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Regression approximation for a partially centralized inventory system considering transportation costs
Computers & Industrial Engineering 56 (2009) 1169–1176
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Regression approximation for a partially centralized inventory system considering transportation costs Dong-Ju Lee a, In-Jae Jeong b,* a b
Department of Industrial & Systems Engineering at Kongju National University, Kongju, Korea 182 Shinkwan-dong, Kongju, Chungnam 314-701, Republic of Korea Department of Industrial Engineering, Hanyang University, Seoul, Korea 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Republic of Korea
a r t i c l e
i n f o
Article history: Received 2 May 2007 Received in revised form 17 December 2007 Accepted 9 June 2008 Available online 14 June 2008 Keywords: Inventory centralization Transportation cost Regression approximation
a b s t r a c t Inventory centralization for multiple stores with stochastic demands reduces costs by establishing and maintaining a central ordering/distribution point. However the inventory centralization may increase the transportation costs since either the customer must travel more to reach the product, or the central warehouse must ship the product over longer distance to reach the customer. In this paper, we study a partially centralized inventory system where multiple central warehouses exist and a central warehouse fulfills the aggregated demand of stores. We want to determine the number, the location of central warehouses and an assignment of central warehouses and a set of stores. The objective is the minimization of the sum of warehouse costs and transportation cost. With the help of the regression approximation of cost function, we transform the original problem to more manageable facility location problems. Regression analysis shows that the approximated cost function is close to the original one for normally distributed demands. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Consider the continuous review order quantity-reorder point (Q, r) inventory system. We have a set of stores which establish their own warehouses and maintain inventory to satisfy customers’ stochastic demand. Store fills customer orders and distributes the product directly to customers which is a Completely Decentralized Inventory System (CDIS). In a Completely Centralized Inventory System (CCIS), instead of running individual stores, a single central warehouse fulfills all the demands occurring in stores. This is termed as, inventory centralization. By running a central warehouse, we can reduce the variability of demands by compensating the underestimated demand of one store with the overestimated demand of another store. Eventually, the reduction of the variability of demands eliminates the unnecessary safety stock and this minimizes the inventory holding cost. The cost reduction by inventory centralization is theoretically proved by Gerchak and Gupta (1991). Therefore in this research line, researchers have focused on the fair allocation of benefits of inventory centralization to stores such that no store will have an incentive to run individual warehouse. Hartman, Dror, and Shaked (2000) and Hartman and Dror (2003) study the benefit allocation of inventory centralization using game theory. However, as Simchi-Levi, Kaminsky, and Simchi-Levi (2003) mentioned, there exist tradeoffs between inventory * Corresponding author. Tel.: +82 2 2220 0412. E-mail addresses: [email protected] (D.-J. Lee), [email protected] (I.-J. Jeong). 0360-8352/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.06.005
costs and transportation costs in inventory centralization. The inventory centralization may incur additional transportation costs since the distribution point has been changed from local stores to central warehouses. Meanwhile the inventory cost can be decreased as the inventory is centralized due to risk-pooling and economies of scale. This paper consider this kind of Partially Centralized Inventory System (PCIS) considering transportation cost where decision variables are the number, the location of central warehouses and the assignment of central warehouses under assumption that each store and warehouse apply (Q, r) inventory policy. The objective of the problem is the minimization of the sum of warehouse costs (i.e., ordering cost, inventory cost and shortage cost) and transportation cost. A few researches on PCIS considering transportation cost have focused on (Q, r) with service level problem which is similar to (Q, r) but in this problem, reorder point is not a decision variable since it is determined by given service level. Service level is a probability that a demand or aggregate demand of stores is met during lead time. Thus the problem can be formulated as EOQ type model with transportation cost. Das and Tyagi (1997) are the first who consider service level in order to determine optimal degree of inventory centralization. Miranda and Garrido (2004) propose a Lagrangian relaxation based heuristics using subgradient method to solve the problem. Sourirajan, Ozsen, and Uzsoy (2007) consider a two-stage supply chain with a production facility and multiple retailers. They incorporate the service level in the problem to calculate safety stock. Teo et al. (2001) study impact on warehouse investment cost and inventory costs when stores are consolidated into one warehouse. They consider sit-
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uations where demands follow nonidentical Poisson process and each store uses (Q, r) inventory policy but transportation costs are not included in their model. The warehouse cost function of PCIS is very complex due to the (Q, r) policy. We first seek to approximate the cost function using regression models that are function of mean and variance of lead time demand. With the help of the approximated cost function, we transform the original PCIS with transportation cost problem to a more manageable problem that is, uncapacitated facility location problem (UFLP). The objective of this research is not to improve or develop an efficient algorithm to solve UFLP but to appropriate the cost function of PCIS with regression model so that PCIS can be solved with existing algorithms for UFLP. This paper is organized as follows. The mathematical formulation of CDIS, CCIS and PCIS are described in Section 2. Proposed regression models will be explained in Section 3. Section 4 shows the regression analysis for the cost approximation. Finally, conclusion is given in Section 5.
Hartman and Dror (1996) and Robinson (1993) examined the fair allocation of joint inventory costs among the participating stores for CCIS. 2.2. Formulation of PCIS
Notation j
xij yj Sj {i | xij = 1} Qj, rj Zj
Dj ¼
2. Problem description In this section, we review traditional CDIS and the solution procedure to find optimal ordering policies. Also we explain the relationship between CDIS and CCIS when there is no transportation cost. Finally we presents PCIS problem with transportation cost considered in this paper. 2.1. Formulation of CDIS Consider N stores with single-item and annual demand. Let Di and Xi be the annual demand and random lead time demand at store i for the period. Xi is a random variable with probability density function gi(Xi) and cumulative distribution function Gi(Xi) where EðX i Þ ¼ li . Also suppose that each store has an ordering cost A, unit inventory holding cost h and unit shortage cost p. This assumption implies that cost parameters will not change drastically depending on where the customers are served from. This is especially true when stores are located nearby in a country. In a CDIS, the expected cost at store i by replenishing items Qi per period when the inventory level reaches ri can be calculated as follows:
Di Q Di Ri ðr i Þ C i ðQ i ; ri Þ ¼ A þ h i þ r i li þ p Qi Qi 2
ð1Þ
R1 where Ri ðri Þ ¼ E½ðX i ri Þþ ¼ ri ðX i r i Þg i ðxi Þdxi and (Xi-ri)+ = max[0, Xi-ri]. The optimal ordering policy ðQ i ; ri Þ to minimize function (1) can be found by solving the following two equations iteratively until Q i and r i converge (Johnson & Montgomery, 1974).
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Di ðA þ pRi ðri ÞÞ ; h hQ i 1 Gi ðr i Þ ¼ : pDi Q i ¼
ð2Þ ð3Þ
From Eq. (3), it can be shown that p P h since 0 6 Gi ðr i Þ 6 1 and Q i 6 Di . Inventory centralization is to centralize inventory of several stores in order to take advantage of the risk-pooling. In CCIS, one central warehouse orders, stores and distributes the inventories of all individual stores jointly. Let G be the aggregated cumulative P lead time demand distribution with mean l ¼ i li and aggregated P annual demand D ¼ i Di . The optimal ordering policy ðQ ; r Þ for the CCIS can be found similar to Eqs. (1)–(3) using Q, D, R, r instead of Qi, Di, Ri, ri. For CCIS, Gerchak and Gupta (1991) proved that inventory centralization always lower the costs compared to CDIS.
potential central warehouse index j = 1,. . .,N 1; if store i is served from central warehouse j 0; otherwise 1; if storej is selected as a central warehouse location 0; otherwise
P
i2Sj Di
R ðr Þ ¼ E½ðZ j rj Þþ ¼ R j1 j rj ðZ j r j Þg j ðxÞdx u dij tij = udij
a set of stores served from central warehouse j order quantity, reorder point of central warehouse j serving stores i 2 Sj . an aggregated lead time demand with mean P lj ¼ i2Sj li and probability density function gj and cumulative density function Gj. aggregated annual demand for central warehouse j serving stores i 2 Sj . Expected number of units short of a central warehouse j serving stores i 2 Sj in a replenishment cycle. transportation cost per unit item per unit distance. distance from store i to central warehouse j. incremental transportation cost per unit item to customer by changing the distribution point from store i to a central warehouse j, where tii = 0.
PCIS problem is similar to CDIS and CCIS except that PCIS considers the transportation cost due to the change of distribution points. For example, if the inventory of a store i and store j is aggregated to a central warehouse j which is equivalent to the location of store j, we assume that either the customer for store i has to travel more to reach the product, or the central warehouse j has to ship the product over longer distance to reach the customer. Also we assume the incremental transportation cost is proportionate to distance between store i and j. Using the previously mentioned notations, we formulate PCIS as follows: Problem PCIS
P 80 9 1 ! P Di Rj ðr j Þxij = X< X X Q D x i ij j i i @A Ay j þ Min þh li xij þ p tij Di xij þ rj : ; Q 2 Q j j j i i ð4Þ
Subject to xij 6 yj ; 8i; 8j X x ¼ 1; 8i j ij
ð6Þ
8i; 8j 8j
ð7Þ ð8Þ
xij P 0;
ð5Þ
yj 2 f0;1g;
The objective of PCIS is the minimization of the warehouse cost and transportation cost as shown in (4). The warehouse cost consists of setup cost, inventory holding cost and shortage cost. Constraint (5) ensures that a store can not be served from a warehouse unless it is established. Each store must be served by exactly one warehouse by constraint (6). Constraint (7) and (8) ensure the integrality of the variables. In this problem, we want to determine the number of central warehouses and an efficient assignment of central warehouses and a set of stores so that it can minimize total cost. Also optimal order quantity-reorder points of central warehouses must be determined. We assume that the potential location of central warehouses exists among existing stores and all stores have the equal setup cost, holding cost and penalty cost. Let the warehouse cost of central warehouse j denotes
P cj ¼ A
i Di xij
Qj
X Qj þh li xij þ rj 2 i
!
