A network design model considering inventory cost

ISSN 1750-9653, England, UK International Journal of Management Science and Engineering Management, 7(1): 29-35, 2012 http://www.ijmsem.org/

A network design model considering inventory cost from a third party logistics perspective Arash Motaghedi-Larijani1 ∗ , Mohammad-Saied Jabalameli1 , Reza Tavakkoli-Moghaddam2 2

1 Department of Industrial Engineering, Iran University of Science and Technology, Iran Department of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran

(Received 8 August 2011, Revised 14 December 2011, Accepted 14 January 2012)

Abstract. The importance of third party logistics (3PLs) service providers is increasing in supply chain management, particularly in warehousing and transportation services as clients expect 3PLs to improve lead times, fill rates, and inventory levels. Therefore, in a dynamic and uncertain business environment, 3PLs are under pressure to meet client service requirements. Few studies have addressed the problem of logistic network design from viewpoint of 3PLs. The main objective of this paper is to propose a new network design problem from 3PLs perspectives by considering inventory to determine economic order quantity. This paper proposes a new nonlinear mixed integer programming model for integrated logistics network design that in facility location-allocation and inventory location problems. To optimize the model, new auxiliary variables and constraints are added. The model was solved by using DICOPT solver. Keywords: location and allocation models, third party logistics, network design

1 Introduction As the complexity of the supply chain continuously grows and the importance of prompt delivery to end customers increases, independent companies, a third party logistics (3PLs) provider, has emerged to perform all or part of a manufacturer’s product distribution function. The 3PL is an outer provider who manages, controls, and delivers logistics activities on behalf of a manufacturer (Hertz and Alfredsson, 2003 [7]). 3PLs is the utilization of an outside firm to perform some or all of the supply chain functions that an organization requires, which can involve any aspect of logistics and is generally more than outsourcing warehousing or transportation alone. 3PLs integrate more than one function within the overall supply chain (Marasco, 2008 [16]). According to Lieb (1992) [14], 3PLs involve ”the use of external companies to perform logistics functions that have traditionally been performed within an organization. The functions performed by the third party can surround the entire logistics process or selected activities within that process”. Berglund et al. (1999) [2] emphasized the supply of management support in addition to operational activities by providers and the duration of the relationship as follows: “3PLs are activities carried out by a logistics service provider on behalf of a shipper and consists of, at least, management and execution of transportation and warehousing”. In addition, other activities performed by 3PLs are as follows: inventory management, information related activities, such as tracking and tracing, value added activities, such as secondary assembly and installation of products, or even supply chain management. Therefore, 3PLS often have to design an integrated reverse logistics facility systems to satisfy their multi-clients’ forward and reverse ∗

logistics demands. The product return process involves the determination of the number and location or allocation of repair facilities for returned products in such a way that total reverse logistics costs such as warehousing and transportation costs are minimized, capacity of repair facilities are fully utilized, and the convenience of customers who return the products is maximized. By considering previous explanations on the role of 3PLs, logistics network design was found to have a key role in reducing cost for 3PLs. A logistics network is a series of nodes and transportation links, the design of which depends on factors such as the type of products, the range and volume of products, the level of service required, and the number and type of customers. A logistics network must meet a specific set of requirements over a given planning horizon. A good design creates the best network for providing the customer with the right goods, in the right quantity, at the right place, in the right condition, at the right time, and at the right cost. In the reminder of this paper, Section two discusses the 3Ps location and network design. Section three presents a new unique mathematical model which is demonstrate, solved and validated in Section four. Finally conclusions and future research directions are given.

