The Multi-Period Variable Cost and Size Bin Packing ...et al., 2008). Therefore, researchers have...

The Multi-Period Variable Cost and Size Bin Packing Problem with Assignment Cost: Efficient Constructive Heuristics

Teodor Gabriel Crainic Franklin Djeumou Fomeni Walter Rei

July 2019

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Teodor Gabriel Crainic*, Franklin Djeumou Fomeni, Walter Rei Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT) and Department of Management and Technology, Université du Québec à Montréal, P.O. Box 8888, Station Centre-Ville, Montréal, Canada H3C 3P8

Abstract. It is well known that the bin packing problem is one of the most studied

combinatorial optimization problems. This paper proposes two novel variants of this problem

that have practical applications in logistics planning. These two new models are aimed at

minimizing, not only the cost of selecting the bins, but also the costs of assigning the items

to specific bins, which may depend on other criteria than the volumes of the items. We show

that in some practical applications, such as multi-modal logistics systems, the cost of

assigning an item to a bin needs to account for both the physical characteristics of the bins

and other economical attributes of the system. In addition to the new models, we also

develop a total of seven constructive heuristic algorithms for both models. Computational

results are presented, which show the efficiency of these heuristic algorithms as well as

the potential benefits of using the new models in practical applications.

Keywords: Multi-period bin packing, bin packing problem with assignment cost, heuristic,

logistics planning, integer programming.

Acknowledgements. While working on the project, the first author was Adjunct Professor,

Department of Computer Science and Operations Research, Université de Montréal. We

gratefully acknowledge the financial support provided by the Natural Sciences and

Engineering Council of Canada (NSERC), through its Discovery, Acceleration Grant and

Engage programs. We also gratefully acknowledge the support of Fonds de recherche du

Québec – Nature et technologies (FRQNT) through their team research project program.

Thanks are also due to Calcul Quebec and Compute Canada for providing the authors

access to their high-performance computing infrastructure.

Results and views expressed in this publication are the sole responsibility of the authors and do not necessarily reflect those of CIRRELT. Les résultats et opinions contenus dans cette publication ne reflètent pas nécessairement la position du CIRRELT et n'engagent pas sa responsabilité. _____________________________ * Corresponding author: [email protected]

Dépôt légal – Bibliothèque et Archives nationales du Québec Bibliothèque et Archives Canada, 2019

© Crainic, Djeumou Fomeni, Rei and CIRRELT, 2019

1 Introduction

Retailers and wholesalers are constantly faced with the need of moving goods acrosstheir distribution networks. This could be the need to ship goods from their warehousesto customers who have made the purchases, or the need to ship goods to some of theiroutlets for stock replenishment. The costs involved in such movements of goods representa significant part of the distribution and operating costs of these companies (Monczkaet al., 2008). Therefore, researchers have developed decision support tools, based on theuse of bin packing models, to reduce such distribution costs (Perboli et al., 2014).

Given a set of items and a set of bins, the traditional bin packing problem (BPP) aimsat loading all the items into the smallest possible number of bins, while ensuring that thepacking of each bin is feasible (Martello and Toth, 1990; Dyckhoff, 1990; Wascher et al.,2007). Various applications of the BPP can be encountered in the planning of telecommu-nication, transportation, production and logistics/supply chain systems (Crainic et al.,2011). In logistics systems planning, the items may represent the goods that need tobe moved, while the bins represent the transportation mediums available (containers,ships, trains, etc.). The traditional BPP assumes that all the bins are identical, thereforefocusing on the physical characteristics of the problem (e.g., packing constraints), while,ignoring other important considerations such as the variability in the costs of the bins,their sizes, and the economical characteristics of the items (Crainic et al., 2016).

A number of contributions address some of the above issues. One such contributionconsidered bins of different sizes (Wascher et al., 2007), yielding the class of variable sizebin packing problems (VSBPP), which takes into account containers of different sizes oreven different transportation modes. However, this variant of BPP still assumes thatall the bins have the same costs, which depend on their volumes. While this mightmake sense in theory, in practical applications, the cost of a bin depends not only on itsvolume, but also on many other external factors such as the type of the bin (e.g., regular,open top, thermal or refrigerated container) (Crainic et al., 2011). This has resulted inthe class of variable cost and size bin packing problems (VCSBPP) (Crainic et al., 2011;Monaci, 2002; Pisinger and Sigurd, 2005; Alves and de Carvalho, 2007). Another variantof the BPP, which generalizes the class of VCSBPP, has also been introduced in theliterature as the generalized bin packing problem (GBPP) (Baldi et al., 2012, 2014). Thelatter accounts for the fact that not all the items have to be loaded. In other words, theset of given items is composed of two subsets, one of compulsory items that have to beloaded into bins, and one of non-compulsory, which can either be loaded into the binsfor an additional profit or can be left behind.