P þp
i Di Rj ðr j Þxij
Qj
:
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Given cj for all j, PCIS can be formulated as a well-known UFLP as follows:
Min
X
cj yj þ
j
XX i
t ij Di xij
ð9Þ
j
s.t. constraints (5)–(8). Therefore a careful estimation of cj will transform PCIS to a more manageable UFLP. In the literature, there has been an attempt to approximate the warehouse cost function for CDIS. Das (1985) propose two strategies to approximate the loss integral Ri(ri) by an exponential function and a quadratic function. Reorder point and order quantity are obtained in a non-iterative way. In this paper, we consider PCIS with transportation cost and estimate the warehouse cost directly using the mean and the variance of lead time demand without calculating reorder point and order quantity. We will explain the approximation of cost function using regression models in the following sections. The solution of UFLP determines the assignment of stores to central warehouses (i.e., Sj is given for all j). Then the optimal ordering policy ðQ j ; rj Þ of the central warehouse j that minimizes cj can be found similar to Eqs. (2) and (3) using Qj, Dj, Rj, rj instead of Qi, Di, Ri, ri as follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 i2Sj Di ðA þ pRj ðr j ÞÞ Qj ¼ ; h hQ j : 1 Gj ðr j Þ ¼ P p i2Sj Di
Fig. 1. Scatter plot of aggregated mean and warehouse cost.
ð10Þ ð11Þ
3. Regression approximation Without loss of generality, we assume that the annual demand is proportionate to the mean of lead time demand, that is li where L is the lead time (week) for inventory replenDi ¼ 52 weeks L ishment. The total number of possible assignment of store to warehouse is 2N 1 since a store can either be included in the assignment or not. Let ak be the set of stores that are included in assignment k where k = 1,. . .,2N 1. For assignment ak the warehouse cost ck can be calculated as follows:
P
ck ¼ A
i2ak Di Q k
X Q k þh li þ r k 2 i2a k
!
P
þp
Fig. 2. Scatter plot of aggregated standard deviation and warehouse cost.
i2ak Di Rk ðr k Þ Qk
where Q k and rk can be found using i 2 ak instead of i 2 Sj in Eqs (10) P and (11). In order to figure out the relationship between i2ak li and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ck also P r2 and ck , we randomly generate an example problem i2ak i as follows: consider N = 10 problem where the lead time demand of a store is iid and normally distributed with L = 3 weeks. The mean li and standard deviation ri are randomly selected from U(100, 1000) and U(10, 100), respectively. ffiffiffiffiffiffiffiffiffiffiffiffi qP P Figs. 1 and 2 show the relationship of ( i2ak li ; ck Þandð i2ak r2i ck ), respectively, for all possible assignments, ak. It seems that the aggregated mean (standard deviation) of store has nonlinear (linear) relation with the warehouse cost. Therefore, we propose four different types regression models that are functions of mean and variance of lead time demands as follows:
Model 1 : cj ¼ a þ b
X
li xij
i
Model 2 : cj ¼ a
X i
Model 3 : cj ¼ a þ b1
!b
li xij X i
Model 4 : cj ¼ a
X i
ð12Þ ð13Þ
qX ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi li xij þ b2 r2 x i i ij
li xij
!b 1
qX ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ b2 r2 x i i ij
where a, b, b1 and b2 are regression parameters. Regression model 1 and 2 consider only the mean of lead time demand meanwhile model 3 and 4 incorporate the variance of lead time demand into the model 1 and 2 for the estimation of the warehouse cost. In the following subsections, we will show that original PCIS problem can be transformed into well-known UFLP using the proposed regression models. 3.1. Regression model 1 Model 1 implies that if store j is selected as a central warehouse and Sj is given, the warehouse cost is approximated as P cj ¼ a þ b i2S li which is a linear function of aggregated demand j of stores that are served by warehouse j. Using the regression model 1, we approximate the objective function as follows:
Min
X
(
aþb
j
ð14Þ
¼
X j
ð15Þ
X i
ayj þ
!
li xij yj þ
X
) t ij Di xij
i
XX ðbli þ t ij Di Þxij i
j
Considering constraint (5) and (8), we can replace the nonlinear term, xijyj with xij in the objective function. Therefore we pro-
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pose an approximated mathematical model of PCIS problem as follows: Formulation 1 (F1)
Min
X
ayj þ
j
XX ðbli þ t ij Di Þxij i
ð16Þ
j
s.t. constraints (5)–(8). F1 is a special case of a well-known UFLP where the opening costs are the same for all warehouses. Once the problem has been approximated as UFLP, we can apply existing heuristics such as greedy heuristic (Daskin, 1995), dual approach (Erlenkotter, 1978), the Lagrangian method (Beasley, 1993) and tabu search (Sun, 2006).
as xij except that we have one more subscript k for dummy facilities. P Thus xij can be replaced with k xijk without any restriction. Define yjk as
8 > < 1 : if dummy facility k of facility j is served as central yjk ¼ warehouse > : 0 : otherwise
Then the objective function can be approximated as follows:
8 X< j
xijk ¼
1 : if store i is assigned to dummy facility k of facility j 0 : otherwise
such that store i must be assigned to exactly one dummy facility i.e., PP 8i. The new decision variable xijk is exactly the same j k xijk ¼ 1;
_
c j (e j )
¼
Min
fj2
Fig. 3. Piecewise linear approximation of function cj ðej Þ.
k
fjk yjk þ
j
k
i
j
k
i
j
k
ljk li þ t ij Di xijk
XXX ðljk li þ tij Di Þxijk i
k
j
ð17Þ
k
ð18Þ ð19Þ
k
8i; 8j; 8k 8j; 8k
xijk P 0;
ð20Þ
yjk 2 f0; 1g;
ð21Þ
It is important to note that the formulation of F2 is exactly the same as that of UFLP other than the subscript k to differentiate dummy facilities at each facility. Constraint (18) ensures that store i cannot be assigned to dummy facility k unless it is opened. Constraint (19) implies that a store must be assigned to exactly one dummy facility. The formulation does not restrict solutions such that multiple dummy facilities are selected at one facility. However this constraint is not necessary since only single dummy facility whose cost is located on the lower envelop of segments will be selected (Krarup and Pruzan (1983)). 3.3. Regression model 3 Regression model 3 approximates the aggregated warehouse qP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 cost as cj ¼ a þ b1 i lii xij þ b2 i ri xij which is a function of aggregated mean and variance of lead time demand that are served by warehouse j. Similar to model 2, the nonlinear term qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 ffi b2 i ri xij in regression model can be approximated as a piecepffiffiffiffi wise linear function. Define a function cj ðej Þ ¼ b2 ej where P 2 ej ¼ i ri xij . cj ðej Þ is approximated with straight lines where kth line has y-axis intercept gjk with slope mjk. Using the same definition of xijk and yjk as model 2, we can transform the objective function as follows:
(
a þ b1 XX j
þ
ej
fjk yjk þ
i
X X X
s:t: xijk 6 yjk ; 8i; 8j; 8k X X xijk ¼ 1; 8i j
ffi
lj2
lj1
j
j
þ
fj1=0
k
XX
j
lj3
j
XX
The approximated mathematical model can be described as follows:Formulation 2 (F2)
X
fj3
9 =
X
a li xij yj þ tij Di xij : ; i i XX XXX XXX ffi fjk yjk þ ljk li xijk þ tij Di xijk
3.2. Regression model 2 Model 2 approximates the cost function as a nonlinear funcP tion of mean of lead time demand as cj ¼ að i2Sj li Þb . There are some researches which attempt to solve UFLP with nonlinear cost function. Holmberg (1999) incorporate the convex transportation cost to the objective function of UFLP. Two methods such as a modified Bender’s decomposition method and a branch and bound method with dual ascent are applied. Wu et al. (2006) consider the general opening cost and apply a Lagrangian heuristic algorithm. Hajiaghayi et al. (2003) develop a error bound guaranteeing heuristic for a concave opening cost function. Harkness and ReVelle (2003) consider the convex production cost function in the problem. A branch and bound approach with linear programming is applied to solve the problem. Krarup and Pruzan (1983) propose the piecewise linear approximation to solve concave opening cost function. Using Karup and Pruzan’s approximation method and regression model 2, we will show that the problem can also be transformed to UFLP similar to model 1. P P b Define a function cj ðej Þ ¼ að i li xij Þ where ej ¼ i li xij . The nonlinear function cj ðej Þ and the piecewise linear functions are depicted in Fig. 3. In Fig. 3, cj ðej Þ is approximated with three straight lines where kth line has y-axis intercept fjk with slope ljk with fjk < fjk+1 and ljk > ljk+1. As Krarup and Pruzan (1983) have mentioned, we can regard fjk as a cost of opening a dummy facility k of facility j. Total cost for warehouse j is given by the lower envelop of the line segments. Define xijk as
!b
X
X
li xij þ b2
i
i
ayjk þ b1
k
XXX i
j
k
i
j
k
! ) rX ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X r2i xij yj þ tij Di xij
XXX i
j
mjk r2i xijk þ
i
li xijk þ
j
k
XXX i
j
k
XX k
t ij Di xijk ¼
X X X b1 li þ mjk r2i þ t ij Di xijk
g jk yjk XX ða þ g jk Þyjk j
k
Therefore we can formulate PCIS as another UFLP as follows: Formulation 3 (F3)
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XX XXX Min ða þ g jk Þyjk þ ðb1 li þ mjk r2i þ t ij Di Þxijk j
i
k
j
generated from U(100, 1000) and U(10, 100), respectively. Let the lead time, L = 3 weeks then the annual demand is determined as follows:
k
s.t. constraints (18)–(21).