2 Literature review Despite a wide body of research in the various domains of 3PLs, there are few that focus on 3PLs network design. Min et al. (2004) [18] proposed a nonlinear mixed integer programming model and a genetic algorithm that could solve a reverse logistics problem involving product returns. Ko et al. (2005) [10] presented a mixed integer nonlinear programming model for the design of a dynamic integrated distribution network that simultaneously accounted for the integra-

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A. Motaghedi-Larijani & M. Jabalameli & R. Tavakkoli-Moghaddam: A network design model considering inventory cost

tion of the optimization of the forward and return network. In further research, Ko et al. (2006) [11] proposed a hybrid algorithm to design a distribution network for 3PLs by applying a simulation approach and to the performance of the warehouses. Min et al. (2008) [17] presented a mixed-integer programming model and a genetic algorithm that solved a reverse logistics problem involving the location and allocation of repair facilities for 3PLs. This model optimally created a reverse logistics network linking repair facilities, warehouses, and manufacturing facilities. Zhang et al. (2007) [30] proposed a fuzzy chance constrained programming model for a remanufacturing logistics network design problem from a 3PLs perspective under an uncertain environment. Zhang et al. (2007) [31] presented a model and algorithm for a third party logistics supplier to design a hazardous waste reverse logistics network under a fuzzy uncertain environment. Ke et al. (2008) [9] presented a fuzzy multi-objective programming model for a 3PLs network design which studied the integration of the forward and reverse logistics demand for 3PLS’ clients, with multi-commodity, multi-customer and a capacitated facility location-allocation problem. The full literature review for 3PLs network design is summarized in Tab. 1. For research in inventory-location. Barhona and Jensen (1998) [1] proposed an integer programming model for plant location with inventory costs and solved it using a decomposition method. Nozick and Turnquist (1998) [19] analyzed a method for including inventory costs within a fixed- charge facility location model. Erlebacr and Meller (2000) [4] proposed a location-inventory model for a two-level network design using continuous approximation as well as the construction number and bounding heuristics and incorporated risk pooling to reduce holding costs. Nozick and Turnquist (2001) [20] developed an analysis procedure for the location of DCs that integrated facility costs, inventory costs, transportation costs and service responsiveness. The various inventory-location models can be divided in two categories: 1) models that applied a risk pooling approach, (e.g. Daskin et al., 2002 [3]; Teo et al., 2001 [26]; Wang et al., 2010 [27]). All of the articles in this category considered that customer demand to be stochastic with a Poisson distribution function 2) models that did not apply risk pooling (e.g. Ghezavati et al., 2009 [6]; Shen et al., 2003 [22]; Shu et al., 2010 [23]; Wang et al., 2006 [28]). Teo et al. (2001) [26] presented an analytical modeling approach to study the impact on facility location cost and inventory costs when several distribution centers (DCs) are merged into a central DC. Daskin et al. (2002) [3] incorporated the inventory problem and the distribution location model by applying risk po0ling and assumed demand had a normal distribution and that the ordering quantity was economic. The objective function of this model included transportation costs, inventory and safety stock inventory costs and fixed facility costs and using a Lagrangian solution algorithm the problem was solved. Shen et al. (2003) [22] employed risk pooling in an inventory location problem and proposed a new model that determined the location of distribution centers and the assignment of retailers to the distribution centers to minimize total fixed distributioncenter location costs, inventory costs at the distribution centers, transport costs from the distribution center to the retailers, and safety stock inventory costs at the distribution centers. This model considered only one supplier, ( Q, r) policy, holding and ordering cost. Shu et al. (2005) [24] solved a general pricing problem in Shen et al. (2003) [22] efficiently. Shen (2005) [21] further

