Author's Accepted Manuscript An improved lagrangian relaxation-based heuristic for a joint location-inventory pro- blem Ali Diabat, Olga Battaïa, Dima Nazzal PII: S0305-0548(14)00061-6 DOI: http://dx.doi.org/10.1016/j.cor.2014.03.006 Reference: CAOR3519 To appear in: Computers & Operations Research Cite this article as: Ali Diabat, Olga Battaïa, Dima Nazzal, An improved lagrangian relaxation-based heuristic for a joint location-inventory problem, Computers & Operations Research, http://dx.doi.org/10.1016/j.cor.2014.03.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. www.elsevier.com/locate/caor
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Author's Accepted Manuscript
An improved lagrangian relaxation-basedheuristic for a joint location-inventory pro-blem
Cite this article as: Ali Diabat, Olga Battaïa, Dima Nazzal, An improvedlagrangian relaxation-based heuristic for a joint location-inventory problem,Computers & Operations Research, http://dx.doi.org/10.1016/j.cor.2014.03.006
This is a PDF file of an unedited manuscript that has been accepted forpublication. As a service to our customers we are providing this early version ofthe manuscript. The manuscript will undergo copyediting, typesetting, andreview of the resulting galley proof before it is published in its final citable form.Please note that during the production process errors may be discovered whichcould affect the content, and all legal disclaimers that apply to the journalpertain.
An Improved Lagrangian Relaxation-based Heuristic for a JointLocation-Inventory Problem
Ali Diabata,∗, Olga Battaıab, Dima Nazzalc
aMasdar Institute of Science and Technology, Abu Dhabi, United Arab EmiratesbHenri Fayol Institute, Ecole Nationale Superieure des Mines de Saint-Etienne, France
cGeorgia Institute of Technology, Atlanta, Georgia 30332, USA
Abstract
We consider a multi-echelon joint inventory-location (MJIL) problem that makes location, orderassignment, and inventory decisions simultaneously. The model deals with the distribution of asingle commodity from a single manufacturer to a set of retailers through a set of sites wheredistribution centers can be located. The retailers face deterministic demand and hold workinginventory. The distribution centers order a single commodity from the manufacturer at regularintervals and distribute the product to the retailers. The distribution centers also hold workinginventory representing product that has been ordered from the manufacturer but has not been yetrequested by any of the retailers. Lateral supply among the distribution centers is not allowed.The problem is formulated as a nonlinear mixed-integer program, which is shown to be NP-hard.This problem has recently attracted attention, and a number of different solution approacheshave been proposed to solve it. In this paper, we present a Lagrangian relaxation-based heuristicthat is capable of efficiently solving large-size instances of the problem. A computational studydemonstrates that our heuristic solution procedure is efficient, and yields optimal or near-optimalsolutions.
Supply chain management (SCM) involves a group of organizations that perform the variousprocesses that are required to manage the flow of products at the lowest possible cost and highestdegree of customer satisfaction. The chain typically begins with raw materials and ends withthe finished product that is delivered to the customer. The supply chain includes the manufac-turer, transporters, warehouses, retailers, and customers themselves. Within each organization,the supply chain includes all functions involved in satisfying customer demand. Supply chaindecision phases are classified into the following three categories based on the frequency with
Preprint submitted to Computers & Operations Research March 19, 2014
which they are made and the time frame over which a decision phase has an impact: 1) strategicdecisions, which impact the firm over several years, for example the locations of distributioncenters; 2) tactical decisions, which are usually made one to four times a year, for example de-termining transportation and inventory policies; and 3) operational decisions, which are usuallymade on a daily basis, for example scheduling and routing decisions; see Chopra and Meindl [6]and Simchi-Levi et al. [31]. In today’s competitive environment, only efficient supply chains thatintegrate decisions in the various phases can survive.
Inventory management and facility location are two major issues in the efficient design ofa supply chain network; see Gunasekaran et al. [16, 17] and Stevens [34]. However, literatureon supply chain optimization has traditionally considered these issues independently not onlybecause of different planning horizons but principally because of the computational complex-ity of the joint optimization problem. Indeed, facility location problems are typically NP-hardcombinatorial optimization problems, and the majority of inventory management problems areformulated as nonlinear programming problems. Combining such two problems lead to moredifficult NP-hard problems that are usually nonconvex, and therefore cannot be easily solved tooptimality using exact optimization methods. However, such an integration offers a possibilityto considerably improve the supply chain management and reduce the costs.