Di ¼
3.4. Regression model 4 Similar reasoning can be applied to regression model 4. The cost qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P P 2 function can be approximated as cj ¼ að i2Sj li Þb1 þ b2 i2Sj ri . Using the same definition of model 2 and model 3, we approximate the nonlinear function of mean (standard deviation) with kth straight line has y-axis intercept fjk(gjk) with slope ljk(mjk). Let yjk(ujk ) and xijk(vijk ) denote the decision variables for the approximation of the nonlinear function of mean (standard deviation). Then the objective function value can be described as follows:
80 9 1 !b 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = X X< X X @a li xij þ b2 r2i xij Ayj þ tij Di xij i : ; j i i XX XXX XX ffi fjk yjk þ ljk li xijk þ g jk ujk j
þ
i
k
XXX i
j
k
i
j
k
j
mjk r2i vijk þ
j
k
XXX i
j
k
t ij Di xijk ¼
XX j
k
XXX XX þ ðljk li þ t ij Di Þxijk þ g jk ujk þ
XXX i
j
j
Gap ¼
k
mjk r2i vijk
k
XX j
þ
fjk yjk þ
i
j
XXX XX ðljk li þ t ij Di Þxijk þ g jk ujk i
j
mjk r
2 i vijk
k
XXX
k
j
k
k
s.t. constraints (18)–(21).
X
y k jk
X
x k ijk
¼
X
ujk ;
8j
ð22Þ
vijk ;
8i; 8j
ð23Þ
k
¼
X
Table 1 shows the six problem types that are considered in this paper. For experimental purposes, we consider low and high level of parameter, A, p and h. The low level of a parameter is generated from U($100, $5000) and the high level from U($5001, $10000), respectively. For each problem type, we randomly generate 10 problems and apply four regression models to determine the most appropriate approximation of warehouse cost function. Let Opt be the optimal objective value obtained by total enumeration of all possible solutions of the original PCIS problem. Also let App* be the optimal objective value obtained using the approximated cost function. In order to measure the quality of the approximation of cost using regression models, we propose the percent gap from the approximated cost to the original cost as follows:
App Opt 100 Opt maxðcj ; cj Þ minðcj ; cj Þ 100 Gapj ¼ maxðcj ; cj Þ
fjk yjk
k
Using the above objective function, we can formulate the problem as follows: Formulation 4 (F4)
Min
k
8i; 8j; 8k 8j; 8k
ujk 2 f0; 1g;
4. Regression analysis We consider N = 10 problem where X-coordinate and Y-coordinate of stores are randomly generated from U(0, 500 km) and the distance between two store are measured by Euclidean distance. We assume that u = $0.03 per item per km and lead time demands are iid and normally distributed with li and ri that are
ð25Þ
(1) In terms of performance measures, model 4 outperforms other models with maximum Gapj = 9.3%, and maximum Gap* = 0.3% for all types of problems. (2) Incorporating the nonlinear term of the mean of lead time demand is important in estimating warehouse cost since model 2, 4 are better than model 1, 3. It is interesting to see that Miranda and Garrido (2004) report similar results for PCIS problem with service level consideration. In their paper, they represent the warehouse cost as a nonlinear function of mean and standard deviation of lead time demand. (3) Model 4 shows better performance compared to model 2 since model 2 shows maximum Gapj = 31.6%, and maximum Gap* = 1.1%. However model 2 leads to a
k
Constraint (22) and (23) ensure that once store j is selected as a warehouse the nonlinear term of mean and standard deviation must be incorporated into the objective function simultaneously. We may apply Lagrangian Relaxation method by relaxing constraint (22) and (23) so that F4 can be decomposed into two independent UFLP problems. We do not investigate the solution procedure for F4 in depth since developing an efficient algorithm is not our purpose in this research. Our concern is to determine appropriate regression model for the approximation of the original problem.
ð24Þ
Gap* measures the quality of the approximation for the optimal solution of the original problem and Gapj measures the percent deviation of estimated cost from original cost. The results are shown in Tables 2–7. Minimum, average and maximum of Gapj are shown in tables along with Gap* for each problem. In order to determine regression parameters, a, b, b1 and b2, we use statistical software SAS 8.1. Especially, we use PROC REG procedure of SAS for model 1, 2 and 3, PROC NLIN for model 4. PROC NLIN applies the Gauss–Newton method to determine the nonlinear regression estimates. We run the experiments on personal computer with Intel Core2 Duo 1.83 MHz. A review of the results obtained for the six problem types suggests the following conclusions:
vijk 6 ujk ; 8i; 8j; 8k X X vijk ¼ 1; 8i j vijk P 0;
52 weeks l: 3 weeks i
Table 1 Problem types Problem type
A
p
h
1 2 3 4 5 6
U($100, $5000) U($100, $5000) U($100, $5000) U($5001, $10,000) U($5001, $10,000) U($5001, $10,000)
U($1, $50) U($51, $100) U($51, $100) U($1, $50) U($51, $100) U($51, $100)
U($1, $50) U($1, $50) U($51, $100) U($1, $50) U($1, $50) U($51, $100)
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Table 2 Problem type 1: low setup cost, low penalty cost and low holding cost Problem
Model 1
Model 2 *
Gapi (%)
Gap (%)
Gapi (%)
Model 3 *
Gap (%)
Model 4 *
Gapi
Gap (%)
Gap* (%)
Gapi (%)
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
1 2 3 4 5 6 7 8 9 10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
1.9 2.6 1.9 2.4 2.8 2.9 3.7 1.8 2.0 3.3
75.1 66.1 90.7 122.9 77.0 138.3 80.2 64.6 62.9 115.5
0.0 0.0 6.4 1.6 3.5 0.2 5.9 0.0 7.1 7.3
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
7.2 7.5 7.7 8.8 7.5 9.2 7.9 6.7 6.6 9.3
19.5 21.6 21.1 28.2 24.6 29.3 27.0 18.6 18.4 31.6
0.0 0.0 0.0 0.2 0.0 0.5 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
1.8 2.2 1.9 2.0 1.8 2.0 1.6 1.7 1.7 1.9
68.0 60.2 87.9 126.2 73.9 125.8 54.8 65.4 65.1 62.5
0.0 0.0 6.4 1.3 0.0 0.2 5.9 0.0 4.6 2.4
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.1 0.0 0.1 0.1 0.1 0.1 0.0 0.0 0.1
0.2 1.7 0.2 7.7 3.8 2.4 3.8 0.4 1.8 4.2
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Min Avg Max
0.0 0.0 0.0
1.8 2.5 3.7
62.9 89.3 138.3
0.0 3.2 7.3
0.0 0.0 0.0
6.6 7.8 9.3
18.4 24.0 31.6
0.0 0.1 0.5
0.0 0.0 0.0
1.6 1.9 2.2
54.8 79.0 126.2
0.0 2.1 6.4
0.0 0.0 0.0
0.0 0.1 0.1
0.2 2.6 7.7
0.0 0.0 0.0
Table 3 Problem type 2: low setup cost, high penalty cost and low holding cost Problem
Model 1
Model 2
Gapi (%)
Gap* (%)
Gapi (%)
Model 3 Gap* (%)
Model 4
Gapi (%)
Gap*
Gapi (%)
Gap* (%)
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
1 2 3 4 5 6 7 8 9 10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.5 2.3 1.9 2.6 2.3 2.1 2.4 2.1 2.2 2.4
89.6 110.3 73.9 83.7 98.6 90.7 118.9 159.7 64.2 84.0
1.9 0.4 3.9 0.9 1.7 0.0 1.2 2.1 4.7 3.6
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
1.3 1.1 0.5 0.9 0.6 0.5 0.5 0.6 1.2 0.9
6.8 5.7 2.8 5.2 3.8 3.8 3.4 2.7 7.4 5.5
0.0 0.0 0.0 0.3 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
1.8 1.8 1.7 2.3 2.2 1.9 2.2 1.9 1.6 2.1
86.2 90.9 65.6 74.2 96.0 89.3 116.5 152.2 64.8 82.4
1.9 0.4 3.9 0.9 1.7 0.0 1.2 2.1 4.7 3.6
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
1.7 1.9 0.3 1.1 1.5 0.3 1.3 0.4 2.4 1.6
0.0% 0.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Min Avg Max
0.0 0.0 0.0
1.9 2.3 2.6
64.2 97.4 159.7
0.0 2.1 4.7
0.0 0.0 0.0
0.5 0.8 1.3
2.7 4.7 7.4
0.0 0.0 0.3
0.0 0.0 0.0
1.6 2.0 2.3
64.8 91.8 152.2
0.0 2.1 4.7
0.0 0.0 0.0
0.0 0.0 0.1
0.3 1.2 2.4
0.0 0.0 0.3
Table 4 Problem type 3: low setup cost, high penalty cost and high holding cost Problem
Model 1
Model 2 *
Gapi (%)
Gap (%)
Gapi (%)
Model 3 *
Gap (%)
Gapi (%)
Model 4 *
Gap (%)
Gapi (%)
Gap* (%)
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
1 2 3 4 5 6 7 8 9 10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.4 2.9 2.9 2.9 3.0 2.5 3.5 2.7 3.8 2.3
130.9 100.3 82.6 94.6 164.9 58.2 130.1 49.5 82.4 131.3
0.3 1.3 3.6 1.8 1.0 3.2 1.1 7.4 0.0 0.4
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.7 1.8 2.0 1.8 1.8 1.6 2.6 1.8 2.7 1.1
2.6 16.8 8.9 8.5 10.7 10.9 11.3 5.6 11.2 5.4
0.0 0.9 0.0 0.4 0.0 0.0 1.1 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.0 1.9 1.7 1.9 1.9 1.6 1.8 1.5 2.4 1.8
102.2 99.5 62.8 78.4 123.0 47.8 91.4 43.7 78.1 107.5
0.3 1.3 0.0 1.8 1.0 0.0 1.1 7.4 0.0 0.4
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.0
0.9 7.2 3.9 2.6 3.1 2.1 9.3 1.0 7.5 1.2
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0
Min Avg Max
0.0 0.0 0.0
2.3 2.9 3.8
49.5 102.5 164.9
0.0 2.0 7.4
0.0 0.0 0.0
0.7 1.8 2.7
2.6 9.2 16.8
0.0 0.2 1.1
0.0 0.0 0.0
1.5 1.8 2.4
43.7 83.4 123.0
0.0 1.3 7.4
0.0 0.0 0.0
0.0 0.1 0.1
0.9 3.9 9.3
0.0 0.0 0.1
more manageable UFLP when it is approximated by piecewise linear programming. Thus model 2 may be selected because of the simplicity of problem. (4) Table 8 shows experiment results for large size problems. In this experiment, we set the number of store 100 and A, h, p are randomly generated from U(1, 10000), U(1, 100), U(1, 100), respectively. In this
large size problem, total possible regression data is 2100. Therefore we cannot calculate Gap* since it is not possible to find an optimal solution. In addition, Gapj cannot be calculated for all possible regression data, thus almost 2000 random samples are selected from 2100. Table 8 also indicates that model 4 outperforms other models with maximum Gapj = 2.3%.
