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Determination of reorder points for spare parts in a two-echelon inventory system: The case of non identical maintenance facilities
458
European Journal of Operational Research 73 (1994)458-464 North-Holland
Theory and Methodology
Determination of reorder points for spare parts in a two-echelon inventory system: The case of non identical maintenance facilities Avraham
Shtub
The Owen Graduate School of Management, Vanderbilt University, Nashville, TN 37203, USA and Department of Industrial Engineering, Tel-Aviv University, Ramat-Aviv 69978, Israel Moti Simon
Israeli Air Force Received March 1992
Abstract: This paper deals with the determination of reorder points in a two-echelon spare parts inventory system. The system consists of a central warehouse that supplies several maintenance facilities. Each maintenance facility faces random demand. Inventory carrying cost is the same in both echelons, but the consequences of stockout vary among the maintenance facilities. An inventory management policy is proposed that determines the reorder point for each facility, so that predetermined requirements concerning relationships between the resulting service levels are met. The inventory system described in this paper is typical to class A (in a Pareto or ABC analysis) of spare parts inventories, i,e., military. Limited quantities of spare parts are stocked in an inventory system serving several maintenance facilities. The same type of equipment is maintained by each facility. However, due to readiness requirements of the organizational units served by each facility, the required service levels of the centers are not all the same. Keywords: Inventory; Maintenance, Military; Nonlinear Programming; Simulation
I. Introduction A generic two-echelon (two-level) arborescent inventory system consists of a central warehouse
Correspondence to: Dr. A. Shtub, Department of Industrial Engineering, Tel-AvivUniversity, Ramat-Aviv 69978, Israel.
and one or more retailers. Each retailer serves its final customers (its market) while being a customer of the central warehouse. The random demand faced by each retailer depends on the number of customers it serves and the demand of each customer. Although it is possible to manage this type of inventory system by means of single-location inventory models, Muckstadt and
0377-2217/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0377-2217(92)00041-G
A. Shtub, M. Simon / Reorder points for spare parts in a two-echelon system
Thomas (1980) have shown that special multilevel methods can substantially improve the performance of these inventory systems. Two common performance criteria are minimization of cost (capital cost, operating cost and the cost of shortages or backlogs), and maximization of service levels (the measured proportion of demand supplied directly from retailers inventory). Inventory control policies that use such systems are based on a continuous review or a periodical review. In a continuous review system, a replenishment order is considered every time that demand occurs. In a periodical review system a replenishment order is considered every period (a month, a quarter, etc.). The decision taken after each review may be based on local information and objectives (the distributed inventory management approach) or on global information and objectives (the centralized inventory management approach). It is possible to combine the distributed and the centralized approaches. For example, a retailer may order from the warehouse based only on the retailer's inventory status, while the warehouse may order from its supplier based on the status of the whole system. An issue under any combination of inventory management policy and performance criteria is to find the best allocation of safety stock among the warehouse and the retailers. This allocation depends on the structure of the system, costs, lead times, demand distributions, and the performance criteria. Policies aimed at minimizing the cost of operating a single location inventory management system have been studied extensively. The work of Roberts (1962) and Ehrhardt (1979, 1984) represents a significant contribution in this area. Service level considerations for the single location problem have been studied by Tijms and Groenevelt (1984). Their analysis focuses on the service level. Specifically, they present approximations for the reorder point for the single-location inventory problem that achieve a required service level. Inventory management policies for two-echelon systems have been proposed by Clark and Scarf (1960), Sherbrooke (1968), Deuermeyer and Schwarz (1981), Graves (1985), Federgruen and Zipkin (1984), J6nsson and Silver (1987), Jackson (1988), and Savoronos and Zipkin (1988). A
459
heuristic procedure was developed by Rosenbaum (1981) and implemented at Eastman Kodak Company. It was designed to allocate the system's inventory among the echelons in a way that minimizes total company safety stock while ensuring a prespecified service level at each retailer. This study is an extension of the work of Simon (1990) that focused on a two-echelon inventory system for high-value (type A in ABC or Pareto analysis) spare parts. The system is composed of a central warehouse and several maintenance facilities. Each facility performs maintenance work on equipment used by organization units of the same organizations. Since each unit has a priority ranking (e.g., a required level of readiness in a military system), the service level at each adjusted according to the required priority the units serve. Thus, the availability of a required spare part off the shelf at each maintenance facility (the fill rate) should be adjusted based on the predetermined priorities. In the following section, a model of the spare parts inventory system is presented. Next, an inventory management scheme is proposed along with an algorithm designed to estimate the required reorder point at each maintenance facility. Computational experience with the proposed algorithm is presented along with our conclusion at the last section.
