ) the critical stock levels depending on the criticality of users, where m is the total number of users. Stock cannot be delivered if the stock level is lower than the order quantity placed by the equipment user with the highest priority. An order that cannot be met immediately is charged to the distribution center with the stockout cost.

2.2.3 Stocking management at suppliers The distribution center can order spare parts from n suppliers. Hence, suppliers receive orders from the distribution center with certain probabilities. The stockout cost takes place at a certain spare part supplier if an order from a distribution center cannot be fulfilled immediately with its current stock. At this time, suppliers deliver spare parts within a period that includes the regular lead time and tardiness due to stockout. Suppliers can also make spare parts. The production cost is charged when the stock level drops to a pre-specified level.

3. THE SIMULATIONOPTIMIZATION APPROACH Regardless of the stock policies, constraints and objectives in addressing the spare part stocking management, the solution techniques can be generally divided into two classes: mathematical optimization and simulation. The mathematical optimization approach deals mainly with static models, e.g. annual or

average demand. The system simulation approach can resolve the dynamic models and provide preferred settings of decision variables through manipulating the input values of these decision variables. Kabir and Al-Olayan [16] and Kabir and Farrash [17] proposed enumeration methods as the optimization procedure to determine the optimal decision variable settings in inventory policy using simulation technique. These approaches are very time-consuming, and may locate a solution far from the optimum. This study intends to construct an adequate multi-echelon model for the spare part stocking management and to optimize the joint age-based preventive replacement and inventory rationing policy in a multi-echelon spare part logistics network which includes suppliers, distribution center and equipment users. Therefore, it is difficult to construct the mathematical optimization model due to the uncertainties, nonlinear relationships, and spare part flow in the problem. Through simulation, the output response values under a certain combination of decision variables can be obtained. However, the simulation approach cannot efficiently find the (near) optimal settings for these decision variables without auxiliary optimization techniques, e.g. genetic algorithms, tabu search, simulated annealing and scatter search [1, 5, 13, 24, 26]. Simulation has been reported to be a powerful tool for evaluating complex spare part systems, which can not be modeled mathematically. Simulation is usually applied for performance evaluation of alternatives, i.e., it only responses to “what if” questions. Recently, the research of simulation optimization extends the capability of simulation for answering “how to” questions. Answering “how to” questions implies to find the optimal solution for factors with respect to the objectives concerned by the decision makers. In this study, the simulation-optimization procedure includes the following steps: constructing the spare part logistics network by using simulation, simulating the multi-echelon system to calculate the profit, optimizing the simulation model using a search method and analyzing the simulation results. Scatter search has recently been adopted as a search method to optimize simulation models [1]. Scatter search proposed by Glover and Laguna [10] is an efficient optimization method that can escape from the local optimum. As genetic algorithms, the scatter search is designed to operate on a set of solutions, namely a reference set. The new solutions are generated by applying the solution combination procedures. The scatter search solution procedure differs from other heuristics by taking advantage of auxiliary heuristic solution methods, and using strategy rather than randomization, to perform process steps [11, 12]. The mechanisms associated with scatter search are not restricted to a single standardized design.

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Chen: Optimizing Joint Maintenance and Stock Provisioning Policy Scatter search implementation generally includes five methods [9]: Diversification Generation Method (DGM), Improvement Method (IM), Reference Set Update Method (RSUP), Subset Generation Method (SGM) and Solution Combination Method (SCM). The five scatter search methods presented herein are briefly described below. Further details of the implementation of these methods are given in Glover [9]. 1. Diversification Generation Method: A collection of diverse trial solutions are generated using an arbitrary trial solution (or seed solution) as an input. 2. Improvement Method: A trial solution is transformed into one or more improved trial solutions. 3. Reference Set Update Method: This method is used to build and maintain a reference set of the b “best” solutions. The reference solutions are used to generate new solutions. The reference set RefSet consists of both high-quality solutions ( RefSet1 with size b1 ) and diverse solutions ( RefSet 2 with size b2 ). 4. Subset Generation Method: This method operates on the reference set to produce a subset of solutions as a basis for creating solutions by applying SCM. This method generates the following types of subsets - two-element subsets, three-element subsets and four-element subsets. 5. Solution Combination Method: This method transforms a given subset of solutions produced by the SGM into one or more combined solution vectors. For presenting the scatter search procedure, the following notation is defined: the set of solutions generated by DGM; P the set of solutions in the reference set; RefSet

