ORDERING AND WAREHOUSING STRATEGIES FOR MULTI-ITEM …€¦ · (1) Decentralized warehousing...

ORDERING AND WAREHOUSING STRATEGIES FOR MULTI-ITEM MULTI-BRANCH FIRM’S INVENTORY: CLUSTERING APPROACHES

Yu-Chiun CHIOU Assistant Professor Department of Traffic and Transportation Engineering and Management Feng Chia University 100, Wenhwa Rd. Seatwen, Taichung, Taiwan 40724 Fax: +886-4-2452-0678 Email: [email protected]

Lawrence W. LAN Professor Institute of Traffic and Transportation, National Chiao Tung University 4F, 114 Sec.1, Chung-Hsiao W. Rd. Taipei, Taiwan 10012 Fax: +886-2-2349-4953 Email: [email protected]

Abstract: This paper develops optimal inventory models for a firm with multiple items and multiple branches by taking into account the price discount of sizable procurement and the scale economies of centralized warehousing and consolidated transportation. Four inventory strategies are proposed: independent ordering and decentralized warehousing, independent ordering and centralized warehousing, joint ordering and decentralized warehousing, and joint ordering and centralized warehousing. The performances of pure policy (all items subject to adopting only one of the four strategies) and mixed policy (different items allowed to adopting different strategies) are compared. The mixed policy employs genetic stepwise clustering and statistical agglomerative clustering methods to classify items into appropriate clusters and then to determine the best strategy for each cluster. A firm of four branches with fifty heterogeneous items is analyzed. It is found that, in terms of total cost minimization (including ordering, warehousing, transportation and procurement costs), the mixed policy with clustering is superior to pure policy without clustering. For mixed policy, genetic stepwise clustering performs better than statistical agglomerative clustering. Key Words: centralized warehousing, genetic stepwise clustering, joint ordering, multi-item

multi-branch firm, statistical agglomerative clustering. 1. INTRODUCTION Appropriate inventory control plays an important role in both supply and demand chains management. The goal for inventory strategies is normally to maintain an optimal level of stockpiles in coping with the fluctuations of items for manufacturing or demands for products. Without well-designed inventory strategies, however, undesired excess or shortage of the items or products would happen, which could erode the firm’s profit. Therefore, numerous types of inventory control models have been developed over the past decades. The conventional economic ordering quantity (EOQ) model, one of the most commonly used inventory control methods, is developed mainly for single item (or product) and/or for a single firm (without branches). Taking the supply chain management as an example, due to the trend of globalization, more and more manufacturing firms nowadays have several branches in the same region, each of which requires various items in its production process. The conventional EOQ model can surely determine the optimal ordering of each item for each branch independently; however, from the whole firm’s perspective, such optimal ordering in general cannot minimize the overall total costs due to the fact that price discount for sizable procurement and scale

Journal of the Eastern Asia Society for Transportation Studies, Vol. 6, pp. 2809 - 2821, 2005

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economies for consolidated transportation and centralized warehousing do exist. In other words, it could be beneficial to such multi-item multi-branch firms by ordering a certain groups of items at a time and transporting them to appropriate warehousing points. In the multi-item ordering literature, Shu (1971) proposed an optimal ordering model for two items jointly replenished to minimize the total cost of set-up and inventory holding. Silver (1975) modified conventional EOQ model to determine the optimal coordinated replenishment of two or more items and indicated that if multiple items with different set-up times had to be jointly replenished, the average inventory cost and system control cost would be increased and the system flexibility would be lowered. Page and Paul (1976) constructed a multi-item EOQ model by synchronizing the ordering schedule and solved the optimal ordering quantity with Lagrange relaxation. Pirkul and Aras (1985) developed a multi-item EOQ model with consideration of volume discount and employed Lagrange relaxation technique to determine the ordering quantity. Güder and Zydiak (1999) developed a multi-item EOQ model with multiple resource constraints and also utilized Lagrange relaxation technique to solve the model with three types of ordering cycles -- independent cycle, fixed cycle and non-stationary cycle. Mondal and Maiti (2002) proposed multi-item fuzzy EOQ model and employed genetic algorithm to resolve the optimal ordering quantity. Most of the abovementioned works have been devoted to the EOQ modeling for multi-item ordering for a single firm without branches, very little has dealt with the integration of multi-item ordering for a firm with multiple branches by considering the scale economies of centralized warehousing and consolidated transportation as well as the price discount of joint procurement. In practice, the heterogeneous items can remarkably diversify across their prices, quantities demanded, volume sizes, etc. In other words, not all of the heterogeneous items are suitable for joint procurement or centralized warehousing or consolidated transporting. Therefore, previous EOQ modeling for multi-items ordering without appropriate clustering based on price, quantity demanded or volume size may not be practically useful. To fill up such gaps, this paper attempts to develop multi-item multi-branch firm’s EOQ models with appropriate clustering approaches to determine the optimal ordering and warehousing strategies. The core logic is to minimize the overall total costs, including ordering, warehousing, transportation and procurement costs, by simultaneously considering the scale economies of warehousing and transportation as well as the concession discount for large ordering. Two clustering methods, genetic stepwise clustering and statistic agglomerative clustering, are incorporated into the proposed EOQ models to determine the coordinated inventory strategies for the items in each cluster. The remaining parts of the paper are organized as follows. Section 2 identifies the ordering and warehousing alternatives, strategies, and policies as well as the total costs associated with different strategies. Section 3 introduces the statistical agglomerative clustering and genetic stepwise clustering methods. Section 4 carries out a case analysis for a firm of four branches with fifty items. Section 5 summarizes the findings and discusses the possible extensions.