It can be noted that in all the above variants of the BPP, the main focus is onselecting the bins at the lowest cost and loading them with any items. This means thatall these variants assume that the cost of loading an item into a bin depends only onthe item’s volume. This is a serious limitation, since other than the item’s volume, the


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cost of assigning an item to a bin may be influenced by many other components relatedto the item and to the bins, as will be discussed in Section 2. For example, in logisticssystems where items have to be shipped from one geographical region to another, it isnot sufficient to only focus on loading the items onto the bins. Indeed, these items mayhave to be transported in certain conditions and may be expected to be delivered withinsome time interval. It is therefore important to account for the type of the bin, which inthis case may be associated with a specific transportation mode, as well as for the traveltime of the corresponding bin when assigning items to bins.

The main objective of this paper is to bridge this gap in the literature and present anew variant of BPP, which generalizes all the existing variants. More precisely, we presenta BPP setting wherein the cost of assigning an item to a bin is explicitly accounted for andmay, or may not, depend solely on the item’s volume. In our settings, we show that thiscost depends on some characteristics of the items and of the bins, such as their types andthe time components related to both the bins and the items. We designate this versionof BPP, the multi-period variable cost and size bin packing problem with assignment costs(MVCSBPP-AC). A sub-variant of the MVCSBPP-AC, which explicitly considers thecost of loading an item to a bin as a function of the item’s and bin’s characteristics,but without the time component, is also studied in this paper. The latter variant willbe referred to as the variable cost and size bin packing problem with assignment costs(VCSBPP-AC). We will propose two mathematical models for these two problems.

In addition to the description and mathematical formulation of these two variantsof BPP, we will propose a set of constructive heuristic algorithms to solve these prob-lems. We propose three heuristic algorithms for the VCSBPP-AC and four heuristicalgorithms for the MVCSBPP-AC. In logistics settings, BPPs are continuously solvedboth to perform tactical capacity planning and to conduct transportation and storageoperations (Crainic et al., 2011). Therefore, when optimization methods are applied inthese settings to provide decision support, it is imperative that the developed methodsbe able to very rapidly find good-quality solutions to the considered BPPs. As will becomputationally shown in Section 6, the proposed heuristic algorithms are such methods,which make them appropriate for both practical applications or to be used in more so-phisticated solution strategies (either within metaheuristics or to provide initial solutionsfor commercial solvers).

To properly assess the efficiency of the proposed heuristics, we carry out a thoroughcomputational experiment with a set of 1300 problem instances of both VCSBPP-AC andMVCSBPP-AC. The objective of these experiments are threefold. Firstly, we evaluate thequality of the upper bounds provided by the proposed heuristic algorithms. This qualitywill be measured in terms of computational time and then in terms of the optimality gap.Secondly, we assess the ability of the heuristics solutions to improve the performance ofcommercial solvers if used as warm-start solutions. In other words, we set the objectivefunction value obtained by the heuristic algorithm as the initial upper bound within the

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commercial solver, so that every branch with upper bound larger than the pre-set valuewill automatically be pruned. Finally, we measure the impact of the MVCSBPP-ACmodel in terms of total cost reduction by comparison to a myopic approach which solvesthe same instances without considering the multi-period setting.

The results of the computational experiments show that depending on the struc-ture of the problem, some instances of these two problems can be solved by commercialsolvers within an acceptable amount of time. While commercial solvers struggle to solvea large number of these instances, the proposed constructive heuristics is able to pro-vide great trade-offs between computational time and optimality gap regardless of thestructure of the problem at hand. The heuristics prove to have the ability to quicklygenerate good quality feasible solutions to the problems that can either be used directlyin practice, or, as a starting solution to warm-start commercial solvers. Finally, com-paring the MVCSBPP-AC solutions with those of a myopic approach that consists ofsolving VCSBPP-AC instances for each time period independently, i.e., without consid-ering multi-period, resulted in that one is able to reduce the operational costs by up toabout 82% using the MVCSBPP-AC model.

The rest of this paper is structured as follows. In Section 2, we provide an in-depth description of the proposed novel variants of BPP. The corresponding mathematicalmodels are then presented in Section 3. Section 4 describes the three heuristic algorithmsfor the VCSBPP-AC model, while the four heuristic algorithms for the MVCSBPP-AC model are presented in Section 5. The results and analyses of the computationalexperiments carried out are discussed in Section 6. Finally, Section 7 provides someconcluding remarks as well as give some insights for potential extensions of this work.