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D.-J. Lee, I.-J. Jeong / Computers & Industrial Engineering 56 (2009) 1169–1176 Table 5 Problem type 4: high setup cost, low penalty cost and low holding cost Problem
Model 1
Model 2 *
Gapi (%)
Gap (%)
Gapi
Model 3 Gap
*
Model 4 *
Gapi (%)
Gap (%)
Gapi (%)
Gap* (%)
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
1 2 3 4 5 6 7 8 9 10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.1 2.1 2.4 2.2 2.4 2.5 2.5 2.2 2.1 2.2
137.8 87.4 138.0 84.0 88.3 144.1 92.0 72.8 104.5 94.1
5.1 3.3 0.0 2.3 0.0 0.0 2.7 3.7 1.3 3.7
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.6 0.1 0.8 0.6 0.6 0.5 0.3 0.2 0.4 0.2
1.8 0.7 2.3 2.4 2.5 1.9 1.1 0.7 0.9 1.1
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.0 2.0 2.2 2.1 2.2 2.3 2.4 2.1 2.0 2.2
126.6 84.7 135.6 83.7 85.9 131.1 84.3 68.3 96.6 94.9
5.1 3.3 0.0 2.3 0.0 0.0 2.7 3.7 1.3 3.7
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.1 0.0 0.1 0.1 0.0 0.0 0.0 0.0
0.7 0.3 3.7 2.3 1.6 2.1 0.9 0.0 0.8 0.8
0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Min Avg Max
0.0 0.0 0.0
2.1 2.3 2.5
72.8 104.3 144.1
0.0 2.2 5.1
0.0 0.0 0.0
0.1 0.4 0.8
0.7 1.5 2.5
0.0 0.0 0.0
0.0 0.0 0.0
2.0 2.1 2.4
68.3 99.2 135.6
0.0 2.2 5.1
0.0 0.0 0.0
0.0 0.0 0.1
0.0 1.3 3.7
0.0 0.0 0.1
Table 6 Problem type 5: high setup cost, high penalty cost and low holding cost Problem
Model 1
Model 2 Gap* (%)
Gapi (%)
Model 3 Gap* (%)
Gapi (%)
Model 4 Gap* (%)
Gapi (%)
Gap* (%)
Gapi (%)
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
1 2 3 4 5 6 7 8 9 10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.3 1.8 2.1 1.9 2.8 2.2 2.2 1.9 2.4 2.0
121.2 46.8 89.8 68.5 131.9 106.4 73.8 60.0 104.8 136.4
0.0 1.9 6.6 0.0 1.6 0.2 0.9 0.8 1.8 1.1
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.7 0.6 0.2 0.6 0.9 0.3 0.6 0.4 1.0 0.5
3.1 3.4 1.0 2.6 4.0 1.6 2.4 1.8 4.9 2.0
0.0 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.2 1.6 2.0 1.7 2.6 2.1 2.0 1.7 2.0 1.8
117.6 45.4 84.4 55.8 129.1 105.6 68.8 40.9 91.8 120.8
0.0 1.9 2.8 0.0 1.6 0.2 0.9 0.8 1.8 1.1
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.1 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0
2.0 0.4 0.1 0.8 2.3 0.9 1.6 0.7 1.0 0.6
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Min Avg Max
0.0 0.0 0.0
1.8 2.2 2.8
46.8 94.0 136.4
0.0 1.5 6.6
0.0 0.0 0.0
0.2 0.6 1.0
1.0 2.7 4.9
0.0 0.0 0.5
0.0 0.0 0.0
1.6 2.0 2.6
40.9 86.0 129.1
0.0 1.1 2.8
0.0 0.0 0.0
0.0 0.0 0.1
0.1 1.0 2.3
0.0 0.0 0.0
Table 7 Problem type 6: high setup cost, high penalty cost and high holding cost Problem
Model 1
Model 2 Gap* (%)
Gapi (%) Min
Avg
Max
1 2 3 4 5 6 7 8 9 10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.3 3.0 2.3 2.6 2.0 2.6 2.3 2.7 2.5 2.2
139.5 117.6 115.4 113.5 72.4 100.2 98.2 97.1 124.2 119.5
Min Avg Max
0.0 0.0 0.0
2.0 2.5 3.0
72.4 109.7 139.5
Model 3 Gap* (%)
Gapi (%) Min
Avg
Max
0.9 0.0 0.9 0.0 0.5 1.1 2.5 0.7 3.1 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.7 1.1 1.0 0.9 0.6 1.1 1.0 0.9 0.9 0.7
1.8 8.2 6.6 3.1 5.2 5.9 3.7 3.1 4.3 3.9
0.0 1.0 3.1
0.0 0.0 0.0
0.6 0.9 1.1
1.8 4.6 8.2
5. Conclusion In this paper we consider the regression approximation of the warehouse cost in partially centralized inventory systems considering transportation costs. The regression approximation of the cost function transforms the original problem into a well-known UFLP. Experimental results show that nonlinear
Model 4 Gap*
Gapi (%) Min
Avg
Max
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.0 2.6 1.9 2.3 1.7 2.1 1.9 2.4 2.3 2.0
118.7 106.4 107.8 102.1 64.6 73.5 91.8 100.3 126.5 115.0
0.0 0.0 0.0
0.0 0.0 0.0
1.7 2.1 2.6
64.6 100.7 126.5
Gap* (%)
Gapi (%) Min
Avg
Max
0.9 0.0 0.9 0.0 0.5 1.1 2.5 0.4 0.8 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.1 0.1 0.1 0.1 0.0 0.1 0.0 0.1 0.1 0.1
1.2 5.9 1.6 1.6 1.0 1.8 0.9 4.0 5.4 3.6
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.7 2.5
0.0 0.0 0.0
0.0 0.1 0.1
0.9 2.7 5.9
0.0 0.0 0.0
regression model of lead time demand and variance provides good approximation of the original cost functions. This paper can be extended in many directions. In terms of solution procedure, we may apply meta heuristics in order to directly solve the PCIS problems. Also the problem with multiple items or the extension to multi-echelon problem would be interesting.
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Table 8 Large size problem Problem
Model 1
Model 2
Gapi (%)
Model 3
Gapi (%)
Model 4
Gapi (%)
Gapi (%)
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
1 2 3 4 5 6 7 8 9 10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
9.8 10.3 9.9 10.5 10.3 9.9 10.5 9.9 10.0 10.2
543.8 441.5 457.6 464.4 457.4 378.1 455.2 348.8 415.8 473.9
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.1 0.5 0.7 2.0 0.4 0.4 0.5 1.7 0.5 0.3
0.2 2.2 5.4 31.9 1.9 2.1 1.8 9.6 3.4 1.1
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
5.2 4.6 4.7 3.1 5.0 4.2 4.8 3.3 4.7 4.6
349.4 186.7 147.5 104.3 228.6 181.7 101.1 64.5 218.9 231.4
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.1 0.1 0.2 0.1 0.1 0.1 0.2 0.1 0.1
1.1 5.7 5.0 9.3 6.4 5.3 6.0 12.3 5.3 6.7
Min Avg Max
0.0 0.0 0.0
9.8 10.1 10.5
348.8 443.7 543.8
0.0 0.0 0.0
0.1 0.7 2.0
0.2 6.0 31.9
0.0 0.0 0.0
3.1 4.4 5.2
64.5 181.4 349.4
0.0 0.0 0.0
0.0 0.1 0.2
1.1 6.3 12.3
References Beasley, J. E. (1993). Lagrangean heuristics for location problems. European Journal of Operational Research, 65, 383–399. Das, C. (1985). On the solution of some approximate continuous review inventory models. Naval Research Logistics Quarterly, 32, 301–313. Das, C., & Tyagi, R. (1997). Role of inventory and transportation costs in determining the optimal degree of centralization. Transportation Research, 33(3), 171–179. Daskin, M. S. (1995). Network and discrete location, models, algorithms, and applications. New York: Wiley. Erlenkotter, D. (1978). A dual-based procedure for uncapacitated facility location. Operations Research, 26, 992–1009. Gerchak, Y., & Gupta (1991). On apportioning costs to customers in centralized continuous review inventory systems. Journal of Operations Management, 10, 546–551. Hajiaghayi, M. T., Mahdian, M., & Mirrokni, V. S. (2003). The facility location problem with general cost functions. Networks, 42(1), 42–47. Harkness, J., & Revelle, C. (2003). Facility location with increasing production costs. European Journal of Operational Research, 145(1), 1–13. Hartman, B. C., & Dror, M. (1996). Cost allocation in continuous-review inventory models. Naval Research Logistics, 43, 549–561. Hartman, B. C., & Dror, M. (2003). Optimizing centralized inventory operations in a cooperative game theory setting. IIE Transactions, 35, 243–257. Hartman, B. C., Dror, M., & Shaked, M. (2000). Cores of inventory centralization games. Games and Economic Behavior, 31, 26–49.