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proposed a multi-commodity supply chain design model for a multi commodity supply chain design model to determine the location of facilities and the assignment of customers to the facilities to minimize the total costs that including inventory costs. Snyder et al. (2007) [25] proposed a stochastic version of the location model with risk pooling (LMRP) that minimized expected total cost (including location, transportation, and inventory costs). They considered some scenarios and applied a Lagrangian-relaxationbased exact algorithm to solve their model. Yao et al. (2010) [29] proposed a joint facility location-allocation and inventory problem with multi products and multi plants and formulated a mixed integer nonlinear programming model that minimized expected total cost by considering the satisfaction of demand weighted average customer lead time and cycle service level. An iterative heuristic method was developed for the problem which supposed that inventory policy is a fixed period order and each customer had stochastic demand. Wang et al. (2010) [27] proposed location model of risk pooling with variable construction costs to construct location model of risking pool with variable construction costs and a compensation policy. Wang et al. (2006) [28] proposed a new locationinventory model as a bi-level programming problem which considered a supply chain with one supplier, one businessto-consumer (B2C) firm and multiple DCs to study product return location and inventory policy simultaneously for product. Mak and Shen [15] presented a model for the design of a two-echelon spare parts inventory system consisting of a central plant and a number of service centers each serving a set of customers with stochastic demand. They modeled this as a queuing system and considered fixed location cost, shipping costs, and inventory holding costs. Jeet et al. (2009) [8] analyzed and developed solution techniques for a network design and inventory stocking problem using a one-for-one replenishment policy for inventory replacement. Whereas their proposed model was originally a non-convex problem, an equivalent convex model was developed based on the linearized model using lower and upper bounding techniques. Ghezavati et al. (2009) [6] proposed a new nonlinear integer programming model for of distribution networks in a supply chain system considering service level constraints to optimize the total expected cost that included facility location and inventory costs including ordering costs, holding costs, inventory lost costs but not safety stock costs. This research considered an ( Q, R) inventory policy and a service level constraint to prevent inventory loss in distribution centers (DCs). Also, they assumed that customer’s demand was stochastic with a Poisson distribution function and that DCs had coverage radius constraints. Shu et al. (2010) [23] addressed a scenario-based two-stage stochastic model for an inventory-location problem under an uncertain environment that minimized the expected systemwide multi-echelon inventory, transportation, and facility location costs over an infinite planning horizon. In the literature, most articles considered only a single product and there are few articles that look at multiple products (e.g. Liao and Hsieh, 2009 [12]; Shen, 2005 [21]; Yao et al., 2010 [29]). Except for Erlebacr and Meller (2000) [4] all articles considered a two-level network design and one plant. Some papers considered a capacitated constraint (e.g. Liao et al., 2010 [13]; Marasco, 2008 [16]). However, in the extensive inventory-location research, only Gebennini et al. (2009) [5] studied a multi period model. This paper

International Journal of Management Science and Engineering Management, 7(1): 29-35, 2012

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Table 1 Illustrating of literature review Ke et. al. (2005) [10] Planning horizon Forward logistic Backward logistic

Ko et. al. (2006) [11]

n

n

-

-

n

-

n

n

n

n

n

Demand return rate and transportation cost are fuzzy Holding cost is included in objective function

Demand return rate and transportation cost are fuzzy Holding cost is included in objective function Warehouse and production capacity

Demand return rate and transportation cost are fuzzy Holding cost is included in objective function

-

Warehouse and Capacity production constraints capacity Function Single objective type Hybrid Solution optimization/ method simulation approach

Deterministic

-

Single period

Ke et al. (2008) [9]

n

-

Single period

Zhang et al. (2007) [31]

Dynamic

Deterministic

Dynamic

Zhang et al. (2007) [30]

Dynamic

Problems Deterministic parameters

Inventory problem

Min et. al. (2004) [18]

Single period

Warehouse and Warehouse and production production capacity capacity

Warehouse capacity

Single objective Single objective

Multi objective Multi objective Multi objective

Heuristic genetic algorithm

Heuristic Genetic algorithm genetic algorithm


Warehouse capacity


Fig. 1 Supply chain structure used for inventory location

proposes a capacitated multi product model and considers a three-level network design.

3 Problem formulation In this section, the problem is explained and details given for the mathematical model. In section one 3PLs manage the warehouse. A three level supply chain structure is illustrated in Fig. 1. Because there is a long term relationships between the factory and the warehouse and a short term relationship between the warehouse and demand node in this paper the

economic ordering quantity from warehouse to factory is examined. This model involves shipping, inventory handling, lost sales, and ordering costs. Index sets are as follows: P : Index of clients’ product types; I : Index of production sites; J : Index of existing warehouses and new potential sites; L : Index of existing repair facilities and new potential sites; K : The index of fixed customer locations; T : The index of time periods.