2. The model
Consider a two-echelon inventory system composed of a central warehouse and n > 1 maintenance facilities. Spare parts are stocked at the warehouse and at the facilities. The central warehouse operates under a periodic review (T, S) policy. Every T periods the warehouse orders the difference between S and the inventory position of the entire system. The inventory position of the system includes the inventory at each facility, warehouse and pipeline. Each maintenance facility j operates under a continuous review (st, St) system. An order is issued when the inventory position of facility j drops below its specific reorder point st. The order size equals the difference between St and the actual inventory position. The warehouse orders from an outside supplier with infinite supply. Orders arrive after a
460
A. Shtub, M. Simon / Reorder points for spare parts in a two-echelon system
fixed lead time L w. The maintenance facilities orders are supplied from the warehouse inventory according to a First In First Out (FIFO) discipline. The lead time of supply between the warehouse and facility j, Lt, is fixed and deterministic as long as the warehouse does not run into shortages. In case of shortages at the warehouse the facility's lead time is composed of the deterministic component Lj and a random component. The length of the random component depends on the time it takes to fill previous maintenance facility's orders. Shortages at the facilities level cause a reduction in readiness as equipment has to wait until the facility gets new supplies from the warehouse. The organizational units served by the maintenance facilities differ in their demand patterns. The first moment/z/ and the second moment o) of the periodical demand faced by facility j can be approximated based on past data. It is known, however, that the coefficient of variation, o)//z/, is larger than one for all facilities. This fact makes inventory management policies based on a Poisson distribution like Metric (Sherbroke, 1968) not appropriate. Furthermore, since the units served by the facility are not equally important and the costs of stockout are not known, available inventory management policies are not appropriate for the system under consideration.
3. Spare parts inventory management policy
The following policy is suggested for the above two-echelon inventory system: Rank the maintenance facilities in a list by order of the relative priority of units they serve so that the facility with the lowest priority is the first one on the list. Assign relative weights a t to the facilities. The weight of the first facility on the list is a 1 = 1. The relative weight of any other facility is the required ratio between the service level of this facility and the service level of the first facility on the list. To illustrate this last point consider a system with three maintenance facilities. Based on readiness considerations, management recommend that the second facility achieve a service level 10% higher than the first one, while the third facility achieve a service level 5% higher than the
second one. The appropriate values of aj in this case are a l = 1, a 2 = 1.1, a 3 = 1.155. These values satisfy the ranking requirements as shown below: a2 a1
1.1
--
1
= 1.1,
a3
1.155
a2
1.1
1.05.
Suppose that the actual service level of the first facility/3~ is 85% then the service levels of the second and third facilities (/32 and 133) should be /31= 1.1 × 8 5 = 9 3 . 5 % ; /33= 1.155 × 8 5 = 98.175%. Note that/3j = min(aj/3~, 1.0). Since this weighting scheme holds as long as the highest service level achieved is less than 100%. If S is high enough to allow for service levels of 100%, the weights should be ignored for those facilities achieving a service level of 100%. For example, an acceptable solution to the previous problem is given by the three service levels 90%, 99%, 100%. In this case the ratio between the service levels of the first and second facilities is 1.1 as required. The ratio between the service levels of facilities 1 and 3 is not 1.155 since for the given value of S the third facility could achieve a service level of 100%. The suggested policy provides a framework for setting the reorder points of the maintenance facilities in the multi-echelon inventory system under study. There is a need for an algorithm by which the required values of a t can be translated into specific values of the reorder points st. An algorithm based on a search procedure, math program and simulation is discussed next.
4. Implementation of the proposed policy - the algorithm
Discussion: The proposed algorithm is based on the following points: 1. Studies of two echelon systems have shown that finding an optimal solution for a system that faces exogenous and stochastic demand at the retailers level, is usually intractable (Federgruen and Zipkin, 1984). Consequently, most research in this area is directed towards finding near-optimal policies that are easy to compute and implement.