Holmberg, K. (1999). Exact solution methods for uncapacitated location problems with convex transportation costs. European Journal of Operational Research, 14, 127–140. Johnson, A., & Montgomery, D. C. (1974). Operations research in production planning, scheduling, and inventory control. New York: Wiley. Krarup, J., & Pruzan, P. M. (1983). The simple plant location problem: Survey and synthesis. European Journal of Operational Research, 12, 36–81. Miranda, P. A., & Garrido, R. A. (2004). Incorporating inventory control decisions into a strategic distribution network design model with stochastic demand. Transportation Research Part E, 40, 183–207. Robinson, L. W. (1993). A comment on Gerchak and Gupta’s ‘on apportioning costs to customers in centralized continuous review inventory systems’. Journal of Operations Management, 11, 99–102. Simchi-Levi, D., Kaminsky, P., & Simchi-Levi, E. (2003). Designing & managing the supply chain: Concepts, strategies & case studies (2nd ed.). McGraw-Hill. Sourirajan, K., Ozsen, L., & Uzsoy, R. (2007). A single-product network design model with lead time and safety stock considerations. IIE Transactions, 39, 411–424. Sun, M. (2006). Solving the uncapacitated facility location problem using tabu search. Computers & Operations Research, 33(9), 1589–2563. Teo, C. P., Ou, J., & Goh, M. (2001). Impact on inventory costs with consolidation of distribution centers. IIE Transactions, 33, 99–110. Wu, L. Y., Zhang, X. S., & Zhang, J. L. (2006). Capacitated facility location problem with general setup cost. European Journal of Operational Research, 33, 1226–1241.
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Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Regression approximation for a partially centralized inventory system considering transportation costs Dong-Ju Lee a, In-Jae Jeong b,* a b
Department of Industrial & Systems Engineering at Kongju National University, Kongju, Korea 182 Shinkwan-dong, Kongju, Chungnam 314-701, Republic of Korea Department of Industrial Engineering, Hanyang University, Seoul, Korea 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Republic of Korea
a r t i c l e
i n f o
Article history: Received 2 May 2007 Received in revised form 17 December 2007 Accepted 9 June 2008 Available online 14 June 2008 Keywords: Inventory centralization Transportation cost Regression approximation
a b s t r a c t Inventory centralization for multiple stores with stochastic demands reduces costs by establishing and maintaining a central ordering/distribution point. However the inventory centralization may increase the transportation costs since either the customer must travel more to reach the product, or the central warehouse must ship the product over longer distance to reach the customer. In this paper, we study a partially centralized inventory system where multiple central warehouses exist and a central warehouse fulfills the aggregated demand of stores. We want to determine the number, the location of central warehouses and an assignment of central warehouses and a set of stores. The objective is the minimization of the sum of warehouse costs and transportation cost. With the help of the regression approximation of cost function, we transform the original problem to more manageable facility location problems. Regression analysis shows that the approximated cost function is close to the original one for normally distributed demands. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Consider the continuous review order quantity-reorder point (Q, r) inventory system. We have a set of stores which establish their own warehouses and maintain inventory to satisfy customers’ stochastic demand. Store fills customer orders and distributes the product directly to customers which is a Completely Decentralized Inventory System (CDIS). In a Completely Centralized Inventory System (CCIS), instead of running individual stores, a single central warehouse fulfills all the demands occurring in stores. This is termed as, inventory centralization. By running a central warehouse, we can reduce the variability of demands by compensating the underestimated demand of one store with the overestimated demand of another store. Eventually, the reduction of the variability of demands eliminates the unnecessary safety stock and this minimizes the inventory holding cost. The cost reduction by inventory centralization is theoretically proved by Gerchak and Gupta (1991). Therefore in this research line, researchers have focused on the fair allocation of benefits of inventory centralization to stores such that no store will have an incentive to run individual warehouse. Hartman, Dror, and Shaked (2000) and Hartman and Dror (2003) study the benefit allocation of inventory centralization using game theory. However, as Simchi-Levi, Kaminsky, and Simchi-Levi (2003) mentioned, there exist tradeoffs between inventory * Corresponding author. Tel.: +82 2 2220 0412. E-mail addresses: [email protected] (D.-J. Lee), [email protected] (I.-J. Jeong). 0360-8352/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.06.005
costs and transportation costs in inventory centralization. The inventory centralization may incur additional transportation costs since the distribution point has been changed from local stores to central warehouses. Meanwhile the inventory cost can be decreased as the inventory is centralized due to risk-pooling and economies of scale. This paper consider this kind of Partially Centralized Inventory System (PCIS) considering transportation cost where decision variables are the number, the location of central warehouses and the assignment of central warehouses under assumption that each store and warehouse apply (Q, r) inventory policy. The objective of the problem is the minimization of the sum of warehouse costs (i.e., ordering cost, inventory cost and shortage cost) and transportation cost. A few researches on PCIS considering transportation cost have focused on (Q, r) with service level problem which is similar to (Q, r) but in this problem, reorder point is not a decision variable since it is determined by given service level. Service level is a probability that a demand or aggregate demand of stores is met during lead time. Thus the problem can be formulated as EOQ type model with transportation cost. Das and Tyagi (1997) are the first who consider service level in order to determine optimal degree of inventory centralization. Miranda and Garrido (2004) propose a Lagrangian relaxation based heuristics using subgradient method to solve the problem. Sourirajan, Ozsen, and Uzsoy (2007) consider a two-stage supply chain with a production facility and multiple retailers. They incorporate the service level in the problem to calculate safety stock. Teo et al. (2001) study impact on warehouse investment cost and inventory costs when stores are consolidated into one warehouse. They consider sit-
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uations where demands follow nonidentical Poisson process and each store uses (Q, r) inventory policy but transportation costs are not included in their model. The warehouse cost function of PCIS is very complex due to the (Q, r) policy. We first seek to approximate the cost function using regression models that are function of mean and variance of lead time demand. With the help of the approximated cost function, we transform the original PCIS with transportation cost problem to a more manageable problem that is, uncapacitated facility location problem (UFLP). The objective of this research is not to improve or develop an efficient algorithm to solve UFLP but to appropriate the cost function of PCIS with regression model so that PCIS can be solved with existing algorithms for UFLP. This paper is organized as follows. The mathematical formulation of CDIS, CCIS and PCIS are described in Section 2. Proposed regression models will be explained in Section 3. Section 4 shows the regression analysis for the cost approximation. Finally, conclusion is given in Section 5.
Hartman and Dror (1996) and Robinson (1993) examined the fair allocation of joint inventory costs among the participating stores for CCIS. 2.2. Formulation of PCIS
Notation j
xij yj Sj {i | xij = 1} Qj, rj Zj
Dj ¼
2. Problem description In this section, we review traditional CDIS and the solution procedure to find optimal ordering policies. Also we explain the relationship between CDIS and CCIS when there is no transportation cost. Finally we presents PCIS problem with transportation cost considered in this paper. 2.1. Formulation of CDIS Consider N stores with single-item and annual demand. Let Di and Xi be the annual demand and random lead time demand at store i for the period. Xi is a random variable with probability density function gi(Xi) and cumulative distribution function Gi(Xi) where EðX i Þ ¼ li . Also suppose that each store has an ordering cost A, unit inventory holding cost h and unit shortage cost p. This assumption implies that cost parameters will not change drastically depending on where the customers are served from. This is especially true when stores are located nearby in a country. In a CDIS, the expected cost at store i by replenishing items Qi per period when the inventory level reaches ri can be calculated as follows:
Di Q Di Ri ðr i Þ C i ðQ i ; ri Þ ¼ A þ h i þ r i li þ p Qi Qi 2
ð1Þ
R1 where Ri ðri Þ ¼ E½ðX i ri Þþ ¼ ri ðX i r i Þg i ðxi Þdxi and (Xi-ri)+ = max[0, Xi-ri]. The optimal ordering policy ðQ i ; ri Þ to minimize function (1) can be found by solving the following two equations iteratively until Q i and r i converge (Johnson & Montgomery, 1974).
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Di ðA þ pRi ðri ÞÞ ; h hQ i 1 Gi ðr i Þ ¼ : pDi Q i ¼
ð2Þ ð3Þ
From Eq. (3), it can be shown that p P h since 0 6 Gi ðr i Þ 6 1 and Q i 6 Di . Inventory centralization is to centralize inventory of several stores in order to take advantage of the risk-pooling. In CCIS, one central warehouse orders, stores and distributes the inventories of all individual stores jointly. Let G be the aggregated cumulative P lead time demand distribution with mean l ¼ i li and aggregated P annual demand D ¼ i Di . The optimal ordering policy ðQ ; r Þ for the CCIS can be found similar to Eqs. (1)–(3) using Q, D, R, r instead of Qi, Di, Ri, ri. For CCIS, Gerchak and Gupta (1991) proved that inventory centralization always lower the costs compared to CDIS.
potential central warehouse index j = 1,. . .,N 1; if store i is served from central warehouse j 0; otherwise 1; if storej is selected as a central warehouse location 0; otherwise
P
i2Sj Di
R ðr Þ ¼ E½ðZ j rj Þþ ¼ R j1 j rj ðZ j r j Þg j ðxÞdx u dij tij = udij
a set of stores served from central warehouse j order quantity, reorder point of central warehouse j serving stores i 2 Sj . an aggregated lead time demand with mean P lj ¼ i2Sj li and probability density function gj and cumulative density function Gj. aggregated annual demand for central warehouse j serving stores i 2 Sj . Expected number of units short of a central warehouse j serving stores i 2 Sj in a replenishment cycle. transportation cost per unit item per unit distance. distance from store i to central warehouse j. incremental transportation cost per unit item to customer by changing the distribution point from store i to a central warehouse j, where tii = 0.