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The assumptions of this problem are the following: (1) Capacity constraints for vehicles do not exist. (2) There are no capacity constraints for repair centers. (3) One type of vehicle is used to carry all products from the manufacturer i to warehouse j. (4) Demand is deterministic. (5) There is the lost sales cost. (6) The planning horizon is the same for all customers. (7) All the returned products at the end of period are delivered to the manufacturer. (8) The warehouse inventory is linearly reduced and is suddenly entered. (9) Inventory control costs for reverse logistics are negligible. (10) For each warehouse, in each period there are number of ordering cycles. (11) The term contract is the same for all customers. (12) At the start of each period, there is no facility. (13) If a warehouse is increased in a period, the warehouse is increased a predefined percentage. Most research in this area supposed that customer demand can only be assigned to a single warehouse or distribution center. This simplifying assumption is not real in practice. In this paper, each customer can be served from more than one warehouse and each warehouse can receive services from more than one plant. The parameter notations are as follows: mc pit : The maximum production capacity of plant i for product p at period t; i ∈ I, t ∈ T, p ∈ P; mc jt : The maximum capacity of warehouse j at period t; j ∈ J, t ∈ T; β p : Per unit storage capacity by product p, p ∈ P; d pkt : Demand of product p at customer k in period t; p ∈ P, k ∈ K, t ∈ T; p pkt : The percent of returns of product p from customer k in period t; p ∈ P, k ∈ K, t ∈ T; Bt : Assigned budget in period t; t ∈ T. Cost parameter A pi jt : fixed cost of ordering product p from warehouse j to manufacturer i in period t; i ∈ I, j ∈ J, t ∈ T; h p jt : Unit inventory holding cost of product p in warehouse j in period t; p ∈ P, j ∈ J, t ∈ T; s pkt : Unit cost of lost sales of product p at customer k in period t; p ∈ P, k ∈ K, t ∈ T; f swjt : The fixed setup cost for installing warehouse j in period t; j ∈ J, t ∈ T; f ewjt : The fixed cost of expanding warehouse size j in period t; j ∈ J, t ∈ T; CC w jt : Cost of closing warehouse j in period t; j ∈ J, t ∈ T; f srlt : The setup cost for installing a repair center l in period t; l ∈ L, t ∈ T; CCltr : Cost of closing repair center l in period t; l ∈ L, t ∈ T; wr jt : Savings associated with opening an integrated warehouse-repair facility in period t; j ∈ J, t ∈ T; f c pi jt : The unit variable cost for the transportation of product p from manufacturer i to warehouse j in period t; p ∈ P, i ∈ I, j ∈ J, t ∈ T; crpklt : The unit variable cost for the transportation of returned product p from customer k to repair center j in period t; p ∈ P, k ∈ K, l ∈ L, t ∈ T; f c p jkt : The unit variable cost for the transportation of product p from warehouse j to customer k in


period t; p ∈ P, j ∈ J, k ∈ K, t ∈ T; crplit : The unit variable cost for the transportation of return product p from repair center l to manufacturer i in period t; p ∈ P, l ∈ L, i ∈ I, t ∈ T. The decision variables for the problem are as follows: f X pi jt : The amount of transported product p from manufacturer i to warehouse j in period t; p ∈ P, i ∈ I, j ∈ J, t ∈ T; X rpklt : The amount of returned product p from customer k to repair center l in period t; p ∈ P, k ∈ K, l ∈ L, t ∈ T; f X p jkt : The amount of transported product p from warehouse j to manufacturer i in period t; p ∈ P, j ∈ J, k ∈ K, t ∈ T; X rplit : The amount of returned product p from repair center l to repair center l in period t; p ∈ P, l ∈ L, i ∈ I, t ∈ T; Q pi jt : The amount of ordering quantity from warehouse j to producer i in period t; 1, if warehouse j is open in period t, j ∈ J Z jt : 0, otherwise 1, if warehouse j is expanded in period t, j ∈ J WE jt : 0, otherwise 1, if repair center l is expanded in period t, l ∈ L Glt : 0, otherwise Now the mathematical models are as follows: Minimize