It is worth mentioning a real-world example, in the interest of demonstrating how the strategiclevel decision of facility location (which does not necessarily refer to an actual location of a newfacility) can be successfully integrated with the tactical level inventory decisions, to providebetter solutions and lead to improved performance. The motivation behind initial work on jointinventory location problems arose from the problem of producing and distributing blood plateletsfor a blood bank in Chicago and it was addressed by Daskin et al. [8], Shen et al. [23] andOzsen et al. [27, 28]. This particular bank distributes blood to more than thirty hospitals inthe region and the inventory cost of the platelets is high, due to specific conditions that shouldbe maintained at all times, such as the frequent agitation of the platelets or the temperaturethat must be kept between 20 and 24 degrees Celsius. Furthermore, the expiration of the bloodplatelets a few days after they are collected is another important factor to be considered. Eachhospital stored its own platelet inventory and this independent inventory and location policyled to platelets going to waste after expiration in certain hospitals, while others ran out verysoon. After re-addressing this problem as a joint location-inventory model, by enforcing somehospitals to serve as distribution centers and others as retailers, the efficiency and usage of theplatelets greatly improved. The same example motivated Le et al. [22] to jointly study inventorydecisions and routing decisions for perishable goods. Therefore, many real-world problems canbe formulated as location-inventory problems without including any “real” location decisions foropening new warehouses. The existence of the location-type decision variables, in addition tothe inventory decision variables, is the only reason for calling such problems location-inventoryproblems.
The objective of the present paper is to develop an efficient Lagrangian-based heuristic forsuch an integrated problem, which we call a large-scale multi-echelon joint inventory-locationproblem and which we refer to as the MJIL problem.
The paper is organized as follows: Section 2 reviews existing joint inventory-location models;Section 3 introduces the MJIL problem; Section 4 introduces the new Lagrangian relaxationalgorithm for solving the MJIL problem; Section 5 presents computational studies; and Section6 discusses future research directions.
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2. Literature Review
Integrated supply chain network design involves several core components, among which arefacility location and inventory management. Most literature on supply chain optimization hastraditionally considered facility location decisions and inventory management decisions inde-pendently: Amiri [1], Daskin et al. [9], Hindi and Pienkosz [18], Pirkul and Jayaraman [29],and Tsiakis et al. [40] focused on location decisions, while Axsater [2], Jones and Riley [20],Muckstadt and Roundy [25], Svoronos and Zipkin [35], and Wee and Yang [41] focused on in-ventory management decisions. Only recently, integrated models have attracted the attention ofresearchers.
Barahona and Jensen [3] introduced a large-scale integer programming formulation for alocation-inventory model, and used Dantzig-Wolfe decomposition to solve the linear program-ming relaxation of this problem. Because the standard implementation of the Dantzig-Wolfedecomposition algorithm was too slow, the authors used subgradient optimization to improve therate of convergence of their solution procedure. Although they included ordering and inventorycosts in their model, they considered these costs only for one echelon. Thus, their model rep-resents the integration of a location model with an economic order quantity (EOQ) model; seeNahmias [26].
Erlebacher and Meller [11] developed an analytical joint location-inventory model. The gen-eral version of their problem is NP-hard, and is therefore difficult to solve, so they developeda heuristic algorithm which performs well on their test problems. They consider ordering andinventory costs at the distribution centers but these costs are omitted at the retailer level. Teo etal. [37] used an analytical modeling approach to study the impact on facility investments and in-ventory costs when several distribution centers are consolidated into a central distribution center,but did not consider transportation costs in their formulation.
Transportation costs are considered in most recent models, as found in the work of Tancrezet al. [36], in which the authors study the integrated location-inventory problem for three-levelsupply networks, comprising of suppliers, distribution centers and retailers. The non-linear con-tinuous formulation includes transportation, fixed, handling and inventory holding costs and theauthors develop a heuristic to solve the problem, which performs efficiently. Keskin and Uster[21] similarly address the three-stage supply network, in terms of modeling assumptions andconsiderations; however, as a solution methodology they develop both a local search and a Sim-ulated Annealing (SA) algorithm and conclude that SA leads to better quality solutions and lowerrun times. As in the current paper, both aforementioned works assume single sourcing.
Cetinkaya et al. [4] further consider that transportation costs are subject to truck and cargocapacity, leading to a need for explicit cargo cost modelling. On the other hand, they considera two-stage distribution system with DCs and retailers and they take advantage of certain struc-tural properties to reduce to a simpler non-linear formulation, which leads to efficient solvingof the problem. Contrary to the discrete models seen in the papers mentioned thus far, Tsao etal. [22] develop a continuous approximation approach, with the motivation of solving larger-scale problems. They conduct a sensitivity analysis with respect to parameter values to providemanagement insights.
Daskin et al. [8] and Shen et al. [23] developed a location-inventory model with risk pooling(LRMP). LMRP is formulated as a nonlinear integer programming problem that incorporatesinventory costs at the distribution centers. Ozsen et al. [27, 28] and Sourirajan et al. [32, 33]proposed two different extensions to the model, presented by Daskin et al. [8] and Shen et al.[23]. Table 1 provides a straightforward comparison of several of the published papers, in terms
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of distinct model assumptions, decision made and solution methodology.