RefSet1 RefSet 2 PSize b b1 b2 MaxIter

the subset of the reference set that contains the best solutions as measured by the objective function value; the subset of the reference set that contains the diverse solutions as measured by the minimum distance criterion; the size of the set of solutions generated by the DGM; the size of the reference set; the size of the high quality subset; the size of the diverse subset;

maximum number of iterations. With the abovementioned mechanisms, the SS-based approach for the economics problems is described as follows [9, 11, 20]: Step 1. Diverse solutions generation for the formulated model (a) Start with P = ∅ . Use DGM to

build a solution and apply IM. Let X be the resulting solution. (b) If X ⊄ P then P = P ∪ X , otherwise discard X . (c) Repeat this step until

P = PSize . Step 2. Reference set update (a) Use RSUM to construct

RefSet = { X 1 , X 2 ,⋅ ⋅ ⋅, X b } with the “best” solution in P . (b) Order the solutions in RefSet with respect to their objective function value F ( X ) such that

X 1 is the best solution and X b the worst. (c) Set New_Solutions = TRUE. Step 3. Solution subsets generation (a) If New_Solutions = TRUE, then proceed to (b). Otherwise, proceed to Step 6. (b) Generate new solution subsets by SGM. Set New_Solutions = FALSE. Step 4. Solution combination (a) If New_Subsets ≠ ∅ , then proceed to (b). Otherwise, proceed to Step 6. (b) Select the next subset denoted by y in New_Subsets . Apply SCM to y to generate one or more new trial solutions. (c) Apply IM to the trial solutions. Step 5. Reference set update and check (a) Apply RSUM. (b) If RefSet has changed, then set

New_Solutions = TRUE. (c) Delete y from New_Subsets . Go back to Step 3. Step 6. Terminate and return the final solution. The scatter search procedure continues in a loop that applies the Solution Combination Method followed by the Improvement Method and the Reference Update Method. This procedure terminates when the reference set does not change and all of the subsets have already been subjected to the Solution Combination Method. At this point, the Diversification Generation Method is used to construct a new RefSet 2 and the search is restarted. The foregoing procedure is executed MaxIter times. A more detailed description of scatter search can be found in

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references [11, 12, 20]. Recently, the applications of scatter search in the area of meta-heuristics have become more plentiful (e.g., [4, 22]).

4. THE IMPLEMENTATION 4.1 The example background A multi-echelon system for spare parts that comprises two equipment users, one distribution center and two suppliers is adopted in this study. In this system, one type of spare part is considered and the two users install the same type of equipment, but not the same quantity. Moreover, only one item is installed on each piece of equipment and equipment is classified into two classes: higher and lower priority. To the spare part distribution center, the equipment has higher priority if the quantity of equipment used in a certain facility is relativity large. The continuous review ( s , S ) model, an inventory rationing policy, and an age-based preventive replacement policy are used in equipment users. In distribution centers, a continuous review ( s , S ) inventory model and an inventory rationing policy are utilized. As to suppliers, the continuous review ( s , S ) inventory model is applied. To demonstrate the advantages of employing the above joint policy (Scenario 1), this study simulates three alternative scenarios for comparison purposes. They are as follows. Scenario 2: Equipment users only utilize the age-based preventive replacement policy combined with the continuous review ( s , S ) policy. Scenario 3: Equipment users utilize only the inventory rationing policy combined with the continuous review ( s , S ) policy. Scenario 4: Equipment users utilize neither the age-based preventive replacement policy nor the inventory rationing policy. They apply only the continuous review ( s , S ) policy. In Scenarios 2-4, distribution center and suppliers utilize the same policy as in Scenario 1. The parameter settings and initial values of decision variables in the spare part logistics network are listed in Tables 1 and 2. The equipment quantities with higher priority and lower priority used by the users with higher priority are assumed to be 40 and 160, respectively. The users with lower priority use 10 equipment with higher priority and 40 equipment with lower priority. The simulation model for the spare part logistics network is built using Arena [18]. Each scenario is simulated with 2190 days (6 years) for 20 replications. The initial stock levels at the equipment users

and spare part distribution centers are set to 1 and 0, respectively. The simulation results for the initial 365 days (the first year) are not collected for analysis such that stocking levels can reach the regular levels in a steady condition. Hence, the daily total profit is the average value calculated according to the simulation results gathered from the last 1825 (5 years) in the 20 replications.