f

f

f

f

∑ ∑ ∑ ∑ Xpi jt · c pi jt + ∑ ∑ ∑ Xpkkt · c p jkt

t=1

j

p

t

+∑∑∑ p

l

· crpklt

k

+∑∑ t

j

X rpklt

p

l

∑

i

X rplit

· crplit

i

f

h p jt

2

p

j

k

+∑∑∑

∑ Q pi jt

p

+∑∑∑∑ t

i

j

p

X pi jt Q pi jt

f

+ ∑ ∑ ∑ s pkt (d pkt − ∑ X p jkt ) t

k

p

· A pi jt (1)

j

Subject to:

∑[ f swjt · Z jt · (1 − Z jt−1 ) + CCwjt · Z jt (1 − Z jt+1 ) j

+ f ewjt · WE jt ] + ∑[ f srlt · Glt · (1 − Glt−1 ) l

+ CCltr

· Glt · (1 + Glt+1 )] −

∑

wr jt · Z jt · Glt ≤ Bt ,

j=l =1

∀t ∈ T,

∑ j

f X pi jt

(2)

≤ mc pit , ∀ p ∈ P, i ∈ I, t ∈ T,

∑ ∑ β p × Q pi jt ≤ p

i

mcwjt +

t

∑

(3)

WE jx

× 0.1 × Z jt ,

x=1

(4)


∀ j ∈ J, t ∈ T, f

∑ Xp jkt ≤ d pkt ,

∀ p ∈ P, k ∈ K, t ∈ T, f

∀ p ∈ P, k ∈ K, t ∈ T,

X pi jt

(6)

Q pi jt

(7)

j

l

f X p jkt ,

∀ p ∈ P, j ∈ J, t ∈ T,

(8)

∑ Xrpklt = ∑ Xrplit ,

∀ p ∈ P, j ∈ J, t ∈ T,

(9)

∑ i

f X pi jt

=∑ k

i

k

X pi jt ≤ M × Z jt , ∀ p ∈ P, i ∈ I, j ∈ J, t ∈ T,

f

(5)

j

∑ Xrpklt ≥ p pkt ∑ Xp jkt ,

33

(10)

· A pi jt .

If Q pi jt is zero, this term is considered as be undefined. In order to solve this problem, a set of auxiliary variables and two sets of auxiliary constraints are defined. The set of auxiliary variables is as follow: ( 1, if manufacturing plant i serve production p to warehouse j in period t y pi jt = 0, otherwise Auxiliary variable are as follow:

(11)

y pi jt ≤ Q pi jt ,

(17)

X p jkt ≤ M × Z jt , ∀ p ∈ P, j ∈ J, k ∈ K, t ∈ T,

(18)

X rpklt ≤ M × Glt , ∀ p ∈ P, k ∈ K, l ∈ L, t ∈ T,

Q pi jt ≤ y pi jt × M.

(12)

X rplit ≤ M × Glt , ∀ p ∈ P, l ∈ L, i ∈ I, t ∈ T,

(13)

Constraint (17) enures that if Q pi jt is zero, y pi jt should also be zero. This means that if the ordering quantity of product p from warehouse j to manufacturing plant i is zero, manufacturing plant i cannot send production p to warehouse j in period t. Constraint (18) enures that if Q pi jt is not zero, y pi jt should be one which means that if the ordering quantity of product p from warehouse j to manufacturing plant i is not zero, manufacturing plant i should send production p to warehouse j in period t. This mentioned term is changed to the new term:

f

f

X pi jt ≤ M × Q pi jt , ∀ p ∈ P, i ∈ I, j ∈ J, t ∈ T, Q pi jt ≤

f X pi jt ,

∀ p, i, j, t,

WE jt ∈ Z jt , ∀ j ∈ J, t ∈ T.