Table 1: Comparison of relevant published papers
Paper Model Features Decision Variables SolutionMethodology
ThisPaper • Ordering, inventory and transportation
costs• Multi-echelon• Single sourcing
• Average inventory level at retailer• Average inventory level at DC• Order-quantity at retailer• Cycle time of retailer• Cycle-time of DC
Lagrangianrelaxation-based heuris-tic
[3] • Ordering and inventory costs• Single echelon
• Whether or not a plant is opened• Customer-plant assignment• Whether or not a customer that is assigned
[11] • Ordering and inventory costs• Two-level distribution system
• Number of DCs• Location of DCs• If DC is open• If DC serves certain customer grid• Average distance from DC to customer• Demand shipped• Distance from plant to DC
Stylized ana-lytical model,heuristics
[37] • Stochastic demands at customer locations• Warehouse consolidation
• Location of DC• Assignment of demand location to DC
Consolidationstrategy
[36] • Possibility for direct flow between cus-tomer and factory
• Multiple sourcing
• Flows between customers, factories, DCs• Size of shipments sent• Total flow passing from every DC
Iterativeheuristic
[4] • Two stage distribution system• Explicit modeling of inventory replen-
ishment, holding and transportationcosts
• Set of open DCs• Assignment of open DCs
Analytical so-lution
[39] • Multiple sourcing • Location of DCs• Assignment of retailers to DCs
Continuousapproxima-tion
In this paper we study a model that considers ordering and inventory costs at both the dis-tribution center and retailer echelons. This model, which was studied by Teo and Shu [38],Shu [30], and Diabat et al. [10], is unlike the models mentioned above, which consider theordering and inventory costs at only one level of the supply chain network. Considering order-ing and inventory costs at two echelons in the model poses additional challenges. Teo and Shu[38] developed a column generation-based algorithm for solving this model that was capable offinding an optimal or near-optimal solution for small to moderate size instances. Shu [30] pre-sented a greedy heuristic to solve large-scale instances of the problem. Later on, Diabat et al.[10] developed a basic Lagrangian relaxation-based heuristic for this problem. Results from the
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aforementioned works that employ Lagrangian relaxation-based heuristics demonstrate that thismethod seems promising for such integrated problems. Therefore, in this paper, a new sophis-ticated Lagrangian relaxation-based heuristic with efficient lower and upper bounding schemesis presented. Our computational studies demonstrate that the proposed method is efficient, andyields optimal or near-optimal solutions in relatively very small computational times.
3. Problem formulation
The formulation addresses the delivery of a single product from a manufacturer to distributioncenters, which can be opened in multiple locations, and from there to multiple retailers. Singlesourcing is assumed, according to which a single distribution center covers the total demand ofany given retailer. On the retailers’ side the demand is deterministic and they hold working inven-tory, which is defined as the product delivered to the retailer by the distribution center, withouthaving yet been requested by end-customers. As far as distribution centers are concerned, theyreceive a single commodity only from the manufacturer, as lateral supply is prohibited, at fre-quent time periods, in order to then redistribute to retailers, and they also hold working inventoryof product from the manufacturer, that has not yet been ordered from retailers. Figure 1 servesas a representation of the described system.
There are four main cost components in this system: (i) fixed-order cost: the cost of placingan order, independent of the size of the order, (ii) unit-inventory cost: the cost of holding oneunit of commodity for one unit of time, (iii) unit-shipping cost: the cost of shipping one unit ofcommodity between facilities, and (iv) fixed-location cost: the cost associated with establishingand operating a distribution center. The objective of the formulation is to decide: (1) the numberof distribution centers to establish; (2) their locations; (3) the sets of retailers assigned to eachdistribution center; and (4) the size and timing of orders for each facility, with the aim of mini-mizing the sum of inventory, shipping, ordering, and location costs while satisfying end-customerdemand.
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Figure 1: MJIL supply chain network.
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To formulate the problem, Diabat et al. [10] introduced the following notation:
SetsI : set of retailers, indexed by i
J : set of potential distribution center locations, indexed by j
� j : subset of retailers that are assigned to the distribution center at location j
Parametersdi : demand rate of retailer i
f j : fixed cost of establishing and operating a distribution center at location j
si j : unit-shipping cost to retailer i from distribution center j
s j : unit-shipping cost from the manufacturer to distribution center j
hi : unit-inventory cost per unit of time at retailer i
ki : fixed-order cost at retailer i
h j : unit-inventory cost at distribution center j, per unit of timek j : fixed-order cost at distribution center j
where (1) represents the objective function that minimizes the sum of inventory, shipping,ordering, and location costs while satisfying end-customer demand. We have defined bi j =
(si j + s j)di, ci j =12
(hi − h j)di, and ei j =12
h jdi. Observe that even when retailer i is not assignedto a distribution center at location j, the variable Ti j will be assigned a value. However, this valuedoes not play a role in the objective function as it is multiplied by Yi j = 0.