A. Shtub, M. Simon / Reorder points for spare parts in a two-echelon system
lem with a math program and a simulation of the two-echelon inventory system. The solution procedure is based on the observation that as long as the central warehouse does not experience any shortages, the T & G procedure applies to each maintenance facility. That is, it is possible to use this procedure to calculate the reorder point of a facility for a desired fill rate. The fill rate each facility can achieve in case of shortages at the warehouse depends on the maximum total inventory in the system S and the
2. The required performance criterion, maximization of maintenance facilities fill rate, is in accordance with a common practice that emphasizes service level rather than assuming given stockout costs. However, under this performance criterion, even for a single-location inventory system, an optimal policy is intractable. Therefore, approximation for the reorder point are suggested by Tijms and Groenevelt (1984). Based on these points, the proposed solution is a heuristic that integrates Tijms and Groenevelt (1984) (T & G) results for the single location prob-
I. SIMULATE THE SY STEM-OBTAIN SUPPLY UPPER BOUND
2. SOLVE PI OBTAIN DESIRED SERVICE LEVELS
4. FIXTHIS
461
] YES
SOLUTION AS A
~
,OW BOU
I
OF a l IS
I
so:: rN uPP BO D
NO 7. SOLVE P1 FOR FFAR=(FUB+FLB)/2
t Figure 1. Flow diagram of the solution procedure
462
A. Shtub, M. Simon / Reorderpoints for spareparts in a two-echelon system
reorder points (and hence the desired fill rates) of the other facilities. If equal service levels are desired for all facilities, it is possible to find an upper bound on this service level by assuming that Lj. = 0 and %---1 for all j (i.e., the warehouse supplies customers demands directly). The proportion of total demand supplied directly from stock in the corresponding simulation run is an upper bound on the facility service level for the case of equally important maintenance facilities. To account for the difference between desired service levels, a math program was developed as part of the proposed algorithm. This math program distributes the total number of backlogged units in the system, according to the desired proportion between fill rates among the maintenance facilities. Based on the solution to the math program the fill rate /3j for each facility is estimated. In the next step, the reorder point that yields the estimated fill rate is approximated using the T & G model only if the central warehouse does not run into shortages. This is tested by simulation. Otherwise, if the resulting fill rates are lower than the math program solution, the proportional number of units supplied off the shelf on the simulation run serves as a lower bound on the number of units the system can supply off the shelf. Therefore, a bisection search procedure is used to calculate a new set of desired fill rates. The procedure iterates in this manner and in each step, either a new lower bound, or a new upper bound, is computed. The search stops when in two consecutive steps, the difference in fill rates in the simulation is smaller than a predetermined threshold value e. The values of reorder points that caused the highest proportion of units supplied off the shelf in the whole system constitute the solution to the problem. The flow diagram of the solution procedure is presented in Figure 1. To explain the details of the procedure, the following notation is used: j ~ J = Index set of maintenance facilities; j = 1,...,n. T = Time between consecutive reviews of the warehouse inventory level. S = Upper limit on total system inventory. Si - sj --- Minimum order size for facility j. sj = Reorder point of facility j.
L w ---Outside supplier to warehouse deterministic lead time. Lj = Deterministic component of warehouse to maintenance facility's lead time. /3s. = Desired fill rate (fraction of demand supplied off the shelf). /3* = Fill rate of facility j as estimated by the simulation. aj = Desired weight for the fill rate of facility j; O/1 = 1,/3j = min(cei/31, 1). /zj = Expected demand per period of facility j. %. = Standard deviation of demand per period of facility j. e 1 = A predetermined threshold value. e = A predetermined threshold value. FUB = Fill rate upper bound. FLB = Fill rate lower bound. The details of the solution procedure in Figure 1 are as follows: Step 1. Obtain an upper bound on the fill rate of the system, FUB, by simulation of the following system: The structure is Multiechelon (ME); for each facility j = 1. . . . . n, t~-,% are known and % is equal to one. The centers lead times Lj are all equal to zero; L w is known and the warehouse policy parameters (T, S) are given. Note that this is the same as single-echelon system. Step 2. Find the calculated values of /3s. for each facility such that the constraints given by the % are preserved and that the mean number of units supplied to customers off the shelf is equal to FTAR, a target value. Initially F T A R is set equal to the number of units supplied off the shelf in the simulation run in Step 1. The/3j-values are found by solving the following program, P1:
(P1) n
max ~ / 3 j j=l
(1)
subject to ~] ~j ./3j = F'FAR, j=l By = min(t~y31, 1)
(2) for all j.
(3)
Problem P1 is a non-linear program. To solve
A Shtub, M. Simon / Reorder points for spare parts in a two-echelon system
P, it is possible to exploit its special structure as follows: Let /3j =%./31 for all j > 2 and a 1 = 1. From constraint (2) we get
~ ILj/3j= ~ ~jO'j/3l=/31~ ].£jOlj j=l
j=l
j-1
= FTAR,
/3, =
]dbjaj
FTAR/
.