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2. INVENTORY STRATEGIES 2.1 Definitions of Alternatives, Strategies and Policies This paper defines the ordering and warehousing alternatives for a multi-item multi-branch firm as follows: (1) Decentralized warehousing alternative: Each branch stores the items of material in its own warehouse. All items are shipped directly from the suppliers to the branch warehouse. (2) Centralized warehousing alternative: All branches store the items of material in a common warehouse center. All items are shipped from the suppliers to the warehouse center and then distributed to each branch on demand. Under this inventory alterative, a hub-spoke distribution network is formed. The trunk link is from suppliers to warehouse center and the feeder link connects the warehouse center with all branches. (3) Independent ordering alternative: Each branch orders the items basing on its own demand and the suppliers handle these orders independently. (4) Joint ordering alternative: The items are ordered by the firm’s headquarter in consideration of the demand of all branches and all items. Based on the combination of the aforementioned four ordering/warehousing alternatives, four inventory strategies are further identified below and Figure 1 depicts the concepts of these four inventory strategies. (1) Independent ordering and decentralized warehousing strategy, hereinafter denoted as S1 strategy. (2) Independent ordering and centralized warehousing strategy, denoted as S2 strategy. (3) Joint ordering and decentralized warehousing strategy, denoted as S3 strategy. (4) Joint ordering and centralized warehousing strategy, denoted as S4 strategy. Furthermore, this paper defines two different inventory policies as follows: (1) Pure policy: All items are subject to adopting only one of the abovementioned four inventory strategies. (2) Mixed policy: Different items are allowed to adopting different inventory strategies. In this paper, the mixed policy will classify the items into several clusters and then determine the best strategy for each cluster.


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: suppliers of items : branches : warehousing center : ordering flows of items : distribution flows of single item : distribution flows of multiple items

Figure 1. The Concepts of Four Inventory Strategies

2.2 Cost Formulation The total costs associated with different inventory strategies mainly comprise four parts: ordering (administrative) cost, inventory (holding) cost, delivery (transportation) cost, and procurement (items themselves) cost. The cost formulation for these four inventory strategies and their solving algorithms are addressed as follows. 2.2.1 Independent ordering and decentralized warehousing (S1 strategy) The total cost of S1 strategy can be expressed as follows:

( )∑ ∑∑∑ +=i j

λijiji

i j ij

rdvβr

αTC 1

211 ∑∑

+

i j

λijijiji

ij clrdv

rδ

211

( ) ( )∑ ∑ ++

−+

−

i jij

rdiji εrεgdp

λijij 2

211

21 3

(1)

where, TC: the total cost of inventory (dollars/year). rij: the decision variable representing the ordering ratio (per order quantity/annual quantity demanded) for ith item of jth branch. dij : the quantity demanded for ith item of jth branch (piece/year). di: total quantity demanded for ith item (piece/year). vi : the unit volume (length by width by height) for ith item (cm3/piece). lij : the distance from supplier of ith item to jth branch (kilometer). yij: the demand ratio for ith item of jth branch (yij=dij/di). pi: the procurement price of ith item. α: per ordering cost (dollars/order). β: the unit volume holding cost (dollars/cm3/year). δ1: the unit cost for transportation from the suppliers to each branch’s warehouse (dollars/truck-kilometer). ε: the interest rate. λ1: the coefficient of scale economy in warehousing (0<λ1<1). λ2: the coefficient of scale economy in transportation (0<λ2<1). λ3: the coefficient of price discount for sizable