4.2 Computational results The optimization tool OptQuest for Arena [1] is applied next to optimize the simulated spare part logistics network. All of the decision variables are restricted to integers with a step size of one. Table 3 summarizes the allowable ranges for these decision variables. The following constraints are set:

Cb − C r ≥ 1

(5)

Cb − C c ≥ 1

(6)

Db − Dr ≥ 1

(7)

Db − Dc ≥ 1

(8)

Sb − Sr ≥ 1

(9)

where

C b : maximum stocking level at the equipment user; C r : reorder point at the equipment user;

C c : critical stock level at the equipment user; Db : maximum stock level at the spare part distribution center; Dr : reorder point at the spare part distribution center; Dc : critical stock level at the spare part distribution center; S b : maximum stock level at the spare part supplier; S r : re-production point at the spare part supplier. Simulation models are conducted with the objective of maximizing the average daily total profit in the spare part logistics network. The total number of simulations (total number of decision variable combinations) are set to 950 (excluding the initial solution). Table 4 summarizes the optimization results.

Chen: Optimizing Joint Maintenance and Stock Provisioning Policy According to Table 4, the average daily total profit in the spare part logistics network in Scenario 1, where the equipment users utilize both the age-based preventive replacement and inventory rationing policies, achieves 1847.51. The average profit is better than that for Scenarios 2, 3 and 4, which are 1796.60, 1581.89, 1545.47, respectively. The profits, revenues and costs for equipment users, spare part distribution center and spare part suppliers are illustrated in Table 5. The detailed cost data and related information for each member in the spare part logistics network are listed in Tables 6-10.

4.3 Discussions of simulation results According to Table 5, the equipment user with higher priority has lower total revenue in Scenario 1. Observing Table 4, the preventive replacement age is shorter, i.e. the demand for spare parts is larger, and the reorder point is closer to the critical stock level, thus bringing a higher stockout probability for equipment with higher priority. Meanwhile, this makes the average daily spare part cost at the equipment user with higher priority in Scenario 1 is larger than Scenario 2. In this scenario, however, the failure replacement cost is lower due to the shorter preventive replacement age, as shown in Table 6. For the equipment user with lower priority, the total revenue is lower based on Table 5, which implies that the corresponding stockout probability for spare parts is relatively high. From Table 4, the preventive replacement age is shorter, which means that the demand for spare parts is larger. Moreover, the reorder point is equal to the critical stock level. Hence, the total revenue tends to be lower because of the higher stockout probability for spare parts. For the equipment user with higher priority, the average daily costs in Scenarios 1 and 2, where the preventive replacement policy is utilized, are smaller than the costs in Scenarios 3 and 4. The total profits in Scenarios 1 and 2 are better than those obtained in Scenarios 3 and 4 because the preventive replacement cost is much less than the failure replacement cost. In Scenario 2, an item might fail before the preventive replacement due to the long preventive replacement age for the equipment user with lower priority. Moreover, the stockout cost is relatively large because of the lower maximum stock level. Hence, the average daily cost is similar to the cost in Scenario 4, where the age-based preventive replacement policy is not used. The total cost is larger than that in Scenario 3 because the inventory rationing policy is not used in Scenario 2, thus raising the total stockout cost for equipment with higher priority. From the above analysis, the total profits for equipment users in Scenario 1 can be improved further by optimizing the reorder point and critical stock