j=
Initially let /3j = afl3p If/3 n _< 1, then stop: current values of/3j are the solution to P,; if/3n > 1, then let /3~ = 1; let F T A R = F T A R - ~zn. Repeat the solution procedure with n = n - 1 facilities. Step 3. Check the feasibility of the solution to P, as follows: First use the T & G method and calculate an initial set of sj, j - 1 . . . . . n, to achieve the service levels /3j from Step 2. These sj will yield the fill rates /3j only if there are no stockouts at the warehouse. Then simulate the ME system with this set of sj. to find the expected fill rates/3*. A solution consisting of a set of sj is defined as feasible if I/3" -/3j} < E1 for all j. If the current solution is feasible, go to Step 4. If the current solution is not feasible, simulate the ME system with sj increased by one for each retailer j where / 3 * - / 3 j > e 1. Continue to increase the appropriate sfvalues until a feasible solution is found or until the mean number of unites supplied to all customers off the shelf in the simulation is smaller than the number supplied in the previous simulation run. In this case the current solution is not feasible - go to Step 5. Step 4. The total number of units supplied off the shelf in the last iteration is the new lower bound on the mean number of units supplied to customers off the shelf, FLB. Go to Step 6. Step 5. Let FUB equal to FTAR. This is the new supply upper bound as it was impossible to achieve this value in Step 3 of the last iteration. Step 6. Check if the algorithm has converged, i.e., if [ F U B - F L B I < e. Then stop. Otherwise go to Step 7. Step 7. Let F T A R = I ( F U B + FLB). Go to Step 2.
463
This algorithm integrates a simulation model with the solution to the nonlinear program P1 and the Tijms-Groenevelt (1984) model to generate feasible solutions. Integration is achieved by a program that uses the three components as subroutines. The program calls each subroutine according to the solution procedure as explained above. The output of each subroutine is transferred automatically to the main program. The result includes the values of sj, the expected service levels/3, and statistics on the demand-supply processes. Study of two-echelon inventory systems with up to 15 maintenance facilities and a variety of service level weights (a j-values) revealed the following points: 1. The assignment of relatively high priorities (high values of a) to some facilities cause a decrease in the system's total number of units supplied off the shelf relative to a system with equal fill rates among the facilities. 2. The procedure requires three to eleven iterations to solve problems with seven to fifteen maintenance facilities. The number of iterations, however, does not depend on the values of aj selected. 3. For all the problems solved it was found that to achieve the best results the central warehouse should carry inventory during the period between two of its consecutive reviews. Thus it is not recommended to stock all of the spare parts inventory at the maintenance facilities.
5. Summary and conclusion The need to maximize the fill rate of a set of maintenance facilities that get spare parts from a central warehouse, while satisfying a required priority among the facilities, is translated into a special type of multiechelon inventory system. Inventories of expensive spare parts (items of type A in ABC analysis) may be limited and the correct allocation of available parts among the central warehouse and the maintenance facilities is important. The proposed algorithm is designed to help management in allocating spare parts inventories; furthermore, the simulation subroutine shows the result of a sequence of allocation decisions giving
464
A. Shtub, M. Simon / Reorder points for spare parts in a two-echelon system
management insight into the effect of its decisions on the demand-supply process. Although the proposed methodology developed for a special system, (T, S) policy at the central warehouse and (s, S) policy at the maintenance facilities, it can be modified (by modifying the simulation subroutine) to handle other inventory management policies.
References Clark, A.J., and Scarf, H.E. (1960), "Optimal policies for a multi-echelon inventory problem", Management Science 6, 475-490. Deuermeyer, B., and Schwarz, L.B. (1981), "A model for the analysis of system service level in warehouse/retailer distribution systems: The identical retailer case", TIMS Studies, Management Science 16, 163-193. Ehrhardt, R. (1979), "The power approximation for computing (s, S) inventory policies", Management Science 25/8, 777-786. Ehrhardt, R. (1984), "(s, S) policies for a dynamic inventory model with stochastic lead times", Operations Research 25/8, 618-622. Federgruen, A., and Zipkin, P. (1983), "Solution techniques for some allocation problems", Mathematical Programming 25, 13-24. Federgruen, A., and Zipkin, P. (1984), "Approximation of dynamic, multilocation production and inventory problems", Management Science 30/1, 69-84.
Graves, S.C. (1985), "A multi-echelon inventory model for a repairable item with one-for-one replenishment", Management Science 31, 1247-1256. Jackson, L.P. (1988), "Stock allocation in a two-echelon distribution system, or 'What to do until your ship comes in'", Management Science 34/7, 880-895. J6nsson, H., and Silver, E.A. (1987), "Analysis of a two-echelon inventory control system with complete redistribution", Management Science 33/2, 215-227. Muckstadt, J.A., and Thomas, L.J. (1980), "Are multi-echelon inventory methods worth implementing in systems with low demand rates?", Management Science 26, 483-494. Roberts, D. (1962), "Approximation to optimal policies in a dynamic inventory model", in: K. Arrow, S. Korlin, and H. Scarf (eds.), Applied Probability and Management Science, Stanford University Press, Stanford, CA. Rosenbaum, B.A. (1981), "Inventory placement in a twoechelon inventory system: An application", TIMS studies, Management Science 10, 195-207. Sherbrooke, C.C. (1968), "Metric: A multi-echelon technique for recoverable item control", Operations Research 16, 122-141. Simon, M. (1990), "Determination of reorder points in a constrained two echelon inventory system for maximum service level", Ms.C Thesis, Tel-Aviv University. Svoronos, A., and Zipkin, P. (1988), "Estimating the performance of multi-level inventory systems", Operations Research 36, 57-72. Tijms, H., and Groenevelt, H. (1984), "Simple approximations for the reorder point in periodic and continous review (s, S) inventory systems with service level constraints", European Journal of Operational Research 17, 175-190. Zipkin, P. (1980), "Simple ranking methods for allocation of one resource", Management Science 26/1, 34-43.