According to constraint (2), each retailer has to be assigned to exactly one distribution center.Constraint (3) assures that a retailer can be assigned to a distribution center only if it is opened.Constraints (4-5) define variables Ti j and T j as positive real numbers. Constraints (6-7) definevariables Xj and Yi j as binary numbers. For more explanation on the formulation of the problem,we refer the reader to Diabat et al. [10].
Let � j be the set of retailers that are assigned to distribution center j. Based on the resultsobtained in [10], we define sub-problem( j) to be the system consisting of the distribution center jand the set of retailers � j. Because of our single-sourcing assumption, the problem decomposesinto |J| sub-problems, each representing a one-distribution center multi-retailer inventory system.The goal of each sub-problem is to find an optimal inventory policy, that is, the size and timing oforders for each facility, so as to minimize the sum of ordering and inventory costs while meetingdemand. To find the subsets � j,∀ j ∈ J, each sub-problem is defined as nonlinear program asshown in [10] and should be solved endogenously and simultaneously with problem (1)-(7), sinceits decisions are interrelated with the ordering decisions Ti j and T j.
If βinv = 0, the MJIL problem (1)-(7) reduces to the uncapacitated fixed-charge location prob-lem (UFLP); see Daskin [7]. Therefore, the MJIL problem is NP-hard. In fact, the nonconvexityof (1)-(7) indicates that it is probably difficult to solve the problem to global optimality. We now
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propose an alternative formulation for the nonlinear mixed integer program (1)-(7) that is easierto work with. For simplicity, we drop the index j on the retailer cycle-time, that is, Ti j is replacedby Ti. We define Zj(� j, T j,Ti, Xj) and Z∗j (� j, Xj) to respectively be the total average inventorycost and the optimal inventory cost of serving all retailers in � j from a distribution center locatedat location j. Formally, we have
Zj(� j, T j,Ti, Xj) =k j
T j
X j +∑i∈� j
(ki
Ti+ ci jTi + ei j max(Ti, T j)
)(8)
and
Z∗j (� j, Xj) = minT j ,Ti
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩k j
T j
X j +∑i∈� j
(ki
Ti+ ci jTi + ei j max(Ti, T j)
)s.t. T j ∈ IR+
Ti ∈ IR+, ∀i ∈ � j
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭ . (9)
By convention we interpret Z∗j (� j) as Z∗j (� j, Xj = 0) when � j = ∅ and as Z∗j (� j, Xj = 1) when� j � ∅.With this convention, the program (1)-(7) can be reformulated as follows:
Yi j − Xj ≤ 0,∀i ∈ I, j ∈ J (12)Xj ∈ {0, 1},∀ j ∈ J (13)Yi j ∈ {0, 1},∀i ∈ I, j ∈ J (14)
where � j = {i ∈ I|Yi j = 1}.
4. Solution Approach
Lagrangian relaxation has shown exceptional success in solving many NP-hard supply chaincombinatorial optimization problems; see for example Chen and Chu [5], Eskigun et al. [12],Jayaraman and Pirkul [19], Min et al. [24], and Pirkul and Jayaraman [29]. Excellent surveysof the computational aspects and applications of Lagrangian relaxations are given by Fisher[13, 14, 15]. In the next two subsections, we describe our approach to solving problem (10)-(14).The proposed solution procedures are based on Lagrangian dual formulations, where Constraints(11) are relaxed. A solution of the Lagrangian dual provides a lower bound on the program (10)-(14). In order to find an upper bound, we use a heuristic that constructs a feasible solution fromthe lower bound solution.
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4.1. Lower BoundsThe Lagrangian dual obtained by relaxing constraints (11) can be written as
s.t. Yi j − Xj ≤ 0,∀i ∈ I, j ∈ J (16)Xj ∈ {0, 1},∀ j ∈ J (17)Yi j ∈ {0, 1},∀i ∈ I, j ∈ J. (18)
Problem (15)-(18) decomposes by location, that is, each distribution center location can beconsidered separately. Denoting the corresponding sub-problem S j(λ) and defining its corre-sponding optimal value as S ∗j(λ), we obtain
minX,Y,�
f jX j +∑i∈I
(βtrnbi j − λi
)Yi j + βinvZ∗j (� j) (19)
s.t. Yi j − Xj ≤ 0,∀i ∈ I (20)Xj ∈ {0, 1} (21)Yi j ∈ {0, 1},∀i ∈ I. (22)
For any given λ, we either set Xj = 0 or Xj = 1 in problem S j(λ). If we set Xj = 0, then Yi j = 0for all i ∈ I, so that � j = ∅ (empty set), and hence Z∗j (� j) = 0. Therefore, S ∗j(λ) = 0. Onthe other hand, if we set Xj = 1, then for each i ∈ I we can select either Yi j = 0 (i � � j) or
Yi j = 1 (i ∈ � j). Since Z∗j(� j
)≥ Z∗j
(� j\{i}
)for any � j ⊆ J, when βtrnbi j − λi > 0 it is never
advantageous to set Yi j = 1 for any i, that is, we must select Yi j = 0, so that i � � j.A more challenging decision to make occurs when Xj = 1 and βtrnbi j − λi < 0. The structure
of subproblem S j(λ), (19)-(22), makes it difficult to decide if it is beneficial to set Yi j = 1 or not,for any i ∈ I. If the term Z∗j (� j) of the objective function was not present in (19), as in the UFLP,then we would simply set Yi j = 1 whenever Xj = 1 and βtrnbi j − λi ≤ 0. However, the existenceof Z∗j (� j) will force the objective value S ∗j(λ) to increase since setting Yi j = 1 implies i ∈ � j.Therefore, although Xj = 1 and βtrnbi j − λi ≤ 0, we might have to set Yi j = 0 if the increase inthe optimal objective value is greater than (βtrnbi j − λi) for any i in � j. It is therefore importantto determine how much of an increase in S ∗j(λ) will be caused by adding i to � j.