(b) S2 strategy (a) S1 strategy

(c) S3 strategy (d) S4 strategy


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procurement (0<λ3<1). g: the coefficient of double exponential function (0＜g＜1). c: the capacity of a truck. Due to the complex nature of equation (1), it is difficult to derive a closed form of scale elasticities in warehousing, transportation and volume discount denoted by λ1, λ2 and λ3, respectively. The first term in Eq. (1) represents the ordering administrative cost, which is proportional to the number of orders per year. The second term stands for the inventory holding cost, which is proportional to the holding period of inventory. The third term represents the transportation cost. The fourth term represents the procurement cost. Exponential functions are assumed for the coefficients λ1, λ2 and λ3 to reflect the scale economies in warehousing, transportation and price discount. The optimal ordering ratio for ith item of jth branch can be obtained by the following first order conditions for all i and j:

( ) 112

11

21 −+−=

∂∂ λ

ijλ

ijiijij

rdvβλrα

rTC

( ) ( )

−+

−+ − 3

2

2

211

211 2

21

-λijij rd

ijiλ

ij

λijiji gdεpr

cldv

λδ

( ) ( )( ) ( ) 02ln41 1

32 3

3

=++− −ij

)(-λijij

*rdiji εrεrd-λ(g)gdp

λijij (2)

Theoretically, a total of n+m first order conditions (n is the number of items, m is the number of branches) must be solved simultaneously; however, the closed form solution is not obviously obtained. Therefore, we employ a quasi-Newton numerical method to solve the optimal ordering ratio. 2.2.2 Independent ordering and centralized warehousing (S2 strategy) The total cost of S2 strategy can be expressed as follows:

∑ ∑ ∑ ∑

+=

i j j

λ

iiji

λj

j

dvrβr

αTC1

1

211

∑∑∑∑

+

+

i j

λjjiji

ji j

λijiji

j clrdv

rδ

clrdv

rδ

22

02

01

11

( ) ( )∑ ∑ ++

−+

i jj

rdiji εrεgdp

-λjij 2

211

21 3 (3)

where, rj: the decision variable representing the ordering ratio of jth branch. For simplicity, each branch is assumed to determine its order independently but will synchronize the ordering for various items. δ2: the unit distribution cost from the warehouse center to individual branch (dollars/ truck-kilometer). li0: the distance from the supplier of ith item to the warehouse center (kilometer). l0j: the distance from warehouse center to jth branch (kilometer). The first, second and fifth terms of Eq. (3) are the same as Eq. (1). The third term is the trunk transportation cost, i.e. the transportation cost from the suppliers to the warehouse center. The fourth term is the distribution cost, i.e. the transportation cost from the warehouse center to branches. The optimal ordering ratio of jth branch can be determined by the following first order conditions for all j:

112

1

1

21 −

+−=

∂∂ ∑ λ

j

λ

iiji

jj

rdvβλrnα

rTC

( ) ( ) 2022

2021

2

2

2

2

11 −−

−+

−+ ∑∑ λ

j

λ

i

jijiλj

λ

i

iiji rcldv

λδrc

ldvλδ

( )

−+ ∑

i

rdiji

-λjijgdpε

3

211

21 ( )( ) ( ) ( )( ) 02ln

41 12

33

3

=

++− ∑ −

i

-λjij

rdijij rdgdpεrε-λ(g)

-λjij (4)


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A total of m first order conditions must be solved simultaneously; however, for similar reason, the quasi-Newton numerical method is employed. 2.2.3 Joint ordering and decentralized warehousing (S3 strategy) The total cost of S3 strategy can be expressed as follows:

( )∑∑ +=i

λii

i i

rdvβr

αTC 1

211 21

1

λ

j

ijiijii

i i clrydv

rδ ∑∑

+

( ) ( )iii

rdi εrεdgp

-λii ++

−+ ∑ 2

211

21 3 (5)

where, rj: the decision variable representing the ordering ratio of ith item. In this strategy, the same item demanded by all branches will be ordered jointly. The optimal ordering ratio of ith material can be obtained by solving the following first order conditions for all i:

( ) 112

11

21 −+−=

∂∂ λ

iλ

iiii

rdvβλrα

rTC ( ) ( )

−+

−+ −∑ 3

2

2

211

211 2

21

-λii rd

iiλ

i

λ

j

ijijii gεdprc

lydvλδ

( ) ( )( )( ) ( ) 02ln

41 1

32 33

=++− −i

-λii

rdii εrεrd-λ(g)gdp

-λii (6)

Similarly, a total of n first order conditions need to be solved simultaneously and the quasi-Newton numerical method is employed. 2.2.4 Joint ordering and centralized warehousing (S4 strategy) The total cost of S4 strategy can be expressed as follows:

1

211

λ

iii

irdvβ

rαTC

+= ∑∑

220

10

111

λ

j i

ijjiiλ

i

iii

cyrldv

rδ

crldv

rδ ∑ ∑∑

+

+

( ) ( )εrεdgp ii

rdi

-λi ++

−+ ∑ 2

211

21 3 (7)

where, r: the decision variable standing for the ordering ratio for all items and all branches. In this strategy, all items demanded by all branches are ordered jointly with the same ordering ratio. The optimal ordering ratio can be obtained by:

112

1

1

21 −

+−=

∂∂ ∑ λ

λ

iii rdvβλ

rnα

rTC

( ) ( ) 2022

2021

2

2

2

2

11 −− ∑ ∑∑

−+

−+ λ

j

λ

i

jijiiλλ

i

iii rc

lydvλδr

cldvλδ

( )

−+ ∑

i

rdii

-λigdpε 3

211

21 ( )( ) ( ) ( )( ) 02ln

41 12

333

=

++− ∑ −

i

-λi

rdii rdgdpεrε-λ(g)

-λi (8)

Although there is only one first order condition to be solved, yet the closed form solution for Eq. (8) is still very complicated; thus, the quasi-Newton numerical method is employed. 3. CLUSTERING ALGORITHMS As emphasized, the multi-item multi-branch firm frequently has various items that would


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remarkably diversify in price, quantity demanded, and volume size; not all of which are suitable for joint procurement, centralized warehousing or consolidated transporting. It is likely to be better off if one could cluster the items first and then apply the appropriate ordering and warehousing strategies. In the present paper, both agglomerative hierarchical clustering method and genetic stepwise clustering method (Chiou and Lan, 2001) are employed and their performances are compared. 3.1 Agglomerative Hierarchical Clustering Method Agglomerative hierarchical clustering method (AHCM), one of the most commonly used conventional statistical clustering algorithms, groups heterogeneous items into several mutually exclusive clusters in order to make as higher homogeneity of items within a cluster and as higher heterogeneity of items between clusters. The similarity between items is often defined as Euclidian distance, which can be expressed:

( )21

2

1,

−= ∑

=

p

ttt yxyxd

(9)

where, d(x,y): the Euclidian distance between item x and item y. xt: the tth attribute of item x. yt: the tth attribute of item y. p: the number of attributes. The AHCM algorithm is briefly depicted as follows (Chiou and Lan, 2001): Step1: Start with N clusters, each containing a single item, that is, Si

(1)={Oi}, i=1,…,N. An NN× symmetric matrix increments of an objective function MF={ ijF∆ , i,j=1,…,N,

ji≠ }, where ijF∆ represents the incremental of objective value in case that ith cluster and jth cluster are fused into a single cluster. Let k=1.