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level. In Scenario 2, shortening the preventive replacement age can further increase the total profit for the equipment user with lower priority. For the spare part distribution center, the average daily revenue in Scenarios 1 and 2 are larger than those in Scenarios 3 and 4. This is because both the replacement frequency and spare parts demand are larger using the age-based preventive replacement policy at equipment users, as shown in Tables 6 and 7. Therefore, this causes an increase in the average daily sales, cost, and revenue at the spare part distribution center. Furthermore, the average daily profits at the distribution center in Scenarios 1 and 2, where the age-based preventive replacement policy is utilized, are better than those in Scenarios 3 and 4 because of the increase in sales revenues. The total profit for the spare part distribution center in Scenario 1 is more than that attained in Scenario 2. This reveals that the average daily profit at the spare part distribution center is higher if equipment users apply the inventory rationing policy. The average daily revenue of the distribution center in Scenario 3 is larger than that in Scenario 4. This is because the stock level in Scenario 4 is higher than that in Scenario 3 such that the inventory holding cost increases in Scenario 4, as shown in Table 8. For spare part suppliers, the average daily revenues in Scenarios 1 and 2, where the equipment users use the preventive replacement policy, are higher than those obtained in Scenarios 3 and 4, as shown in Table 5. The average replacement frequency for items, i.e., the demand for spare parts, is larger if the equipment users use the preventive replacement policy, thus increasing the average daily sales, revenue and profits for the spare part suppliers. Furthermore, the average daily profits for the spare part suppliers in Scenarios 1 and 3 are better than those achieved in Scenarios 2 and 4. This implies that the revenue can be increased when the equipment users utilize the preventive replacement policy, thus raising the average daily profits for the spare parts suppliers. The profit for the spare parts supplier in Scenario 1 is superior to that acquired in Scenario 2. This indicates that the average daily profit can be more if the equipment users apply the inventory rationing policy. The profit for the spare parts supplier in Scenario 3 is worse than that in Scenario 4, which indicates that the average daily profit is less if the equipment users apply the inventory rationing policy. This is because the stockout probability for equipment with lower priority is larger, thus prolonging the preventive replacement age and decreasing the spare parts demand when the equipment users apply the inventory rationing policy, as shown in Tables 6 and 7.

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Table 1. Parameter settings. Parameters Equipment user Distribution center Production profit of an equipment ($/unit time) (with higher priority, with lower pri25, 15 N/A ority) Ordering cost of spare parts ($/time) 12.5 12.5 Unit purchasing cost of spare part ($/piece) 500 380 Setup cost for producing spare part ($/time) N/A N/A Unit production cost of spare part ($/piece) N/A N/A Age-based preventive replacement cost of 10 N/A spare part ($/piece) Failure replacement cost of spare part 590 N/A ($/piece) Unit holding cost of spare part ($/(piece× 2 1.5 unit time)) Unit stockout cost of spare part ($/(piece× unit time)) (equipment with higher priority, 1800, 10 10 with lower equipment priority) Failure time of spare part (day) WEIB(100,2) N/A Replenishment lead time of spare part (day) WEIB(10,3.2) WEIB(10,3.2) Production time of spare part (day) N/A N/A Table 2. Initial values of decision variables. Decision variables Scenario 1 Scenario 2 Reorder point 40 40 Users with higher prior- Maximum stocking level 80 80 ity Preventive replacement age 80 80 Critical stocking level 0 N/A Reorder point 10 10 Users with lower prior- Maximum stocking level 20 20 ity Preventive replacement age 80 80 Critical stocking level 0 N/A Reorder point 150 150 Distribution center Maximum stocking level 200 200 Critical stocking level 100 100 Re-production level 50 50 Suppliers Maximum stocking level 100 100 Members

Members Users with higher priority

Users with lower priority

Distribution center Suppliers

Table 3. Allowable ranges of decision variables. Decision variables Reorder point Maximum stocking level Preventive replacement age Critical stocking level Reorder point Maximum stocking level Preventive replacement age Critical stocking level Reorder point Maximum stocking level Critical stocking level Re-production level Maximum stocking level

Supplier N/A N/A N/A 12.5 250 N/A N/A 1 10 N/A N/A WEIB(10,3.2)

Scenario 3 40 80 N/A 0 10 20 N/A 0 80 100 50 50 100

Scenario 4 40 80 N/A N/A 10 20 N/A N/A 80 100 50 50 100

Ranges [20, 60] [50, 90] [60, 110] [0, 20] [5, 20] [15, 330] [60, 110] [0, 10] [20, 150] [50, 200] [0, 100] [20, 90] [50, 150]