Definition 4.1. The jth marginal inventory cost of retailer i, denoted by Mi
(� j
)for any i ∈ � j, is
the difference in the optimal average inventory costs of sub-problem( j) between serving retailer
i or not. That is, Mi
(� j
)= Z∗j
(� j
)− Z∗j
(� j\{i}
)for any i ∈ � j and any � j ⊆ I where Z∗j (∅) = 0.
Proposition 4.1. Let � j ⊆ I and let i ∈ � j. The jth marginal inventory cost of retailer i is lower
bounded by√
2kidihi; i.e. Mi
(� j
)≥√
2kidihi.10
Proof of Proposition 4.1. Assume T j is fixed. Define
gi j(Ti, T j) =ki
Ti+
12
(hi − h j)diTi +12
h jdi max(Ti, T j). (23)
Let ∂gi j(Ti, T j) be the subgradient of gi j with respect to its first argument, which we will abbre-viate as ∂gi j(Ti). Clearly, the function gi j(Ti, T j) is a convex function of its first argument overTi > 0 since it is the sum of three functions that are convex in Ti for fixed T j; it follows that theminimum of gi j over Ti > 0 will be attained at any point where 0 ∈ ∂gi j. We have:
∂gi j(Ti, T j) =
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩− ki
T 2i+ 1
2 (hi − h j)di if Ti < T j
− ki
T 2i+ 1
2 (hi − h j)di,− ki
T 2i+ 1
2 hidi if Ti = T j
− ki
T 2i+ 1
2 hidi if Ti > T j
(24)
There are three possibilities:
Case 1: If T j ≤√
2kihidi
, then define T ∗i =√
2kidihi
. It then follows that T ∗i > T j and hence that
∂gi j(T ∗i ) = 0, and thus that T ∗i minimizes gi j(·, T j).
Case 2: If T j ≥√
2ki
(hi−h j)di, then define T ∗i =
√2ki
(hi−h j)di. It then follows that T ∗i < T j and hence
that ∂gi j(T ∗i ) = 0, and thus that T ∗i minimizes gi j(·, T j).
Case 3: If√
2kihidi≤ T j ≤
√2ki
(hi−h j)di, then define T ∗i = T j. It then follows that 0 ∈ ∂gi j(T ∗i ), and
thus that T ∗i minimizes gi j(·, T j).
It follows from the above that gi j(·, T j) attains a minimum at
T ∗i =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
√2kidihi
if T j ≤√
2kihidi
T j if√
2kihidi≤ T j ≤
√2ki
(hi−h j)di√2ki
(hi−h j)diif T j ≥
√2ki
(hi−h j)di
(25)
Letgi(T j) = min
Ti>0gi j(Ti, T j)
be the value of gi j at the minimum, then
gi(T j) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
√2kidihi if T j <
√2ki
dihi
ki
T j
+12
dihiT j if
√2ki
dihi≤ T j ≤
√2ki
(hi − h j)di√2ki(hi − h j)di +
12
h jdiT j if T j >
√2ki
(hi − h j)di.
(26)
11
Since gi(T j) is continuous and nondecreasing in T j, it follows that
minT j
{gi(T j)
}=√
2kidihi.
Without loss of generality, select retailer i ∈ � j. We prove next that Mi
(� j
)≥√
2kidihi, for any
� j ⊆ I.
Mi
(� j
)= Z∗j
(� j
)− Z∗j
(� j\{i}
)= min
T j ,Ti
{Zj(� j, T j,Ti, Xj)
}−min
T j ,Ti
{Zj(� j\{i}, T j,Ti, Xj)
}
= minT j ,Ti
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩k j
T j
X j +∑i∈� j
(ki
Ti+ ci jTi + ei j max(Ti, T j)
)⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭− min
T j ,Ti
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩k j
T j
X j +∑
i∈� j\{i}
(ki
Ti+ ci jTi + ei j max(Ti, T j)
)⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭≥ min
T j ,Ti
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ki
Ti+ ci jTi + ei j max(Ti, T j)
⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭+ min
T j ,Ti
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩k j
T j
X j +∑
i∈� j\{i}
(ki
Ti+ ci jTi + ei j max(Ti, T j)
)⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭− min
T j ,Ti
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩k j
T j
X j +∑
i∈� j\{i}
(ki
Ti+ ci jTi + ei j max(Ti, T j)
)⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭= min
T j
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩minTi
⎧⎪⎪⎪⎨⎪⎪⎪⎩ ki
Ti+ ci jTi + ei j max(Ti, T j)
⎫⎪⎪⎪⎬⎪⎪⎪⎭⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭
= minT j
{gi(T j)
}=√
2kidihi.