PCIS problem is similar to CDIS and CCIS except that PCIS considers the transportation cost due to the change of distribution points. For example, if the inventory of a store i and store j is aggregated to a central warehouse j which is equivalent to the location of store j, we assume that either the customer for store i has to travel more to reach the product, or the central warehouse j has to ship the product over longer distance to reach the customer. Also we assume the incremental transportation cost is proportionate to distance between store i and j. Using the previously mentioned notations, we formulate PCIS as follows: Problem PCIS
P 80 9 1 ! P Di Rj ðr j Þxij = X< X X Q D x i ij j i i @A Ay j þ Min þh li xij þ p tij Di xij þ rj : ; Q 2 Q j j j i i ð4Þ
Subject to xij 6 yj ; 8i; 8j X x ¼ 1; 8i j ij
ð6Þ
8i; 8j 8j
ð7Þ ð8Þ
xij P 0;
ð5Þ
yj 2 f0;1g;
The objective of PCIS is the minimization of the warehouse cost and transportation cost as shown in (4). The warehouse cost consists of setup cost, inventory holding cost and shortage cost. Constraint (5) ensures that a store can not be served from a warehouse unless it is established. Each store must be served by exactly one warehouse by constraint (6). Constraint (7) and (8) ensure the integrality of the variables. In this problem, we want to determine the number of central warehouses and an efficient assignment of central warehouses and a set of stores so that it can minimize total cost. Also optimal order quantity-reorder points of central warehouses must be determined. We assume that the potential location of central warehouses exists among existing stores and all stores have the equal setup cost, holding cost and penalty cost. Let the warehouse cost of central warehouse j denotes
P cj ¼ A
i Di xij
Qj
X Qj þh li xij þ rj 2 i
!
P þp
i Di Rj ðr j Þxij
Qj
:
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Given cj for all j, PCIS can be formulated as a well-known UFLP as follows:
Min
X
cj yj þ
j
XX i
t ij Di xij
ð9Þ
j
s.t. constraints (5)–(8). Therefore a careful estimation of cj will transform PCIS to a more manageable UFLP. In the literature, there has been an attempt to approximate the warehouse cost function for CDIS. Das (1985) propose two strategies to approximate the loss integral Ri(ri) by an exponential function and a quadratic function. Reorder point and order quantity are obtained in a non-iterative way. In this paper, we consider PCIS with transportation cost and estimate the warehouse cost directly using the mean and the variance of lead time demand without calculating reorder point and order quantity. We will explain the approximation of cost function using regression models in the following sections. The solution of UFLP determines the assignment of stores to central warehouses (i.e., Sj is given for all j). Then the optimal ordering policy ðQ j ; rj Þ of the central warehouse j that minimizes cj can be found similar to Eqs. (2) and (3) using Qj, Dj, Rj, rj instead of Qi, Di, Ri, ri as follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 i2Sj Di ðA þ pRj ðr j ÞÞ Qj ¼ ; h hQ j : 1 Gj ðr j Þ ¼ P p i2Sj Di
Fig. 1. Scatter plot of aggregated mean and warehouse cost.
ð10Þ ð11Þ
3. Regression approximation Without loss of generality, we assume that the annual demand is proportionate to the mean of lead time demand, that is li where L is the lead time (week) for inventory replenDi ¼ 52 weeks L ishment. The total number of possible assignment of store to warehouse is 2N 1 since a store can either be included in the assignment or not. Let ak be the set of stores that are included in assignment k where k = 1,. . .,2N 1. For assignment ak the warehouse cost ck can be calculated as follows:
P
ck ¼ A
i2ak Di Q k
X Q k þh li þ r k 2 i2a k
!
P
þp
Fig. 2. Scatter plot of aggregated standard deviation and warehouse cost.
i2ak Di Rk ðr k Þ Qk
where Q k and rk can be found using i 2 ak instead of i 2 Sj in Eqs (10) P and (11). In order to figure out the relationship between i2ak li and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ck also P r2 and ck , we randomly generate an example problem i2ak i as follows: consider N = 10 problem where the lead time demand of a store is iid and normally distributed with L = 3 weeks. The mean li and standard deviation ri are randomly selected from U(100, 1000) and U(10, 100), respectively. ffiffiffiffiffiffiffiffiffiffiffiffi qP P Figs. 1 and 2 show the relationship of ( i2ak li ; ck Þandð i2ak r2i ck ), respectively, for all possible assignments, ak. It seems that the aggregated mean (standard deviation) of store has nonlinear (linear) relation with the warehouse cost. Therefore, we propose four different types regression models that are functions of mean and variance of lead time demands as follows:
Model 1 : cj ¼ a þ b
X
li xij
i
Model 2 : cj ¼ a
X i
Model 3 : cj ¼ a þ b1
!b
li xij X i
Model 4 : cj ¼ a
X i
ð12Þ ð13Þ
qX ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi li xij þ b2 r2 x i i ij
li xij
!b 1
qX ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ b2 r2 x i i ij
where a, b, b1 and b2 are regression parameters. Regression model 1 and 2 consider only the mean of lead time demand meanwhile model 3 and 4 incorporate the variance of lead time demand into the model 1 and 2 for the estimation of the warehouse cost. In the following subsections, we will show that original PCIS problem can be transformed into well-known UFLP using the proposed regression models. 3.1. Regression model 1 Model 1 implies that if store j is selected as a central warehouse and Sj is given, the warehouse cost is approximated as P cj ¼ a þ b i2S li which is a linear function of aggregated demand j of stores that are served by warehouse j. Using the regression model 1, we approximate the objective function as follows:
Min
X
(
aþb
j
ð14Þ
¼
X j
ð15Þ
X i
ayj þ
!
li xij yj þ
X
) t ij Di xij
i
XX ðbli þ t ij Di Þxij i
j
Considering constraint (5) and (8), we can replace the nonlinear term, xijyj with xij in the objective function. Therefore we pro-
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pose an approximated mathematical model of PCIS problem as follows: Formulation 1 (F1)
Min
X
ayj þ
j
XX ðbli þ t ij Di Þxij i
ð16Þ
j
s.t. constraints (5)–(8). F1 is a special case of a well-known UFLP where the opening costs are the same for all warehouses. Once the problem has been approximated as UFLP, we can apply existing heuristics such as greedy heuristic (Daskin, 1995), dual approach (Erlenkotter, 1978), the Lagrangian method (Beasley, 1993) and tabu search (Sun, 2006).
as xij except that we have one more subscript k for dummy facilities. P Thus xij can be replaced with k xijk without any restriction. Define yjk as
8 > < 1 : if dummy facility k of facility j is served as central yjk ¼ warehouse > : 0 : otherwise
Then the objective function can be approximated as follows:
8 X< j
xijk ¼
1 : if store i is assigned to dummy facility k of facility j 0 : otherwise
such that store i must be assigned to exactly one dummy facility i.e., PP 8i. The new decision variable xijk is exactly the same j k xijk ¼ 1;
_
c j (e j )
¼
Min
fj2
Fig. 3. Piecewise linear approximation of function cj ðej Þ.
k
fjk yjk þ
j
k
i
j
k
i
j
k
ljk li þ t ij Di xijk
XXX ðljk li þ tij Di Þxijk i
k
j
ð17Þ
k
ð18Þ ð19Þ
k
8i; 8j; 8k 8j; 8k
xijk P 0;
ð20Þ
yjk 2 f0; 1g;
ð21Þ
It is important to note that the formulation of F2 is exactly the same as that of UFLP other than the subscript k to differentiate dummy facilities at each facility. Constraint (18) ensures that store i cannot be assigned to dummy facility k unless it is opened. Constraint (19) implies that a store must be assigned to exactly one dummy facility. The formulation does not restrict solutions such that multiple dummy facilities are selected at one facility. However this constraint is not necessary since only single dummy facility whose cost is located on the lower envelop of segments will be selected (Krarup and Pruzan (1983)). 3.3. Regression model 3 Regression model 3 approximates the aggregated warehouse qP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 cost as cj ¼ a þ b1 i lii xij þ b2 i ri xij which is a function of aggregated mean and variance of lead time demand that are served by warehouse j. Similar to model 2, the nonlinear term qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 ffi b2 i ri xij in regression model can be approximated as a piecepffiffiffiffi wise linear function. Define a function cj ðej Þ ¼ b2 ej where P 2 ej ¼ i ri xij . cj ðej Þ is approximated with straight lines where kth line has y-axis intercept gjk with slope mjk. Using the same definition of xijk and yjk as model 2, we can transform the objective function as follows:
(
a þ b1 XX j
þ
ej
fjk yjk þ
i
X X X
s:t: xijk 6 yjk ; 8i; 8j; 8k X X xijk ¼ 1; 8i j
ffi
lj2
lj1
j
j
þ
fj1=0
k
XX
j
lj3
j
XX
The approximated mathematical model can be described as follows:Formulation 2 (F2)
X
fj3
9 =
X
a li xij yj þ tij Di xij : ; i i XX XXX XXX ffi fjk yjk þ ljk li xijk þ tij Di xijk
3.2. Regression model 2 Model 2 approximates the cost function as a nonlinear funcP tion of mean of lead time demand as cj ¼ að i2Sj li Þb . There are some researches which attempt to solve UFLP with nonlinear cost function. Holmberg (1999) incorporate the convex transportation cost to the objective function of UFLP. Two methods such as a modified Bender’s decomposition method and a branch and bound method with dual ascent are applied. Wu et al. (2006) consider the general opening cost and apply a Lagrangian heuristic algorithm. Hajiaghayi et al. (2003) develop a error bound guaranteeing heuristic for a concave opening cost function. Harkness and ReVelle (2003) consider the convex production cost function in the problem. A branch and bound approach with linear programming is applied to solve the problem. Krarup and Pruzan (1983) propose the piecewise linear approximation to solve concave opening cost function. Using Karup and Pruzan’s approximation method and regression model 2, we will show that the problem can also be transformed to UFLP similar to model 1. P P b Define a function cj ðej Þ ¼ að i li xij Þ where ej ¼ i li xij . The nonlinear function cj ðej Þ and the piecewise linear functions are depicted in Fig. 3. In Fig. 3, cj ðej Þ is approximated with three straight lines where kth line has y-axis intercept fjk with slope ljk with fjk < fjk+1 and ljk > ljk+1. As Krarup and Pruzan (1983) have mentioned, we can regard fjk as a cost of opening a dummy facility k of facility j. Total cost for warehouse j is given by the lower envelop of the line segments. Define xijk as
!b
X
X
li xij þ b2
i
i
ayjk þ b1
k
XXX i
j
k
i
j
k
! ) rX ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X r2i xij yj þ tij Di xij
XXX i
j
mjk r2i xijk þ
i
li xijk þ
j
k
XXX i
j
k
XX k
t ij Di xijk ¼
X X X b1 li þ mjk r2i þ t ij Di xijk
g jk yjk XX ða þ g jk Þyjk j
k
Therefore we can formulate PCIS as another UFLP as follows: Formulation 3 (F3)
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XX XXX Min ða þ g jk Þyjk þ ðb1 li þ mjk r2i þ t ij Di Þxijk j
i
k
j
generated from U(100, 1000) and U(10, 100), respectively. Let the lead time, L = 3 weeks then the annual demand is determined as follows:
k
s.t. constraints (18)–(21).