Step2: If vuF∆ = max{ ijF∆ , i,j=1,…,N-k, ji≠ } and v>u, then let )()()1( k

vk

uk SSS

u∪=+ , S1

(k+1)= S １

(k),…, Su-1(k+1)=Su-1

(k), Su+1(k+1)=Su+1

(k),…, Sv(k+1)=Sv+1

(k),…, SN-v-1(k+1)=SN-v

(k) and Φ=+

−

)1(kkN

S . Calculate the objective value F(X)(k) of the partition. Let k=k＋1. Step3: Repeat Step2 until k=N –1. F(X)*= max{F(X)(k), k=1,…,N-1}. 3.2 Genetic Stepwise Clustering Method Clustering is rapidly becoming computationally intractable as the problem scale increases because of the combinatorial character of the method. The conventional statistical clustering method, such as AHCM, divides items in a procedure approach, thus no good clustering results can be guaranteed. In addition, the attributes of items must be given in advance for proceeding to any statistical clustering method. To overcome this problem, numerous studies have been done by employing genetic algorithms (GAs) to perform the clustering more effectively (e.g., Murthy and Chowdhury, 1996; Tseng and Yang, 2001; Chiou and Lan, 2001). Chiou and Lan (2001) proposed and validated three methods of genetic clustering algorithms: simultaneously clustering method (SICM), stepwise clustering method (STCM), and cluster seed points method (CSPM), of which CSPM is proven to be the most effective, followed by STCM. However, because CSPM needs additional information, such as the Euclidian distance


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between cluster seed points and items, for assigning them into specific clusters, its applications are restrained. In contrast, STCM can perform quite satisfactorily and it does not require additional information to cluster. Thus, this paper utilizes the STCM to cluster the items. STCM successively solves the optimal binary clustering of a cluster until the objective value can no longer be improved. The best strategy and minimum total cost for each cluster would be determined. The objective value can be simply calculated by summing up the minimum total cost of each cluster. A cluster is fathomed when it cannot be further binary clustered to improve the objective value. The framework of STCM is depicted in Figure 2 and its algorithm is briefly depicted as follows (Chiou and Lan, 2001):

Stage 0

Stage 1

Stage 2

Stage k

},1{ )0()0()1( * SOXOSS iii ∈===

},...,1{)0( NiOSS i ===

},0{ )0()0( * SOXOS iii ∈==

},1{ )1()1()2( * SOXOSS iii ∈===},0{ )1()1( * SOXOS iii ∈==

}

,1{)(

)( *

k

k

SO

XOS

i

ii

∈

==Φ=)(k

S

•••

•••

GAs

Determine the best inventory alternativeTotal cost

Summation

Figure 2. Framework of Genetic Stepwise Clustering Method Source: Chiou and Lan (2001)

Step 0: Let S(0) stand for the cluster containing all objects. The problem of optimally

dividing S(0) into two subgroups, namely )0(S and )0(

S , can be formulated as the following 0-1 mathematical programming:

MP (0): Max F(X) (0) (10)

Subject to Xi={0,1} )0(,...,1 Si = (11)

where Xi =1 denotes that ith object of S(0) is grouped in the cluster )0(S , Xi=0

denotes that ith object of S(0) is grouped in the cluster )0(

S . Xi is encoded as the gene of chromosomes (the length of chromosomes is )0(S ), then GAs are

employed to solve MP(0) by maximizing Ｆ(X)(0) to attain the optimal binary


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clustering: },1{ )0()0( * SOXOS iii ∈== and },0{ )0()0( * SOXOS iii ∈== .

Step 1: Let )0()1( SS = and renumber the objects of S(1). Formulate the optimal binary clustering problem of S(1) as MP(1), which is also solved by GAs. F(X)(1)* is the objective value of optimal binary clustering of S(1) under the assumption that the

other cluster ()0(

S ) remaining unchanged. The clustering result is:

},1{ )1()1( * SOXOS iii ∈== and },0{ )1()1( * SOXOS iii ∈== . Three clusters are

formed, they are: )0(

S , )1(S and )1(

S . Ｆ(X)(1)* is the objective value of these three clusters.

Step 2: Let )1()( −= ii SS and solve MP(i) by GAs.

Step 3: Repeat Step2 until Φ=)( k

S , then this branch is fathomed. There is a total of k+1

clusters, that is )0(

S , )1(

S ,…, )1( −k

S and )( kS . )( kS can no longer be divided in the following steps. Ｆ(X)(k)* is the optimal objective value of these k+1 clusters.