Proposition 4.2. Assume that �∗j ⊆ I is the set of retailers that will be assigned to distributioncenter j in the optimal solution to problem S j(λ). Then
βtrnbi j − λi + βinv
√2kidihi ≤ 0 for any i ∈ �∗j .
12
Proof of Proposition 4.2. Since i∗ ∈ �∗j , it follows that∑i∈�∗j
(βtrnbi j − λi
)+ βinvZ∗j
(�∗j)≤∑
i∈�∗j\{i∗}
(βtrnbi j − λi
)+ βinvZ∗j
(�∗j\{i}
).
This inequality can be simplified to(βtrnbi∗ j − λi∗
)+ βinvZ∗j
(�∗j)≤ βinvZ∗j
(�∗j\{i∗}
)≤ βinvZ∗j
(�∗j)− βinv
√2ki∗di∗hi∗ .
Simplifying the above expression, we obtain
0 ≤ −(βtrnbi∗ j − λi∗
)− βinv
√2ki∗di∗hi∗
or equivalently
0 ≥(βtrnbi∗ j − λi∗
)+ βinv
√2ki∗di∗hi∗
which is the desired result.Proposition 4.2 yields the following helpful contrapositive that can be used in designing our
solution algorithm for solving the MJIL problem.
Corollary 1. Assume that�∗j ⊆ I is the set of retailers that will be assigned to distribution centerj in the optimal solution to problem S j(λ). For any i ∈ I,
if βtrnbi j − λi + βinv
√2kidihi > 0 then i � �∗j . (27)
Corollary 1 shows that, in solving subproblem S j(λ), we should choose Yi j = 0 for anyretailer i ∈ I that has a positive value of βtrnbi j − λi + βinv
√2kidihi. Unfortunately, Xj = 1 and
βtrnbi j − λi + βinv√
2kidihi < 0 are not sufficient conditions to require Yi j = 1 for any i ∈ I.However, Corollary 1 shows that we can restrict our search to the set of retailers:
I j =
{i ∈ I s.t. βtrnbi j − λi + βinv
√2kidihi < 0
}.
For those retailers, we sort all i ∈ I j such that δ1 j ≤ δ2 j ≤ · · · ≤ δm j, where m = |I j| and δi j =(βtrnbi j − λi
)+Z∗j ({i}). Then we use a simple greedy heuristic to assign these retailers as outlined
in Algorithm AlgLB below. This algorithm is used to obtain a lower bound on S j(λ).Algorithm AlgLB
STEP 1: Partition set I into two subsets as follows:
I�j ={
i ∈ I s.t. βtrnbi j − λi + βinv
√2kdih ≥ 0
}I j = I − I�j
13
STEP 2: Compute δi j =(βtrnbi j − λi
)+ Z∗j ({i}), for i = 1, . . . ,m.
STEP 3: Form the sets
I(k)j = {� ∈ I j : δ� j is among the k smallest elements of {δi j}i=1,...,m}.
STEP 4: Compute the partial sums
Δ(k)j = βinvZ∗j
(I(k)
j
)+∑i∈I(k)
j
(βtrnbi j − λi
)for k = 1, 2, · · · ,m.
STEP 5: Let k∗ be the value of k that gives the minimum value of Δ(k)j and set
X∗j ={
1, if f j + Δ(k∗)j < 0
0, otherwise.
STEP 6: Set
Y∗i j =
{1, if Xj = 1 and i ∈ I(k∗)
j0, otherwise.
Algorithm AlgLB is applied for every j ∈ J. The objective function (10) evaluated at X∗j , Y∗i jserves as a lower bound on the MJIL problem, (10)-(14).
4.2. Upper Bounds
In most cases, the solution obtained by algorithm AlgLB will be infeasible for the MJILproblem. However, a feasible solution can typically be obtained by using a constructive heuristicon the lower bound solution, as described in the algorithm AlgUB. The resulting feasible solutionprovides an upper bound on (10)-(14). The problem that algorithm AlgUB addresses is: givenan initial solution X, Y (0) that satisfies (12)-(14) but not necessarily (11), how does one extendthis solution to a good quality feasible solution Y? Since (11) implies that a feasible solution Ycontains exactly a single “1” in each row, in order to satisfy (11) we must insert a “1” into eachrow that does not contain a “1”, and delete all “1”’s but one from each row that contains two ormore “1”’s. The algorithm we propose is given below.