European Journal of Operational Research 73 (1994)458-464 North-Holland
Theory and Methodology
Determination of reorder points for spare parts in a two-echelon inventory system: The case of non identical maintenance facilities Avraham
Shtub
The Owen Graduate School of Management, Vanderbilt University, Nashville, TN 37203, USA and Department of Industrial Engineering, Tel-Aviv University, Ramat-Aviv 69978, Israel Moti Simon
Israeli Air Force Received March 1992
Abstract: This paper deals with the determination of reorder points in a two-echelon spare parts inventory system. The system consists of a central warehouse that supplies several maintenance facilities. Each maintenance facility faces random demand. Inventory carrying cost is the same in both echelons, but the consequences of stockout vary among the maintenance facilities. An inventory management policy is proposed that determines the reorder point for each facility, so that predetermined requirements concerning relationships between the resulting service levels are met. The inventory system described in this paper is typical to class A (in a Pareto or ABC analysis) of spare parts inventories, i,e., military. Limited quantities of spare parts are stocked in an inventory system serving several maintenance facilities. The same type of equipment is maintained by each facility. However, due to readiness requirements of the organizational units served by each facility, the required service levels of the centers are not all the same. Keywords: Inventory; Maintenance, Military; Nonlinear Programming; Simulation
I. Introduction A generic two-echelon (two-level) arborescent inventory system consists of a central warehouse
Correspondence to: Dr. A. Shtub, Department of Industrial Engineering, Tel-AvivUniversity, Ramat-Aviv 69978, Israel.
and one or more retailers. Each retailer serves its final customers (its market) while being a customer of the central warehouse. The random demand faced by each retailer depends on the number of customers it serves and the demand of each customer. Although it is possible to manage this type of inventory system by means of single-location inventory models, Muckstadt and
0377-2217/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0377-2217(92)00041-G
A. Shtub, M. Simon / Reorder points for spare parts in a two-echelon system
Thomas (1980) have shown that special multilevel methods can substantially improve the performance of these inventory systems. Two common performance criteria are minimization of cost (capital cost, operating cost and the cost of shortages or backlogs), and maximization of service levels (the measured proportion of demand supplied directly from retailers inventory). Inventory control policies that use such systems are based on a continuous review or a periodical review. In a continuous review system, a replenishment order is considered every time that demand occurs. In a periodical review system a replenishment order is considered every period (a month, a quarter, etc.). The decision taken after each review may be based on local information and objectives (the distributed inventory management approach) or on global information and objectives (the centralized inventory management approach). It is possible to combine the distributed and the centralized approaches. For example, a retailer may order from the warehouse based only on the retailer's inventory status, while the warehouse may order from its supplier based on the status of the whole system. An issue under any combination of inventory management policy and performance criteria is to find the best allocation of safety stock among the warehouse and the retailers. This allocation depends on the structure of the system, costs, lead times, demand distributions, and the performance criteria. Policies aimed at minimizing the cost of operating a single location inventory management system have been studied extensively. The work of Roberts (1962) and Ehrhardt (1979, 1984) represents a significant contribution in this area. Service level considerations for the single location problem have been studied by Tijms and Groenevelt (1984). Their analysis focuses on the service level. Specifically, they present approximations for the reorder point for the single-location inventory problem that achieve a required service level. Inventory management policies for two-echelon systems have been proposed by Clark and Scarf (1960), Sherbrooke (1968), Deuermeyer and Schwarz (1981), Graves (1985), Federgruen and Zipkin (1984), J6nsson and Silver (1987), Jackson (1988), and Savoronos and Zipkin (1988). A
459
heuristic procedure was developed by Rosenbaum (1981) and implemented at Eastman Kodak Company. It was designed to allocate the system's inventory among the echelons in a way that minimizes total company safety stock while ensuring a prespecified service level at each retailer. This study is an extension of the work of Simon (1990) that focused on a two-echelon inventory system for high-value (type A in ABC or Pareto analysis) spare parts. The system is composed of a central warehouse and several maintenance facilities. Each facility performs maintenance work on equipment used by organization units of the same organizations. Since each unit has a priority ranking (e.g., a required level of readiness in a military system), the service level at each adjusted according to the required priority the units serve. Thus, the availability of a required spare part off the shelf at each maintenance facility (the fill rate) should be adjusted based on the predetermined priorities. In the following section, a model of the spare parts inventory system is presented. Next, an inventory management scheme is proposed along with an algorithm designed to estimate the required reorder point at each maintenance facility. Computational experience with the proposed algorithm is presented along with our conclusion at the last section.