Step 4: Choose one of the remaining branches to be binarily clustered by repeating Steps 2

and 3 until it is fathomed. Step 5: If all branches are fathomed, then stop. The clusters formed are the optimal

clustering result of STCM. Otherwise, go to Step 4. 4. CASE STUDY 4.1 Data To compare the overall total costs associated with different inventory policies (with and without clustering), a case study is conducted on a X manufacturer, which produces semiconductors in Taiwan, with four branches, A, B, C and D. A total of 50 items of input factors are required for the production process. Without loss of generality, the warehousing center is assumed located at branch A. The distances from branches B, C and D to the warehouse center (i.e., branch A) are 15, 8 and 24 kilometers, respectively. The volume size, price, and quantity demanded for these 50 items are detailed in Table 1, in which large coefficients of variation (CV) of the three attributes indicate remarkable heterogeneity of the items, which implies that clustering of items might be beneficial. Due to the confidential reason, some parameters of the cost models are assumed: per ordering administrative cost =10,000 dollars; unit inventory holding cost =10 dollars/cm3-year; unit transportation cost =100 dollars/truck-kilometer; unit distribution cost =50 dollars/truck-kilometer; truck capacity =3,000 cm3; annual interest rate =5%; coefficient of scale economy in warehousing=0.9; coefficient of scale economy in transportation=0.9; coefficient of price discount for sizable procurement=0.1; g=0.01; crossover rate=0.9; mutation rate=0.01.


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Table 1. Characteristics of Items Required for X Manufacturer Production Process Quantity demanded

(piece/year) Distance from the suppliers

(kilometer) Item number

Volume size

(cm3)

Price (dollars) A B C D A B C D

1 250 5,000 20,000 16,000 19,500 22,500 108 103 110 1252 150 8,000 600 250 340 800 38 47 33 553 2 3,500 17,800 13,600 15,500 19,800 79 86 81 764 3 10,000 2,000 1,500 1,800 2,600 66 60 70 835 98 40 12,000 8,000 9,300 16,500 25 34 22 436 120 120 280 340 230 420 123 130 128 1167 1 250 42,000 36,000 39,500 45,500 88 84 92 1068 2 90 3,600 2,400 2,800 3,900 47 58 42 689 12 4,800 32,500 28,500 30,000 36,500 105 113 108 105

10 3 60 5,500 4,000 4,800 6,700 74 70 78 8611 114 5,000 2,000 1,000 1,900 2,500 110 103 112 12312 218 6,600 680 550 440 700 37 46 33 5413 289 3,500 27,800 3,600 5,500 9,800 79 81 87 8614 27 12,000 6,000 3,500 3,800 4,600 67 60 72 8315 13 14,000 12,000 7,000 7,300 16,500 25 34 28 4316 24 120 390 940 670 1,220 113 130 138 11617 186 250 4,200 3,600 9,500 5,500 88 84 92 10218 202 5,090 1,600 1,400 1,800 900 42 58 47 6719 335 480 13,500 18,500 10,000 16,500 125 113 108 10520 95 160 3,500 4,200 4,800 6,500 77 70 78 8621 142 7,800 18,500 14,500 16,300 21,000 38 47 33 5522 8 3,800 990 1,250 740 850 79 86 81 7623 6 12,000 18,500 12,600 11,000 21,000 66 60 70 8324 78 440 2,300 1,400 2,500 1,200 25 34 22 4325 105 720 11,000 7,500 6,800 12,000 123 130 128 11626 41 5,200 190 140 170 220 67 60 72 8327 22 3,600 40,000 35,000 37,300 39,500 25 34 28 4328 12 4,500 600 400 800 900 136 129 138 14029 33 12,500 35,500 25,500 30,500 36,000 88 84 92 10230 127 13,700 2,500 3,000 5,800 6,600 42 58 47 6731 3 8,000 3,800 1,050 2,900 2,390 79 86 81 7632 98 3,550 880 650 1,440 300 66 60 70 8333 120 8,500 17,700 3,500 8,500 6,900 123 131 128 11934 31 240 6,100 13,500 3,800 4,600 77 69 82 8335 52 1,120 2,000 1,000 7,300 6,500 125 115 118 12236 73 14,300 1,390 1,940 3,670 4,220 74 78 79 8637 112 120 5,200 8,600 3,500 6,500 117 109 115 12338 8 2,530 1,600 300 850 700 47 46 48 5639 114 5,090 13,030 18,000 12,800 13,600 79 81 87 8640 218 460 3,600 4,050 4,800 7,330 25 34 22 4341 95 3,500 6,500 3,100 5,200 6,900 69 73 78 8042 42 12,600 3,000 1,050 1,980 1,500 60 64 56 5543 58 10,300 650 450 540 300 23 30 27 2244 86 720 7,800 3,600 5,500 9,800 82 89 88 7745 78 2,850 8,200 8,500 9,800 4,600 60 66 69 7346 229 3,500 1,200 7,000 7,300 6,500 121 111 118 11047 138 10,800 90 3,940 670 2,220 77 69 82 7348 2 340 4,000 2,600 7,500 8,500 155 145 148 14249 23 7,120 700 1,400 2,800 1,900 75 73 70 8450 93 2,250 12,500 15,500 13,000 6,500 127 122 115 125