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Algorithm AlgUB
STEP 1: Initialize the matrix Y and the list LFor i = 1, . . . , I:
• If the ith row of Y (0) has exactly one “1”:
– Set Yi, j = Y (0)i, j for 1 ≤ j ≤ J, i.e. set the ith row of Y equal to the ith row of
Y (0).
• If the ith row of Y (0) has either no “1”’s or more than one “1“:
– Set Yi, j = 0 for 1 ≤ j ≤ J, i.e. set the ith row of Y equal to zeros.
STEP 2: Initialize the list LFor i = 1, . . . , I:
• If the ith row of Y (0) has no “1”’s:
– For each j such that Xj = 1, add the tuple (i, j,Δg(i, j)) to the list L, whereΔg(i, j) is the amount by which the objective function (10) is increased whenthe (i, j)th entry of Y is changed to a “1”.
• If the ith row of Y (0) has more than one “1”:
– For each j such that Y (0)i, j = 1, add the tuple (i, j,Δg(i, j)) to the list L, where
Δg(i, j) is the amount by which the objective function (10) is increased whenthe (i, j)th entry of Y is changed to a “1”.
STEP 3: Update Y and LRepeat the following steps until the list L is empty:
• Find the tuple (i, j,Δg(i, j)) in the list having the smallest value of Δg(i, j). Letthe value of i and j that achieve the minimum be respectively imin, jmin.
• Remove the tuple (imin, jmin,Δg(imin, jmin)) from the list and setYimin, jmin = 1.
• Scan the list and remove any tuple (i, j,Δg(i, j)) with i = imin.
• Scan the list and recalculate Δg(i, j) for any tuple (i, j,Δg(i, j)) with j = jmin.
We note that upon completion of algorithm AlgUB, the pair X, Y will be a feasible solution.Also, at each step, a “1” is inserted in the location (i, j) that brings about the smallest increaseΔg(i, j) in the objective function (10). To clarify Step 3, we note that:
(i) The reason that tuples (i, j,Δg(i, j)) with i = imin are removed in Step 3 is that once Yimin, jmin
has been set to one, we already have one “1” in row imin, so we remove any tuples thatwould place another “1” in this row.
(ii) The reason why Δg(i, j) is recalculated for tuples (i, j,Δg(i, j)) with j = jmin in Step 3 isthat the objective function (10) depends on all entries in the jth column of Y via the termβinvZ∗j (Y, j), and hence when an entry in this column changes, all tuples that would add a“1” to this column must be recalculated.
15
If the greatest observed lower bound is equal to the smallest observed upper bound withinsome pre-specified tolerance, we have found an “optimal” solution to (10)-(14). Otherwise, theLagrange multipliers are updated using subgradient optimization as described in Fisher [13, 14,15] and we repeat algorithms AlgLB and AlgUB until a feasible solution with the desired tol-erance is obtained or the minimum value of the step-size is reached. If, when the Lagrangianprocedure terminates, the best known lower bound is equal to the best known upper bound (towithin some pre-specified tolerance), we have found the optimal solution to the MJIL problem.Otherwise, a branch-and-bound algorithm is used to close the gap, with branching performedon the location variables Xj. At each node of the branch-and-bound tree, the distribution centerselected for branching is the unopened distribution center with the greatest assigned demand; ifall distribution centers in the solution have already been forced open, we branch on an arbitrarilyselected unforced distribution center. The variable is first forced to zero and then to one. Branch-ing is done in a depth-first manner. The tree is fathomed at a given node if the lower bound atthat node is greater than or equal the objective value of the best feasible solution found anywherein the tree to date, or if all distribution centers have been forced open or closed.
5. Computational Results
In this section, we first explain the design of our computational experiments and then summa-rize the results. We tested our heuristic for the MJIL problem on a total 1,750 randomly generatedinstances against the Lagrangian relaxation based algorithm used by Diabat et al. [10]. As inShu [30], the location of the distribution centers and the retailers are uniformly distributed over[0,100]×[0,100]. All transportation costs are assumed to be proportional to the euclidean dis-tance in the plane. Retailers’ fixed-order costs, unit-inventory costs, and demands are generateduniformly in [150,250]. For the distribution centers, fixed-order costs are generated in [300,400]and unit-inventory costs are generated uniformly in (0,100]. We ran the three algorithms on thefollowing pairs (βtrn, βinv): (0.01,1), (0.01,100), (1,0.01), (1,1), (1,100), (100,0.01), and (100,1).The values for these parameters were chosen in this manner to provide a large range of tradeoffsbetween location costs, transportation costs, and inventory costs. For every pair (βtrn, βinv) weran 25 instances for every problem size.
Table 2: Parameters for the Lagrangian relaxation algorithm.