2. The model
Consider a two-echelon inventory system composed of a central warehouse and n > 1 maintenance facilities. Spare parts are stocked at the warehouse and at the facilities. The central warehouse operates under a periodic review (T, S) policy. Every T periods the warehouse orders the difference between S and the inventory position of the entire system. The inventory position of the system includes the inventory at each facility, warehouse and pipeline. Each maintenance facility j operates under a continuous review (st, St) system. An order is issued when the inventory position of facility j drops below its specific reorder point st. The order size equals the difference between St and the actual inventory position. The warehouse orders from an outside supplier with infinite supply. Orders arrive after a
460
A. Shtub, M. Simon / Reorder points for spare parts in a two-echelon system
fixed lead time L w. The maintenance facilities orders are supplied from the warehouse inventory according to a First In First Out (FIFO) discipline. The lead time of supply between the warehouse and facility j, Lt, is fixed and deterministic as long as the warehouse does not run into shortages. In case of shortages at the warehouse the facility's lead time is composed of the deterministic component Lj and a random component. The length of the random component depends on the time it takes to fill previous maintenance facility's orders. Shortages at the facilities level cause a reduction in readiness as equipment has to wait until the facility gets new supplies from the warehouse. The organizational units served by the maintenance facilities differ in their demand patterns. The first moment/z/ and the second moment o) of the periodical demand faced by facility j can be approximated based on past data. It is known, however, that the coefficient of variation, o)//z/, is larger than one for all facilities. This fact makes inventory management policies based on a Poisson distribution like Metric (Sherbroke, 1968) not appropriate. Furthermore, since the units served by the facility are not equally important and the costs of stockout are not known, available inventory management policies are not appropriate for the system under consideration.
3. Spare parts inventory management policy
The following policy is suggested for the above two-echelon inventory system: Rank the maintenance facilities in a list by order of the relative priority of units they serve so that the facility with the lowest priority is the first one on the list. Assign relative weights a t to the facilities. The weight of the first facility on the list is a 1 = 1. The relative weight of any other facility is the required ratio between the service level of this facility and the service level of the first facility on the list. To illustrate this last point consider a system with three maintenance facilities. Based on readiness considerations, management recommend that the second facility achieve a service level 10% higher than the first one, while the third facility achieve a service level 5% higher than the
second one. The appropriate values of aj in this case are a l = 1, a 2 = 1.1, a 3 = 1.155. These values satisfy the ranking requirements as shown below: a2 a1
1.1
--
1
= 1.1,
a3
1.155
a2
1.1
1.05.
Suppose that the actual service level of the first facility/3~ is 85% then the service levels of the second and third facilities (/32 and 133) should be /31= 1.1 × 8 5 = 9 3 . 5 % ; /33= 1.155 × 8 5 = 98.175%. Note that/3j = min(aj/3~, 1.0). Since this weighting scheme holds as long as the highest service level achieved is less than 100%. If S is high enough to allow for service levels of 100%, the weights should be ignored for those facilities achieving a service level of 100%. For example, an acceptable solution to the previous problem is given by the three service levels 90%, 99%, 100%. In this case the ratio between the service levels of the first and second facilities is 1.1 as required. The ratio between the service levels of facilities 1 and 3 is not 1.155 since for the given value of S the third facility could achieve a service level of 100%. The suggested policy provides a framework for setting the reorder points of the maintenance facilities in the multi-echelon inventory system under study. There is a need for an algorithm by which the required values of a t can be translated into specific values of the reorder points st. An algorithm based on a search procedure, math program and simulation is discussed next.
4. Implementation of the proposed policy - the algorithm
Discussion: The proposed algorithm is based on the following points: 1. Studies of two echelon systems have shown that finding an optimal solution for a system that faces exogenous and stochastic demand at the retailers level, is usually intractable (Federgruen and Zipkin, 1984). Consequently, most research in this area is directed towards finding near-optimal policies that are easy to compute and implement.