(14) (15) (16)

The objective function (1) minimizes total transportation costs, holding costs, ordering and lost sales costs. Constraint (2) assures that the total investment costs which includes the costs of opening, closing and expansion of facilities costs, the savings from integrated facilities, and expansion costs, does not exceed the budget. In this model there is no facility at the start of the planning horizon hence Z j,0 is zero . Constraint (3) assures that the manufacturing plant has limited capacity during the contract terms. Constraint (4) is the capacity limitations on the warehouses including expansion size across the time period. In this model it is assumed that the warehouse expansion size is certain in each period. Constraint (6) guarantees that shipping the product to the customer doesn’t exceed customer demand. Constraint (7) ensures that all returned products are sent to the manufacturing plant. Constraint (8) ensures that the total products for each warehouse equals the total product out the same warehouse at each period. This constraint clarifies that inventory there should be no inventory in the warehouse at the end of the period. Similarly Constraint (9) ensures that the total products at each repair center is equal to the total products out of the same repair center at each period. This constraint clarifies that there should be no inventory in the repair center at the end of the period. Constraints (10) to (13) relate to products flows. If the warehouse or repair center aren’t in the candidate location, there are no flows in or out of candidate locations. Note that M is a very big number. Constraint (14) explains that if the ordering quantity from the warehouse to the manufacturer equals to zero, the shipped products from the manufacturer to the warehouse should also be zero. Conversely, constraint (15) explains that if the shipped products from the manufacturer to the warehouse equals zero, the ordering quantity from the warehouse to the manufacturer should also be zero. Finally constraint (16) enures that warehouse capacity can be increased if the warehouse is opened in the same period. 3.1 Auxiliary variables There is a nonlinear term in the objective function that can be undefined. This term is as follow:

f

A pi jt ×

X pi jt Q pi jt + (1 − y pi jt )

× y pi jt .

So, as the new term cannot be undefined, constraint (14) is converted to new constraint: f

X pi jt ≤ M × y − pi jt, ∀ p, i, j, t. Now a new constraint is necessary to complete this new model: y pi jt ≤ Z jt .

(19)

This constraint enures that if warehouses aren’t generated in the candidate location j in period t, the manufacturing plant i cannot send production p to warehouse j in period t.

4 Experimental results To illustrate the validity of the proposed model, ten test problems were generated which are illustrated in Tab. 2 and are solved using Gams 23.6.5 on a vostro 1500 with 2.2 GHz CPU and 2 GB Ram. Between the Gams solvers, there are nine solvers that can solve mixed nonlinear problems. Some of these solvers are illustrated in Fig. 2. The Dicopt solver was selected to solve this model. The test problem parameters were generated randomly using uniform distribution. These input parameters are shown in Tab. 2. The budget was changed for each problem. Various test problems were generated to evaluate this model in Tab. 2. Ten test problems were generated validate the proposed model. Each problem was determined using five notations. These notations are as follows: P : number of products; I : number of plants; J : number of potential sites; k : number of demand nodes; T : number of periods. For example 4/3/5/7/3 specifies a problem using four productions, three plants, five potential sites, seven demand nodes and three periods.


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the inventory cost. Also in this model does not have a commitment to satisfy all customer demand. The contributions of this model are as follow: • It considers an inventory-location model under respective of 3PLs. • It determines the economic ordering quantity with other decision variables concurrently. • It takes into account holding costs, ordering costs and lost sale costs simultaneously. • It considers multi periods and a multi production model. • It proposes a three level model. • And it omits simplifying assumption that in other research limits production receipts to warehouses and customers from only one source. From the presented model it is considered that the following could be further research areas: • The consideration of a nondeterministic parameter such customer demand. • Using meta-heuristic algorithms to solve the problem and and hybridizing them with an exact method. • Incorporating this model into a with vehicle routing problem. Fig. 2 Gams solvers