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Table 4. The optimal solutions in four scenarios. Response and decision variables Scenario 1 Scenario 2 Scenario 3 Scenario 4 Members Total profit in a supply chain 1847.51 1796.60 1581.89 1545.47 Reorder point 20 43 42 39 User with higher Maximum stocking level 69 87 67 68 priority Preventive replacement age 73 97 N/A N/A Critical stocking level 15 N/A 10 N/A Reorder point 8 14 11 11 User with lower Maximum stocking level 24 18 18 21 priority Preventive replacement age 76 103 N/A N/A Critical stocking level 8 N/A 6 N/A Reorder point 71 44 54 73 Distribution center Maximum stocking level 104 74 80 106 Critical stocking level 34 11 5 53 Reproduction point 25 70 59 43 First supplier Maximum stocking level 72 127 82 70 Reproduction point 49 35 74 67 Second supplier Maximum stocking level 68 69 75 72 Table 5. Profits, revenues and costs in the spare part logistics network. Response and decision variables Scenario 1 Scenario 2 Scenario 3 Scenario 4 Total profit of a supply chain 1847.51 1796.60 1581.89 1545.47 Members Total revenue of a supply chain 7572.03 7216.58 6706.41 6729.24 Total cost of a supply chain 5724.52 5419.98 5124.52 5183.77 Profit 911.07 960.88 876.50 871.42 User with higher Revenue 3354.86 3399.78 3395.26 3399.78 priority Cost 2443.79 2438.90 2518.77 2528.37 Profit 214.45 204.30 210.91 202.57 User with lower Revenue 835.59 847.98 843.33 849.33 priority Cost 621.14 643.68 632.42 646.76 Profit 347.81 337.96 262.90 225.66 Distribution center Revenue 1922.41 1685.89 1401.52 1408.88 Cost 1574.61 1347.93 1138.62 1183.21 Profit 240.20 179.40 156.47 166.66 First supplier Revenue 889.12 779.75 654.55 653.90 Cost 648.92 600.35 498.08 487.25 Profit 133.98 114.07 75.11 79.16 Second supplier Revenue 570.05 503.18 411.75 417.34 Cost 436.07 389.11 336.64 338.18 cost in Scenario 1 are much larger than those in SceAs shown in Table 5, the average daily revenues nario 2. for the multi-echelon system in Scenarios 1 and 2 are From the viewpoint of the entire spare part lolarger than those obtained in Scenarios 3 and 4, gistics network, the total profit achieved when both where the preventive replacement policy is not apthe preventive replacement and inventory rationing plied. This is because that the total revenues for the policies are used is better than that acquired when spare part distribution center and suppliers improve only one stock policy is used by the equipment users. greatly even if the revenues for the equipment users The obtained profit when no stock policies are utildo not increase significantly under the preventive ized is the least profitable. In the case of applying one replacement policy of equipment users. The average stock policy, the profit obtained when using the predaily costs of the entire spare part system in Scenarventive policy is better than that obtained when using ios 1 and 2 are greater than the costs in Scenarios 3 the inventory rationing policy. The reason is that the and 4. The reason is that the cost reduction for the preventive replacement policy is aimed at each reequipment users is less than the cost growth at the placement item, thus having more extensive influence. distribution center and suppliers when the preventive However, the inventory rationing policy has a narreplacement policy is utilized by the equipment users. rower influence because it focuses only on the items Because the preventive replacement age in Scenario 1 that cannot be replaced due to spare part stockout. is much shorter than that in Scenario 2, the spare part demand is increased. Hence, both the revenue and