A. Shtub, M. Simon / Reorder points for spare parts in a two-echelon system
lem with a math program and a simulation of the two-echelon inventory system. The solution procedure is based on the observation that as long as the central warehouse does not experience any shortages, the T & G procedure applies to each maintenance facility. That is, it is possible to use this procedure to calculate the reorder point of a facility for a desired fill rate. The fill rate each facility can achieve in case of shortages at the warehouse depends on the maximum total inventory in the system S and the
2. The required performance criterion, maximization of maintenance facilities fill rate, is in accordance with a common practice that emphasizes service level rather than assuming given stockout costs. However, under this performance criterion, even for a single-location inventory system, an optimal policy is intractable. Therefore, approximation for the reorder point are suggested by Tijms and Groenevelt (1984). Based on these points, the proposed solution is a heuristic that integrates Tijms and Groenevelt (1984) (T & G) results for the single location prob-
I. SIMULATE THE SY STEM-OBTAIN SUPPLY UPPER BOUND
2. SOLVE PI OBTAIN DESIRED SERVICE LEVELS
4. FIXTHIS
461
] YES
SOLUTION AS A
~
,OW BOU
I
OF a l IS
I
so:: rN uPP BO D
NO 7. SOLVE P1 FOR FFAR=(FUB+FLB)/2
t Figure 1. Flow diagram of the solution procedure
462
A. Shtub, M. Simon / Reorderpoints for spareparts in a two-echelon system
reorder points (and hence the desired fill rates) of the other facilities. If equal service levels are desired for all facilities, it is possible to find an upper bound on this service level by assuming that Lj. = 0 and %---1 for all j (i.e., the warehouse supplies customers demands directly). The proportion of total demand supplied directly from stock in the corresponding simulation run is an upper bound on the facility service level for the case of equally important maintenance facilities. To account for the difference between desired service levels, a math program was developed as part of the proposed algorithm. This math program distributes the total number of backlogged units in the system, according to the desired proportion between fill rates among the maintenance facilities. Based on the solution to the math program the fill rate /3j for each facility is estimated. In the next step, the reorder point that yields the estimated fill rate is approximated using the T & G model only if the central warehouse does not run into shortages. This is tested by simulation. Otherwise, if the resulting fill rates are lower than the math program solution, the proportional number of units supplied off the shelf on the simulation run serves as a lower bound on the number of units the system can supply off the shelf. Therefore, a bisection search procedure is used to calculate a new set of desired fill rates. The procedure iterates in this manner and in each step, either a new lower bound, or a new upper bound, is computed. The search stops when in two consecutive steps, the difference in fill rates in the simulation is smaller than a predetermined threshold value e. The values of reorder points that caused the highest proportion of units supplied off the shelf in the whole system constitute the solution to the problem. The flow diagram of the solution procedure is presented in Figure 1. To explain the details of the procedure, the following notation is used: j ~ J = Index set of maintenance facilities; j = 1,...,n. T = Time between consecutive reviews of the warehouse inventory level. S = Upper limit on total system inventory. Si - sj --- Minimum order size for facility j. sj = Reorder point of facility j.
L w ---Outside supplier to warehouse deterministic lead time. Lj = Deterministic component of warehouse to maintenance facility's lead time. /3s. = Desired fill rate (fraction of demand supplied off the shelf). /3* = Fill rate of facility j as estimated by the simulation. aj = Desired weight for the fill rate of facility j; O/1 = 1,/3j = min(cei/31, 1). /zj = Expected demand per period of facility j. %. = Standard deviation of demand per period of facility j. e 1 = A predetermined threshold value. e = A predetermined threshold value. FUB = Fill rate upper bound. FLB = Fill rate lower bound. The details of the solution procedure in Figure 1 are as follows: Step 1. Obtain an upper bound on the fill rate of the system, FUB, by simulation of the following system: The structure is Multiechelon (ME); for each facility j = 1. . . . . n, t~-,% are known and % is equal to one. The centers lead times Lj are all equal to zero; L w is known and the warehouse policy parameters (T, S) are given. Note that this is the same as single-echelon system. Step 2. Find the calculated values of /3s. for each facility such that the constraints given by the % are preserved and that the mean number of units supplied to customers off the shelf is equal to FTAR, a target value. Initially F T A R is set equal to the number of units supplied off the shelf in the simulation run in Step 1. The/3j-values are found by solving the following program, P1:
(P1) n
max ~ / 3 j j=l
(1)
subject to ~] ~j ./3j = F'FAR, j=l By = min(t~y31, 1)
(2) for all j.
(3)
Problem P1 is a non-linear program. To solve
A Shtub, M. Simon / Reorder points for spare parts in a two-echelon system
P, it is possible to exploit its special structure as follows: Let /3j =%./31 for all j > 2 and a 1 = 1. From constraint (2) we get
~ ILj/3j= ~ ~jO'j/3l=/31~ ].£jOlj j=l
j=l
j-1
= FTAR,
/3, =
]dbjaj
FTAR/
.