Table 2 Input parameters Parameter Value A pi jt uniform (750, 1000) h p jt uniform (2,4) S pkt uniform (35,50) uniform (4000,8000) mc pit mc jt uniform (2000,3000) βp uniform (1,3) d pkt uniform (750,1000) p pkt uniform (0.1,0.2) uniform (10000,15000) f swjt f ewjt CC wjt f srlt CCltr wr jt f c pi jt f

uniform (1000,1500) uniform (1000,1500) uniform (8000,12000) uniform (750,1250) uniform (5000,7500) uniform (4,6)

c p jkt

uniform (3,4)

crpklt crplit

uniform (1,2) uniform (2,4)

5 Conclusions Because of the increasing importance of outsourcing companies, 3PLs have a key role in logistics services. In 3Pls one thing that should be focused on is logistic network design. Previous research in this area look at this problem but few considered the logistics costs. In this paper, a new model is proposed that seeks to optimize logistic cost. By using a unique inventory-location model. This model determines a warehouses location, the number of products that should be transferred from the manufacturing plants to warehouses and from warehouses to customers, and the production economic ordering quantity that should be ordered by warehouses from manufacturing plants simultaneously to minimize total inventory and transportation costs. Holding costs and lost sale costs are considered in this model as part of


References [1] Barahona, F. and Jensen, D. (1998). Plant location with minimum inventory. Mathematical Programming, 83(1):101–111. [2] Berglund, M., van Laarhoven, V., and et al. (1999). Thirdparty logistics: Is there a future? The International Journal of Logistics Management, 10(1):59–70. [3] Daskin, M., Coullard, C., and Shen, Z. (2002). An inventory-location model: Formulation, solution algorithm and computational results. Annals of Operation Research, 110(1-4):83–106. [4] Erlebacher, S. and Meller, R. (2000). The interaction of location and inventory in designing distribution systems. IIE Transactions, 32:155–166. [5] Gebennini, E., Gamberini, R., and Manzini, R. (2009). An integrated production-distribution model for the dynamic location and allocation problem with safety stock optimization. International Journal of Production Economics, 122:286–304. [6] Ghezavati, V., Jabal-Ameli, M., and Makui, A. (2009). A new heuristic method for distribution networks considering service level constraint and coverage radius. Expert Systems with Applications, 36:5620–5629. [7] Hertz, S. and Alfredsson, M. (2003). Strategic development of third party logistics providers. Industrial Marketing Management, 32:139–149. [8] Jeet, V., Kutanoglu, E., and Partani, A. (2009). Logistics network design with inventory stocking for lowdemand parts: Modeling and optimization. IIE Transactions, 41:389–407. [9] Ke, W., Xiao-jiao, C., and Yong, Z. (2008). Fuzzy multiobject model and solution to design logistics facility network for 3PLS. In Proceedings of the IEEE International Conference on Automation and Logistics, pages 1978–1982. [10] Ko, H. and Evans, G. (2005). A genetic algorithm-based heuristic for the dynamic integrated forward/reverse logistics network for 3PLs. Computers & Operations Research, 34:346–366. [11] Ko, H., Ko, C., and Kim, T. (2006). A hybrid optimization/simulation approach for a distribution network design of 3PLs. Computers & Industrial Engineering, 50:440–449. [12] Liao, S. and Hsieh, C. (2009). A capacitated inventorylocation model: Formulation, solution approach and preliminary computational results. pages 323–332. SpringerVerlag, Berlin Heidelberg, IEA/AIE.


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Table 3 Test problems No. problem

Symbols

Budget

1 2 3 4 5 6 7 8 9 10

2/2/4/5/2 3/3/5/6/2 4/3/5/7/3 5/3/6/8/3 6/4/6/10/4 6/4/6/12/4 7/5/7/14/4 8/5/8/16/5 9/5/9/18/5 10/5/10/20/5

uniform (50000,60000) uniform (55000,65000) uniform (60000,70000) uniform (70000,80000) uniform (80000,100000) uniform (90000,100000) uniform (100000,120000) uniform (120000,140000) uniform (140000,160000) uniform (160000,175000)

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