Di ¼
3.4. Regression model 4 Similar reasoning can be applied to regression model 4. The cost qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P P 2 function can be approximated as cj ¼ að i2Sj li Þb1 þ b2 i2Sj ri . Using the same definition of model 2 and model 3, we approximate the nonlinear function of mean (standard deviation) with kth straight line has y-axis intercept fjk(gjk) with slope ljk(mjk). Let yjk(ujk ) and xijk(vijk ) denote the decision variables for the approximation of the nonlinear function of mean (standard deviation). Then the objective function value can be described as follows:
80 9 1 !b 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = X X< X X @a li xij þ b2 r2i xij Ayj þ tij Di xij i : ; j i i XX XXX XX ffi fjk yjk þ ljk li xijk þ g jk ujk j
þ
i
k
XXX i
j
k
i
j
k
j
mjk r2i vijk þ
j
k
XXX i
j
k
t ij Di xijk ¼
XX j
k
XXX XX þ ðljk li þ t ij Di Þxijk þ g jk ujk þ
XXX i
j
j
Gap ¼
k
mjk r2i vijk
k
XX j
þ
fjk yjk þ
i
j
XXX XX ðljk li þ t ij Di Þxijk þ g jk ujk i
j
mjk r
2 i vijk
k
XXX
k
j
k
k
s.t. constraints (18)–(21).
X
y k jk
X
x k ijk
¼
X
ujk ;
8j
ð22Þ
vijk ;
8i; 8j
ð23Þ
k
¼
X
Table 1 shows the six problem types that are considered in this paper. For experimental purposes, we consider low and high level of parameter, A, p and h. The low level of a parameter is generated from U($100, $5000) and the high level from U($5001, $10000), respectively. For each problem type, we randomly generate 10 problems and apply four regression models to determine the most appropriate approximation of warehouse cost function. Let Opt be the optimal objective value obtained by total enumeration of all possible solutions of the original PCIS problem. Also let App* be the optimal objective value obtained using the approximated cost function. In order to measure the quality of the approximation of cost using regression models, we propose the percent gap from the approximated cost to the original cost as follows:
App Opt 100 Opt maxðcj ; cj Þ minðcj ; cj Þ 100 Gapj ¼ maxðcj ; cj Þ
fjk yjk
k
Using the above objective function, we can formulate the problem as follows: Formulation 4 (F4)
Min
k
8i; 8j; 8k 8j; 8k
ujk 2 f0; 1g;
4. Regression analysis We consider N = 10 problem where X-coordinate and Y-coordinate of stores are randomly generated from U(0, 500 km) and the distance between two store are measured by Euclidean distance. We assume that u = $0.03 per item per km and lead time demands are iid and normally distributed with li and ri that are
ð25Þ
(1) In terms of performance measures, model 4 outperforms other models with maximum Gapj = 9.3%, and maximum Gap* = 0.3% for all types of problems. (2) Incorporating the nonlinear term of the mean of lead time demand is important in estimating warehouse cost since model 2, 4 are better than model 1, 3. It is interesting to see that Miranda and Garrido (2004) report similar results for PCIS problem with service level consideration. In their paper, they represent the warehouse cost as a nonlinear function of mean and standard deviation of lead time demand. (3) Model 4 shows better performance compared to model 2 since model 2 shows maximum Gapj = 31.6%, and maximum Gap* = 1.1%. However model 2 leads to a
k
Constraint (22) and (23) ensure that once store j is selected as a warehouse the nonlinear term of mean and standard deviation must be incorporated into the objective function simultaneously. We may apply Lagrangian Relaxation method by relaxing constraint (22) and (23) so that F4 can be decomposed into two independent UFLP problems. We do not investigate the solution procedure for F4 in depth since developing an efficient algorithm is not our purpose in this research. Our concern is to determine appropriate regression model for the approximation of the original problem.
ð24Þ
Gap* measures the quality of the approximation for the optimal solution of the original problem and Gapj measures the percent deviation of estimated cost from original cost. The results are shown in Tables 2–7. Minimum, average and maximum of Gapj are shown in tables along with Gap* for each problem. In order to determine regression parameters, a, b, b1 and b2, we use statistical software SAS 8.1. Especially, we use PROC REG procedure of SAS for model 1, 2 and 3, PROC NLIN for model 4. PROC NLIN applies the Gauss–Newton method to determine the nonlinear regression estimates. We run the experiments on personal computer with Intel Core2 Duo 1.83 MHz. A review of the results obtained for the six problem types suggests the following conclusions:
vijk 6 ujk ; 8i; 8j; 8k X X vijk ¼ 1; 8i j vijk P 0;
52 weeks l: 3 weeks i
Table 1 Problem types Problem type
A
p
h
1 2 3 4 5 6
U($100, $5000) U($100, $5000) U($100, $5000) U($5001, $10,000) U($5001, $10,000) U($5001, $10,000)
U($1, $50) U($51, $100) U($51, $100) U($1, $50) U($51, $100) U($51, $100)
U($1, $50) U($1, $50) U($51, $100) U($1, $50) U($1, $50) U($51, $100)
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Table 2 Problem type 1: low setup cost, low penalty cost and low holding cost Problem
Model 1
Model 2 *
Gapi (%)
Gap (%)
Gapi (%)
Model 3 *
Gap (%)
Model 4 *
Gapi
Gap (%)
Gap* (%)
Gapi (%)
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
1 2 3 4 5 6 7 8 9 10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
1.9 2.6 1.9 2.4 2.8 2.9 3.7 1.8 2.0 3.3
75.1 66.1 90.7 122.9 77.0 138.3 80.2 64.6 62.9 115.5
0.0 0.0 6.4 1.6 3.5 0.2 5.9 0.0 7.1 7.3
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
7.2 7.5 7.7 8.8 7.5 9.2 7.9 6.7 6.6 9.3
19.5 21.6 21.1 28.2 24.6 29.3 27.0 18.6 18.4 31.6
0.0 0.0 0.0 0.2 0.0 0.5 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
1.8 2.2 1.9 2.0 1.8 2.0 1.6 1.7 1.7 1.9
68.0 60.2 87.9 126.2 73.9 125.8 54.8 65.4 65.1 62.5
0.0 0.0 6.4 1.3 0.0 0.2 5.9 0.0 4.6 2.4
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.1 0.0 0.1 0.1 0.1 0.1 0.0 0.0 0.1
0.2 1.7 0.2 7.7 3.8 2.4 3.8 0.4 1.8 4.2
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Min Avg Max
0.0 0.0 0.0
1.8 2.5 3.7
62.9 89.3 138.3
0.0 3.2 7.3
0.0 0.0 0.0
6.6 7.8 9.3
18.4 24.0 31.6
0.0 0.1 0.5
0.0 0.0 0.0
1.6 1.9 2.2
54.8 79.0 126.2
0.0 2.1 6.4
0.0 0.0 0.0
0.0 0.1 0.1
0.2 2.6 7.7
0.0 0.0 0.0
Table 3 Problem type 2: low setup cost, high penalty cost and low holding cost Problem
Model 1
Model 2
Gapi (%)
Gap* (%)
Gapi (%)
Model 3 Gap* (%)
Model 4
Gapi (%)
Gap*
Gapi (%)
Gap* (%)
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
1 2 3 4 5 6 7 8 9 10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.5 2.3 1.9 2.6 2.3 2.1 2.4 2.1 2.2 2.4
89.6 110.3 73.9 83.7 98.6 90.7 118.9 159.7 64.2 84.0
1.9 0.4 3.9 0.9 1.7 0.0 1.2 2.1 4.7 3.6
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
1.3 1.1 0.5 0.9 0.6 0.5 0.5 0.6 1.2 0.9
6.8 5.7 2.8 5.2 3.8 3.8 3.4 2.7 7.4 5.5
0.0 0.0 0.0 0.3 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
1.8 1.8 1.7 2.3 2.2 1.9 2.2 1.9 1.6 2.1
86.2 90.9 65.6 74.2 96.0 89.3 116.5 152.2 64.8 82.4
1.9 0.4 3.9 0.9 1.7 0.0 1.2 2.1 4.7 3.6
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
1.7 1.9 0.3 1.1 1.5 0.3 1.3 0.4 2.4 1.6
0.0% 0.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Min Avg Max
0.0 0.0 0.0
1.9 2.3 2.6
64.2 97.4 159.7
0.0 2.1 4.7
0.0 0.0 0.0
0.5 0.8 1.3
2.7 4.7 7.4
0.0 0.0 0.3
0.0 0.0 0.0
1.6 2.0 2.3
64.8 91.8 152.2
0.0 2.1 4.7
0.0 0.0 0.0
0.0 0.0 0.1
0.3 1.2 2.4
0.0 0.0 0.3
Table 4 Problem type 3: low setup cost, high penalty cost and high holding cost Problem
Model 1
Model 2 *
Gapi (%)
Gap (%)
Gapi (%)
Model 3 *
Gap (%)
Gapi (%)
Model 4 *
Gap (%)
Gapi (%)
Gap* (%)
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
1 2 3 4 5 6 7 8 9 10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.4 2.9 2.9 2.9 3.0 2.5 3.5 2.7 3.8 2.3
130.9 100.3 82.6 94.6 164.9 58.2 130.1 49.5 82.4 131.3
0.3 1.3 3.6 1.8 1.0 3.2 1.1 7.4 0.0 0.4
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.7 1.8 2.0 1.8 1.8 1.6 2.6 1.8 2.7 1.1
2.6 16.8 8.9 8.5 10.7 10.9 11.3 5.6 11.2 5.4
0.0 0.9 0.0 0.4 0.0 0.0 1.1 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.0 1.9 1.7 1.9 1.9 1.6 1.8 1.5 2.4 1.8
102.2 99.5 62.8 78.4 123.0 47.8 91.4 43.7 78.1 107.5
0.3 1.3 0.0 1.8 1.0 0.0 1.1 7.4 0.0 0.4
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.0
0.9 7.2 3.9 2.6 3.1 2.1 9.3 1.0 7.5 1.2
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0
Min Avg Max
0.0 0.0 0.0
2.3 2.9 3.8
49.5 102.5 164.9
0.0 2.0 7.4
0.0 0.0 0.0
0.7 1.8 2.7
2.6 9.2 16.8
0.0 0.2 1.1
0.0 0.0 0.0
1.5 1.8 2.4
43.7 83.4 123.0
0.0 1.3 7.4
0.0 0.0 0.0
0.0 0.1 0.1
0.9 3.9 9.3
0.0 0.0 0.1
more manageable UFLP when it is approximated by piecewise linear programming. Thus model 2 may be selected because of the simplicity of problem. (4) Table 8 shows experiment results for large size problems. In this experiment, we set the number of store 100 and A, h, p are randomly generated from U(1, 10000), U(1, 100), U(1, 100), respectively. In this
large size problem, total possible regression data is 2100. Therefore we cannot calculate Gap* since it is not possible to find an optimal solution. In addition, Gapj cannot be calculated for all possible regression data, thus almost 2000 random samples are selected from 2100. Table 8 also indicates that model 4 outperforms other models with maximum Gapj = 2.3%.