Mean 87.8 4824.2 8759.4 7118.0 7704.8 9219.4 77.3 78.7 79.0 85.1 SD 82.9 4494.1 10767.6 8913.9 9209.4 10890.4 34.3 30.9 34.5 28.2 CV 0.9 0.9 1.2 1.3 1.2 1.2 0.4 0.4 0.4 0.3

Note: SD: standard deviation. CV: coefficient of variation.


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4.2 Pure Policy The total cost and optimal ordering ratio of four inventory strategies under pure policy are reported in Table 2. Note that S3 strategy is the best with the lowest total cost, followed by S4, and S2 is the worst strategy. S4 strategy has the lowest ordering cost because it has effectively consolidated the orderings of all items for all branches. Despite that S3 strategy adopts the decentralized warehousing alternative, it still has the lowest warehousing cost because the advantage (cost saving) of jointly ordering has outweighed the disadvantage of decentralized warehousing. S4 has the lowest transportation cost because of the significant effect of consolidated transportation. S3 strategy has the lowest procurement cost because of large saving of joint ordering. Although S4 strategy can have the advantage of centralized warehousing, the restraint of the amount of ordering at one time would make it inferior to S3 strategy.

Table 2. Results of Pure Policy (Dollars)

Inventory strategy Cost S1 S2 S3 S4 Ordering cost 4,949,149 5,475,264 1,464,634 1,184,844Warehousing cost 26,864,272 30,029,668 24,970,818 29,891,206Transportation cost 68,119,096 69,873,406 67,833,696 60,818,892Procurement cost 5,809,251,800 6,264,620,000 5,679,327,200 5,687,869,400Total cost 5,909,184,500 6,369,998,300 5,773,596,200 5,779,764,200

4.3 Mixed Policy 4.3.1 Genetic stepwise clustering method The results of mixed policy with genetic stepwise clustering method are reported in Table 3, in which nine clusters are optimally formed. Note that cluster 7 or 9 consists of only one item while cluster 2 or 3 is composed of 11 items. Seven clusters have adopted S3 strategy while the remaining two clusters have adopted S4 strategy. The total cost of this mixed policy is 5,771,112,192 dollars, lower than the pure policy by 2,484,008 dollars (0.04%).

Table 3. Results of Mixed Policy with Genetic Stepwise Clustering Method Cluster Number of

items Item number Strategy Total cost (dollars)

1 9 1, 6, 7, 11, 16, 25, 29, 36, 45 S4 672,845,700

2 11 2, 12, 14, 15, 17, 22, 26, 31, 37, 42, 47 S3 449,855,490

3 11 3, 10, 13, 19, 24, 32, 40, 43, 48, 49, 50 S3 1,430,087,300

4 10 4, 5, 9, 21, 23, 27, 28, 30, 41, 46 S3 2,374,861,300

5 2 8, 33 S4 153,861,810

6 2 18, 35 S3 2,775,092

7 1 20 S3 1,609,256

8 3 34, 39, 44 S3 684,605,700

9 1 38 S3 610,544

Total 50 - - 5,771,112,192


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4.3.2 Agglomerative hierarchical clustering method The optimal number of clusters with genetic stepwise clustering method is nine. To make comparison on the same foundation, the number of clusters with agglomerative hierarchical clustering method is also preset as nine. The results are reported in Table 4. Similarly, the number of items in each cluster ranges from one (cluster 1) to sixteen (cluster 8). Note that five clusters have adopted S3 strategy and the remaining four clusters have adopted S4 strategy. The total cost of this mixed policy with agglomerative hierarchical clustering method is 5,772,427,248 dollars, also lower than the pure policy by 1,168,952 dollars (0.02%) but slightly higher than the mixed policy with genetic stepwise clustering method.