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Table 6. Cost data and related information for the equipment user with higher priority. Cost and related inforPriority of an Scenario 1 Scenario 2 Scenario 3 Scenario 4 mation equipment Average daily holding 50.3987 93.7239 76.4587 78.6617 cost Average daily holding 25.1994 46.8620 38.2294 39.3308 stock Average daily stockout Higher priority 0.0986 4.7342 0 3.9945 cost Lower priority 30.0918 0.1058 3.1586 0.1068 Higher priority 0.00005 0.0026 0 0.0022 Average daily stockout Lower priority 3.0092 0.0106 0.3159 0.0107 Average daily ordering 0.7877 0.7726 1.0620 0.9486 cost Average daily preventive Higher priority 3.8203 2.1145 N/A N/A replacement cost Lower priority 13.4315 8.3852 N/A N/A Average daily failure Higher priority 155.92 196.12 266.03 264.93 replacement cost Lower priority 650.42 787.09 1060.16 1064.49 Average daily preventive Higher priority 0.3820 0.2115 N/A N/A replacement Lower priority 1.3432 0.8385 N/A N/A Average daily failure Higher priority 0.2643 0.3324 0.4509 0.4490 replacement Lower priority 1.1024 1.3341 1.7969 1.8042 Average daily spare part 1546.52 1357.97 1122.71 1127.34 cost Average replenishment 49 44 26.4648 29.6779 Table 7. Cost data and related information for the equipment user with lower priority. Cost and related Priority of an Scenario 1 Scenario 2 Scenario 3 Scenario 4 information equipment Average daily hold24.4453 17.8519 22.2529 24.5335 ing cost Average daily hold12.2226 8.9260 11.1264 12.2668 ing stock Average daily Higher priority 0 41.5726 0 11.3918 stockout cost Lower priority 9.6068 0.9608 4.4479 0.3386 Average daily Higher priority 0 0.0231 0 0.0063 stockout Lower priority 0.9607 0.0961 0.4448 0.0339 Average daily or0.5873 1.0955 0.9849 0.6911 dering cost Average daily preHigher priority 0.8704 0.4540 N/A N/A ventive replacement Lower priority 3.0759 1.7940 N/A N/A cost Average daily failure Higher priority 41.9142 50.4167 66.6134 66.8882 replacement cost Lower priority 168.32 204.20 262.49 265 Average daily preHigher priority 0.0870 0.0454 N/A N/A ventive replacement Lower priority 0.3076 0.1794 N/A N/A Average daily failure Higher priority 0.0710 0.0855 0.1129 0.1134 replacement Lower priority 0.2853 0.3461 0.4449 0.4492 Average daily spare 375.89 327.92 278.81 281.53 part cost Average replenish16 7.4880 7.0744 10.1843 ment

Chen: Optimizing Joint Maintenance and Stock Provisioning Policy Table 8. Cost data and related information for the spare part distribution center Cost and related inforEquipment user or supScenario 1 Scenario 2 Scenario 3 mation plier Average daily holding 93.8095 58.0100 67.0214 cost Average daily holding 62.5369 38.6733 44.6809 stock Equipment user with 9.8537 1.7962 4.1767 Average daily stockout higher priority cost Equipment user with 10.9852 4.4214 0.1901 lower priority Equipment user with 0.9854 0.1796 0.4177 higher priority Average daily stockout Equipment user with 1.0985 0.4421 0.0190 lower priority Average daily ordering 0.7880 0.7726 0.9271 cost Average daily spare part First supplier 889.12 779.75 654.55 cost Second supplier 570.05 503.18 411.75 Average replenishment 60.9109 54.6257 37.8288

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Scenario 4 103.33 68.8841 0.1589 7.6688 0.0159 0.7669 0.8134 653.90 417.34 43.3417

Table 9. Cost data and related information for the first spare part supplier Cost and related information Scenario 1 Scenario 2 Scenario 3 Scenario 4 Average daily holding cost 50.7182 87.8489 64.6649 54.5506 Average daily holding stock 50.7182 87.8489 64.6649 54.5506 Average daily stockout cost 10.9784 0.3274 2.9079 3.1153 Average daily stockout 1.0978 0.0327 0.2908 0.3115 Average daily setup cost 0.4928 0.2798 0.5401 0.4983 Average daily spare part production cost 586.73 511.90 429.97 429.08 Average production 59.3970 91.8572 39.8360 43.2139 Table 10. Cost data and related information for the second spare part supplier Cost and related information Scenario 1 Scenario 2 Scenario 3 Scenario 4 Average daily holding cost 54.6145 57.0714 63.6757 62.3000 Average daily holding stock 54.6145 57.0714 63.6757 62.3000 Average daily stockout cost 5.4118 0.2216 2.1868 0.8236 Average daily stockout 0.5412 0.0222 0.2187 0.0824 Average daily setup cost 0.3158 0.3034 0.3455 0.3168 Average daily spare part production cost 375.73 331.51 270.43 274.74 Average production 59.4437 54.5511 39.2320 43.3247