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D.-J. Lee, I.-J. Jeong / Computers & Industrial Engineering 56 (2009) 1169–1176 Table 5 Problem type 4: high setup cost, low penalty cost and low holding cost Problem
Model 1
Model 2 *
Gapi (%)
Gap (%)
Gapi
Model 3 Gap
*
Model 4 *
Gapi (%)
Gap (%)
Gapi (%)
Gap* (%)
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
1 2 3 4 5 6 7 8 9 10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.1 2.1 2.4 2.2 2.4 2.5 2.5 2.2 2.1 2.2
137.8 87.4 138.0 84.0 88.3 144.1 92.0 72.8 104.5 94.1
5.1 3.3 0.0 2.3 0.0 0.0 2.7 3.7 1.3 3.7
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.6 0.1 0.8 0.6 0.6 0.5 0.3 0.2 0.4 0.2
1.8 0.7 2.3 2.4 2.5 1.9 1.1 0.7 0.9 1.1
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.0 2.0 2.2 2.1 2.2 2.3 2.4 2.1 2.0 2.2
126.6 84.7 135.6 83.7 85.9 131.1 84.3 68.3 96.6 94.9
5.1 3.3 0.0 2.3 0.0 0.0 2.7 3.7 1.3 3.7
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.1 0.0 0.1 0.1 0.0 0.0 0.0 0.0
0.7 0.3 3.7 2.3 1.6 2.1 0.9 0.0 0.8 0.8
0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Min Avg Max
0.0 0.0 0.0
2.1 2.3 2.5
72.8 104.3 144.1
0.0 2.2 5.1
0.0 0.0 0.0
0.1 0.4 0.8
0.7 1.5 2.5
0.0 0.0 0.0
0.0 0.0 0.0
2.0 2.1 2.4
68.3 99.2 135.6
0.0 2.2 5.1
0.0 0.0 0.0
0.0 0.0 0.1
0.0 1.3 3.7
0.0 0.0 0.1
Table 6 Problem type 5: high setup cost, high penalty cost and low holding cost Problem
Model 1
Model 2 Gap* (%)
Gapi (%)
Model 3 Gap* (%)
Gapi (%)
Model 4 Gap* (%)
Gapi (%)
Gap* (%)
Gapi (%)
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
1 2 3 4 5 6 7 8 9 10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.3 1.8 2.1 1.9 2.8 2.2 2.2 1.9 2.4 2.0
121.2 46.8 89.8 68.5 131.9 106.4 73.8 60.0 104.8 136.4
0.0 1.9 6.6 0.0 1.6 0.2 0.9 0.8 1.8 1.1
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.7 0.6 0.2 0.6 0.9 0.3 0.6 0.4 1.0 0.5
3.1 3.4 1.0 2.6 4.0 1.6 2.4 1.8 4.9 2.0
0.0 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.2 1.6 2.0 1.7 2.6 2.1 2.0 1.7 2.0 1.8
117.6 45.4 84.4 55.8 129.1 105.6 68.8 40.9 91.8 120.8
0.0 1.9 2.8 0.0 1.6 0.2 0.9 0.8 1.8 1.1
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.1 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0
2.0 0.4 0.1 0.8 2.3 0.9 1.6 0.7 1.0 0.6
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Min Avg Max
0.0 0.0 0.0
1.8 2.2 2.8
46.8 94.0 136.4
0.0 1.5 6.6
0.0 0.0 0.0
0.2 0.6 1.0
1.0 2.7 4.9
0.0 0.0 0.5
0.0 0.0 0.0
1.6 2.0 2.6
40.9 86.0 129.1
0.0 1.1 2.8
0.0 0.0 0.0
0.0 0.0 0.1
0.1 1.0 2.3
0.0 0.0 0.0
Table 7 Problem type 6: high setup cost, high penalty cost and high holding cost Problem
Model 1
Model 2 Gap* (%)
Gapi (%) Min
Avg
Max
1 2 3 4 5 6 7 8 9 10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.3 3.0 2.3 2.6 2.0 2.6 2.3 2.7 2.5 2.2
139.5 117.6 115.4 113.5 72.4 100.2 98.2 97.1 124.2 119.5
Min Avg Max
0.0 0.0 0.0
2.0 2.5 3.0
72.4 109.7 139.5
Model 3 Gap* (%)
Gapi (%) Min
Avg
Max
0.9 0.0 0.9 0.0 0.5 1.1 2.5 0.7 3.1 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.7 1.1 1.0 0.9 0.6 1.1 1.0 0.9 0.9 0.7
1.8 8.2 6.6 3.1 5.2 5.9 3.7 3.1 4.3 3.9
0.0 1.0 3.1
0.0 0.0 0.0
0.6 0.9 1.1
1.8 4.6 8.2
5. Conclusion In this paper we consider the regression approximation of the warehouse cost in partially centralized inventory systems considering transportation costs. The regression approximation of the cost function transforms the original problem into a well-known UFLP. Experimental results show that nonlinear
Model 4 Gap*
Gapi (%) Min
Avg
Max
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2.0 2.6 1.9 2.3 1.7 2.1 1.9 2.4 2.3 2.0
118.7 106.4 107.8 102.1 64.6 73.5 91.8 100.3 126.5 115.0
0.0 0.0 0.0
0.0 0.0 0.0
1.7 2.1 2.6
64.6 100.7 126.5
Gap* (%)
Gapi (%) Min
Avg
Max
0.9 0.0 0.9 0.0 0.5 1.1 2.5 0.4 0.8 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.1 0.1 0.1 0.1 0.0 0.1 0.0 0.1 0.1 0.1
1.2 5.9 1.6 1.6 1.0 1.8 0.9 4.0 5.4 3.6
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.7 2.5
0.0 0.0 0.0
0.0 0.1 0.1
0.9 2.7 5.9
0.0 0.0 0.0
regression model of lead time demand and variance provides good approximation of the original cost functions. This paper can be extended in many directions. In terms of solution procedure, we may apply meta heuristics in order to directly solve the PCIS problems. Also the problem with multiple items or the extension to multi-echelon problem would be interesting.
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Table 8 Large size problem Problem
Model 1
Model 2
Gapi (%)
Model 3
Gapi (%)
Model 4
Gapi (%)
Gapi (%)
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
Min
Avg
Max
1 2 3 4 5 6 7 8 9 10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
9.8 10.3 9.9 10.5 10.3 9.9 10.5 9.9 10.0 10.2
543.8 441.5 457.6 464.4 457.4 378.1 455.2 348.8 415.8 473.9
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.1 0.5 0.7 2.0 0.4 0.4 0.5 1.7 0.5 0.3
0.2 2.2 5.4 31.9 1.9 2.1 1.8 9.6 3.4 1.1
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
5.2 4.6 4.7 3.1 5.0 4.2 4.8 3.3 4.7 4.6
349.4 186.7 147.5 104.3 228.6 181.7 101.1 64.5 218.9 231.4
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.1 0.1 0.2 0.1 0.1 0.1 0.2 0.1 0.1
1.1 5.7 5.0 9.3 6.4 5.3 6.0 12.3 5.3 6.7
Min Avg Max
0.0 0.0 0.0
9.8 10.1 10.5
348.8 443.7 543.8
0.0 0.0 0.0
0.1 0.7 2.0
0.2 6.0 31.9
0.0 0.0 0.0
3.1 4.4 5.2
64.5 181.4 349.4
0.0 0.0 0.0
0.0 0.1 0.2
1.1 6.3 12.3
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