Parameter Value
Maximum number of iterations at each node 1200Number of non-improving iterations before halving α 12Initial value of α 2Minimum value of α 0.00000001Minimum LB-UB gap 1%Initial value for λi 10μ + 10 fi
The parameters that we used for the Lagrangian relaxation procedure are given in Table 2.The notation μ stands for the average demand for all retailers. We terminated our algorithm whenthe optimality gap was below 1%, or the maximum number of iterations allowed or the minimumvalue of α (the scalar used to calculate the step-size) occurred. For a more detailed explanation
16
of the Lagrangian relaxation parameters, see Daskin [7]. We coded the algorithms in C++ andran them on a 2.26 GHz dual processor Dell Precision T7500 workstation with 12 GB of RAM.
Table 3: Results for (βinv, βtrn) = (0.01, 1)
Problem size Diabat et al. (2013) Lagrangian heuristics
Tables 3 - 9 summarize the results of our computational studies. For Diabat et al. [10], thegap is defined by (ZD
UB − ZDLB)/ZD
LB, where ZDUB and ZD
LB are respectively the best upper boundand best lower bound obtained by the same algorithm. Whereas for the Lagrangian relaxationheuristic, the value gap is defined by (ZH
UB − ZDLB)/ZD
LB, where ZHUB represents the best upper
bound obtained using our new heuristic. The reason for using the best lower bound obtained byDiabat et al. in calculating the gap for our new heuristic is that algorithm AlgLB might give asolution that is not really a lower bound to the MJIL problem and consequently may be higherthan the best known upper bound. However, during our computational experience with the 1,750instances, we never observed this.
17
Table 5: Results for (βinv, βtrn) = (1, 0.01)
Problem size Diabat et al. (2013) Lagrangian heuristics
As can be seen from Tables 3 - 9, our Lagrangian relaxation heuristic is always able to obtaina solution that is within 0.5% of the solution obtained by the Diabat et al. (2013), but in muchsmaller computational times.
6. Conclusion and future research
In this paper we studied an integrated supply chain model that considers facility locationdecisions and inventory decisions simultaneously. The model combines the one-distribution cen-ter multi-retailer inventory problem with the uncapacitated fixed-charge location problem. Themodel aims to determine: (1) the number of distribution centers to establish; (2) their location;(3) the sets of retailers that are assigned to each distribution center; and (4) and the size andtiming of orders for each facility so as to minimize the sum of inventory, shipping, ordering, andlocation costs while satisfying end-customer demand.
Due to the success that Lagrangian relaxation has exhibited in tackling several NP-hard sup-ply chain combinatorial optimization problems, we chose to address the MJIL with a Lagrangianrelaxation-based heuristic. After decomposing the problem by location, we are able to considereach distribution center location separately. A simple greedy heuristic is implemented to assignretailers to each distribution center, and a lower bound is obtained for the problem. Anotheralgorithm is developed to obtain an upper bound to the problem and if the greatest observedlower bound is identical to the lowest observed upper bound, within a pre-specified tolerance,the optimal solution to the problem is found. Otherwise, the values of the Lagrange multipliersare updated by means of subgradient optimization and the lower and upper bound algorithms arerepeated until a feasible solution is reached, that satisfies the given tolerance.
Our computational tests were performed for 1,750 problem instances, based on problems of10 different sizes, and the algorithm was terminated each time when a gap of less than 1% wasachieved, or if the maximum number of iterations was reached. Results demonstrate that theproposed Lagrangian relaxation framework is capable of efficiently producing optimal or near-optimal solutions to the problem. The sub-problems are solved heuristically and this means thatthe lower bound obtained from the constructed algorithm could in fact exceed the best known
20
upper bound. However, this was not observed in any of the problem instances, which proves thatour approach is robust and reliable.
The research presented in this paper can be extended in a number of important ways. Thestructure of the model considered in this paper is such that new constraints or cost componentscan be added easily to the model. The following are some recommendations for future work andresearch directions for enhancing the model: (1) The model can be naturally extended to con-sider multiple products. (2) We have assumed that there is no capacity restriction on the amountof product that can be stored or processed by a facility. We can replace the uncapacitated fixedcharge location problem by the capacitated fixed charge location problem and then integrate thiswith the proposed inventory model. The resulting model would include capacity considerations atthe distribution centers. (3) We can relax the single-sourcing restriction to allow a single retailerto be supplied by more than one distribution center. This relaxation is practical in capacitatedmodels or in models with multiple products. (4) Another important extension to our model is toallow lateral shipments between distribution centers. A distribution center may face a demandthat exceeds its inventory for a certain product that could be shipped from another distributioncenter with excess inventory for that product. Lateral shipments are known to reduce costs inpractice especially when both distribution centers (the provider and the recipient) are owned bythe same firm. Even if these distribution centers belong to different firms, the concept of lateralshipments can still reduce costs because a firm with inventory in excess of demand would gen-erally be willing to sell it at a reduced price. (5) We have assumed direct shipments between thedistribution centers and the retailers. An important extension is to incorporate routing decisions.The resulting model would then become a location-inventory-routing model.
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