5. CONCLUSIONS This study builds a simulation model for stock management in a spare part logistics network. Three echelons, including the equipment users, spare part distribution center, and spare part suppliers, are involved in this logistics network. The ( s , S ) spare part inventory model that combines the age-based preventive replacement and inventory rationing policies is used at the equipment users. At the spare part distribution center, an ( s , S ) spare part inventory model is integrated with the inventory rationing policy. Only the ( s , S ) spare parts inventory model was applied for the spare parts supplier. A simulation-optimization approach is used to obtain the near optimal decision variable values thus maximizing the

total profit in the spare part logistics network. The impacts of different joint maintenance and stock provisioning policies for spare part logistics network are also analyzed. According to the simulation results, the total profit in the spare part logistics network adopting an age-based preventive replacement policy is higher than that obtained in the spare part logistics network without an age-based preventive replacement policy by over 16%. Moreover, the profits of all the members in the spare part logistics network can be increased simultaneously. For equipment users, the total profit in the spare part logistics network using an inventory rationing policy can exceed that achieved in the spare part supply chain that does not utilize such a policy by over 2%. However, only the total profits of the equipment users are improved. The

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total profits, revenues and costs for the spare part distribution center and suppliers decrease. In Scenario 1, where both the age-based preventive replacement and inventory rationing policies are used by the equipment users, the total profit in the spare part logistics network is larger than that obtained in Scenario 2. However, the total profit for equipment users with higher priority in Scenario 1 is smaller than that achieved in Scenario 2. Similarly, the total profit in the spare part logistics network in Scenario 2 is larger than that obtained in Scenario 3. However, the total profit for the equipment users with higher priority in Scenario 2 is smaller than that achieved in Scenario 3. In future research, the information sharing mechanism in the logistics network is not included while constructing the simulation model of spare part logistics network management. In future research, the spare parts stock information at the equipment users or distribution center can be shared with the distribution center or suppliers. In this way, the stock policies that lead to lower total costs at the spare part distribution center or suppliers can be promoted.

ACKNOWLEDGEMENT This work is partially supported by National Science Council, Taiwan, ROC under grant NSC 93-2212-E-027-001.

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ABOUT THE AUTHORS Mu-Chen Chen received his Ph.D. and M.Sc. degrees both in Industrial Engineering and Management from National Chiao Tung University, and his B.S.

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degree in Industrial Engineering from Chung Yuan Christian University. He is currently a Professor in the Department of Business Management, and Institute of Commerce Automation and Management at National Taipei University of Technology, Taiwan. His teaching and research interests include meta-heuristics, simulation analysis, logistics management, quality and maintenance management and data mining. He is also involved with research and industry in a range on NSC (National Science Council, Taiwan) and enterprise projects. Chih-Ming Hsu is currently an Associate Professor in the Department of Business Administration at Ming Hsin University of Science and Technology, Taiwan. He holds a Ph.D. in Industrial Engineering and Management from National Chiao Tung University, Taiwan. His present research interests include quality engineering, optimization methods in industrial applications and data mining applications in CRM. Shih-Wei Chen got his MBA from Institute of Commerce Automation and Management, National Taipei University of Technology, Taipei, Taiwan, ROC. He is currently a System Coordinator at Data Processing Division, Taipei National Tax Administration, Ministry of Finance, Taiwan, ROC.

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多階備用零件物流網路聯合維修與存貨供應政策之最佳化陳穆臻* 國立台北科技大學經營管理系 106 台北市忠孝東路三段一號徐志明明新科技大學企業管理系陳士偉財政部台北市國稅局電子作業科

摘要在機器維修保養策略中，備用零件庫存管理是一重要決策，其能夠維持機器的生產力水準。預防性替換政策主要在規劃最低成本之備用零件替換。若考慮設備之關鍵性，備用零件可以分成關鍵與非關鍵二類需求，系統可以保留一定的庫存提供予關鍵需求，此即為存貨比例政策。備用零件庫存顯著地影響生產系統之績效，特別是在機器當機會造成巨大生產損失的情況下。本研究探討採用聯合預防性替換與存貨比例政策在多階備用零件物流網路之最佳化問題。多階備用零件物流網路包括設備使用廠商、備用零件配送中心與備用零件供應商。整個備用零件物流網路的存貨模式非常複雜，因此，本研究以分散搜尋法為基礎之模擬最佳化方法求算在最大化備用零件供應鏈整體利潤之目標下之近似最佳解。並且，以一個範例驗證聯合維修與存貨供應政策之最佳化。關鍵詞：備用零件，預防替換政策，存貨比例政策，模擬最佳化，分散搜尋法 (*聯絡人：[email protected])