j=
Initially let /3j = afl3p If/3 n _< 1, then stop: current values of/3j are the solution to P,; if/3n > 1, then let /3~ = 1; let F T A R = F T A R - ~zn. Repeat the solution procedure with n = n - 1 facilities. Step 3. Check the feasibility of the solution to P, as follows: First use the T & G method and calculate an initial set of sj, j - 1 . . . . . n, to achieve the service levels /3j from Step 2. These sj will yield the fill rates /3j only if there are no stockouts at the warehouse. Then simulate the ME system with this set of sj. to find the expected fill rates/3*. A solution consisting of a set of sj is defined as feasible if I/3" -/3j} < E1 for all j. If the current solution is feasible, go to Step 4. If the current solution is not feasible, simulate the ME system with sj increased by one for each retailer j where / 3 * - / 3 j > e 1. Continue to increase the appropriate sfvalues until a feasible solution is found or until the mean number of unites supplied to all customers off the shelf in the simulation is smaller than the number supplied in the previous simulation run. In this case the current solution is not feasible - go to Step 5. Step 4. The total number of units supplied off the shelf in the last iteration is the new lower bound on the mean number of units supplied to customers off the shelf, FLB. Go to Step 6. Step 5. Let FUB equal to FTAR. This is the new supply upper bound as it was impossible to achieve this value in Step 3 of the last iteration. Step 6. Check if the algorithm has converged, i.e., if [ F U B - F L B I < e. Then stop. Otherwise go to Step 7. Step 7. Let F T A R = I ( F U B + FLB). Go to Step 2.
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This algorithm integrates a simulation model with the solution to the nonlinear program P1 and the Tijms-Groenevelt (1984) model to generate feasible solutions. Integration is achieved by a program that uses the three components as subroutines. The program calls each subroutine according to the solution procedure as explained above. The output of each subroutine is transferred automatically to the main program. The result includes the values of sj, the expected service levels/3, and statistics on the demand-supply processes. Study of two-echelon inventory systems with up to 15 maintenance facilities and a variety of service level weights (a j-values) revealed the following points: 1. The assignment of relatively high priorities (high values of a) to some facilities cause a decrease in the system's total number of units supplied off the shelf relative to a system with equal fill rates among the facilities. 2. The procedure requires three to eleven iterations to solve problems with seven to fifteen maintenance facilities. The number of iterations, however, does not depend on the values of aj selected. 3. For all the problems solved it was found that to achieve the best results the central warehouse should carry inventory during the period between two of its consecutive reviews. Thus it is not recommended to stock all of the spare parts inventory at the maintenance facilities.
5. Summary and conclusion The need to maximize the fill rate of a set of maintenance facilities that get spare parts from a central warehouse, while satisfying a required priority among the facilities, is translated into a special type of multiechelon inventory system. Inventories of expensive spare parts (items of type A in ABC analysis) may be limited and the correct allocation of available parts among the central warehouse and the maintenance facilities is important. The proposed algorithm is designed to help management in allocating spare parts inventories; furthermore, the simulation subroutine shows the result of a sequence of allocation decisions giving
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management insight into the effect of its decisions on the demand-supply process. Although the proposed methodology developed for a special system, (T, S) policy at the central warehouse and (s, S) policy at the maintenance facilities, it can be modified (by modifying the simulation subroutine) to handle other inventory management policies.
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Graves, S.C. (1985), "A multi-echelon inventory model for a repairable item with one-for-one replenishment", Management Science 31, 1247-1256. Jackson, L.P. (1988), "Stock allocation in a two-echelon distribution system, or 'What to do until your ship comes in'", Management Science 34/7, 880-895. J6nsson, H., and Silver, E.A. (1987), "Analysis of a two-echelon inventory control system with complete redistribution", Management Science 33/2, 215-227. Muckstadt, J.A., and Thomas, L.J. (1980), "Are multi-echelon inventory methods worth implementing in systems with low demand rates?", Management Science 26, 483-494. Roberts, D. (1962), "Approximation to optimal policies in a dynamic inventory model", in: K. Arrow, S. Korlin, and H. Scarf (eds.), Applied Probability and Management Science, Stanford University Press, Stanford, CA. Rosenbaum, B.A. (1981), "Inventory placement in a twoechelon inventory system: An application", TIMS studies, Management Science 10, 195-207. Sherbrooke, C.C. (1968), "Metric: A multi-echelon technique for recoverable item control", Operations Research 16, 122-141. Simon, M. (1990), "Determination of reorder points in a constrained two echelon inventory system for maximum service level", Ms.C Thesis, Tel-Aviv University. Svoronos, A., and Zipkin, P. (1988), "Estimating the performance of multi-level inventory systems", Operations Research 36, 57-72. Tijms, H., and Groenevelt, H. (1984), "Simple approximations for the reorder point in periodic and continous review (s, S) inventory systems with service level constraints", European Journal of Operational Research 17, 175-190. Zipkin, P. (1980), "Simple ranking methods for allocation of one resource", Management Science 26/1, 34-43.