Table 4. Results of Mixed Policy with Agglomerative Hierarchical Clustering Method Cluster Number

of items Item number Strategy Total cost (dollars)

1 1 1 S4 395,106,020

2 4 2, 11, 12,31 S4 122,404,580

3 5 3, 13, 19, 23, 33 S3 1,009,164,900

4 5 4, 14, 24, 26, 43 S3 439,630,880

5 5 5, 15, 21, 25, 45 S3 407,519,870

6 4 6, 16, 32, 36 S4 23,666,228

7 4 7, 9, 27, 29 S4 1,716,356,600

8 16 8, 10, 17, 18, 20, 28, 30, 35, 37, 38, 40, 42, 44, 46, 47, 49 S3 133,165,970

9 6 22, 34, 39, 41, 48, 50 S3 1,525,412,200

Total 50 - - 5,772,427,248

5. CONCLUDING REMARKS This paper develops inventory models for a multi-item multi-branch firm by considering the scale economies of centralized warehousing and consolidated transportation as well as the price discount of sizable procurement. The objectives of the models are to determine the optimal inventory strategies and ordering ratios by minimizing the overall total costs, including ordering, warehousing, transportation, and procurement costs. Four inventory strategies are proposed: independent ordering and decentralized warehousing, independent ordering and centralized warehousing, joint ordering and decentralized warehousing, and joint ordering and centralized warehousing. Two inventory policies, pure and mixed, are further identified based on the inventory strategy or combination of inventory strategies adopted. For the mixed policy, genetic stepwise clustering method and agglomerative hierarchical clustering method are employed to classify items into several exclusive clusters and then the best strategy and optimal ordering ratio are determined in each cluster. A case study for a semiconductor firm in Taiwan with four branches and 50 major items of input factors in the manufacturing process is carried out. The results show that the mixed policy with genetic stepwise clustering method has the lowest total cost, followed by the mixed policy with agglomerative hierarchical clustering method, and the pure policy has the highest total cost. Notice that the procurement cost takes the lion share of the total cost, thus the savings of mixed inventory policy with two clustering approaches over the pure inventory


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policy are only 0.02% to 0.04% of the total cost. Nonetheless, in terms of dollars, the savings for mixed inventory policy with two clustering approaches can reach an order of some millions a year. It still deserves to implement the proposed mixed inventory policy with appropriate clustering. For simplicity and due partially to confidential reason, this study has only very limited access to the cost data for the case manufacturer firm, and thus expresses the scale economies of warehousing and transportation as well as the price discount of joint procurement as exponential function. More practical functional forms that reflect the real scale economies require further investigation. Due to the complexity of the models, the present paper only employs the quasi-Newton numerical method to obtain the nearly optimal solutions. Techniques for solving the models with closed form solutions deserve to explore.

ACKNOWLEDGEMENTS The authors wish to thank Mr. Tony H. Shyu for providing the case study data.

REFERENCES Chiou, Y. C. and Lan, L. W. (2001) Genetic clustering algorithm, European Journal of Operational Research, Vol.135, 413-427.

Güder, F. and Zydiak, J. L. (1999) Ordering policies for multi-item inventory system subject to multiple resource constraints, Computer and Operation Research, Vol. 26, 583-597.

Mondal, S. and Maiti, M. (2002) Multi-item fuzzy EOQ models using genetic algorithm, Computers and Industrial Engineering, Vol.44, 105–117.

Murthy, C. A. and Chowdhury, N. (1996) In search of optimal cluster using genetic algorithms, Pattern Recognition Letters, Vol. 17, 825-832.

Page, E. and Paul, R. J. (1976) Multi-product inventory situation with one restriction, Operational Research Quarterly, Vol. 27, 815-834.

Pirkul, H. and Aras, O. A. (1985) Capacitated multiple item ordering problem with quantity discounts, IIE Transactions, Vol. 17, 206-211.

Shu, F. T. (1971) Economic Ordering Frequency for two items jointly replenished, Management Science, Vol. 17, 406-410.

Silver, E. A. (1975) Modifying the economic order quantity (EOQ) to handle coordinated replenishment of two or more items, Production and Inventory Management Journal (3rd Quarter), 26-38.

Tseng, L. Y. and Yang, S. B. (2001) A genetic approach to the automatic clustering problem, Pattern Recognition, Vol. 34, 415-424.


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ORDERING AND WAREHOUSING STRATEGIES FOR MULTI-ITEM …€¦ · (1) Decentralized warehousing...

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