2 Problem description

We consider a logistics system wherein on one hand, we have a set of items belongingto different stakeholders (e.g., retailers, wholesalers, etc.) that need to be shipped fromone geographical region to another. On the other, we have a set of transportation serviceproviders that participate in the system in the sense that, they form a pool of trans-portation units that could be selected to ship the items. Such a setting can be foundin some collaborative logistics systems (Yilmaz and Savasaneril, 2012) or as part of theoperations of fourth party logistics service providers (4PLs) (Huang et al., 2013).

Given a time horizon (e.g., one week, one month), subdivided into time periods ofequal length (e.g., one day), the problem studied in this paper consists of prescribing aplanning schedule for shipping the set of items from one geographical region to another atthe lowest possible cost using a subset of the available transportation units, also referredto as bins. The items as well as the bins are made available to the system at different

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time periods. The shipment planning prescribes, for each time period, a subset of selectedbins as well as the items to be loaded into each bin, with the aim of minimizing the totalcost over the entire time horizon.

Each item is characterized by its size and its associated time components, whichconsists of the time at which the items appears in the system and is available for shipping,the time window within which the item has to be delivered, and the time period beyondwhich delivery is not acceptable. Once an item is made available to the system, it can beloaded onto a bin that will depart straight away. It can also be delayed to be loaded ontoa bin which will depart at a later time period as long as it can be delivered before thelatest delivery threshold. Finally, it is also possible to ship an item using the services ofnon-participating transportation service providers (e.g., express courier services), whichwill be referred to as the spot market.

On the other hand, the bins are associated with time components representing thetime window within which they are available and can be used for shipping the items,their volumes capacities, their fix costs as well as their travel times, i.e., the amountof time they will need to travel from the departing region to the destination region.This logistics system assumes long distance between the two geographical regions, whichmakes it impossible to neglect the travel time of the bins between the two regions. It isparticularly important to account for such travel time when loading items to the bins,since the delivery time components of the items have to be satisfied. Moreover, eachavailable bin can be selected and loaded with any combination of items that can fit intoit, a.k.a. consolidation.

The overall objective of the problem is to minimize the total cost of shipping theitems, while satisfying the time and physical attributes of both the bins and the items.In terms of costs, there are three main types of costs, namely, the fix cost of selectingeach bin, the cost of shipping items via the spot market and the specific cost of assigningan item to a bin. In practical applications, the latter category of cost may comprise manyother sub-costs which depend on the characteristics of each item and those of the binonto which the item is assigned. More precisely, there may be a direct cost of assigningan item to a bin, which depends on the type of the item (e.g., its weight) and the type ofthe bin (e.g., the transportation mode). There may be a penalty cost for delaying an itemto be shipped later, thus accounting for storage costs. Moreover, an item that arrives atthe destination region before its delivery time window may incur an early delivery penaltycost, since it may require additional storage before it can actually be delivered. Similarly,an item that arrives after its delivery time window, may also incur a late delivery penaltycost.

The novelties in the description of this problem are reflected in the time componentsassociated with the characteristics of both the items and the bins, and in the explicitaccount for the cost of assigning an item to a specific bin. These novelties allow to

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describe new emerging practices in logistics planning (e.g., collaborative logistics planning(Yilmaz and Savasaneril, 2012) or part of the operations of 4PLs (Huang et al., 2013))that cannot be handled with existing mathematical models and for which we proposesuitable mathematical models. It is advantageous to take the time components of theitems and the bins into account and solve the problem over the entire time horizon. Infact, it allows the opportunity of delaying some items that could be assigned to a bin ina later period at a lower cost while still meeting the delivery requirements of the items.

3 Mathematical models

In this section, we firstly propose a mathematical model for the problem described inSection 2. This model is a variant of the BPP which generalizes all the existing variantswith account for two new features, namely, the time component and the relevant cost ofassigning items to bins. Secondly, a variant of this model which does not account for thetime component is also provided in this section.

3.1 Notation

Let T designates the length of the time horizon, i.e., the number of time periods in thetime horizon, with the index of each time periods being t. The goal of the model isto determine an operational planing for shipping the items which minimizes the totalcost for the entire time horizon, while meeting all the requirements of the system. Thedecisions for each time period prescribe the set of bins to be selected, the set of items tobe loaded onto each selected bin, as well as the set of items to be shipped via the spotmarket. We will first present the mathematical notation used throughout the paper,before presenting the general optimization formulation for the entire problem.

1. Notation related to the items:

• I: is the set of all the items available for the full time horizon, with each itembeing indexed by i.

• vi: is the volume of item i ∈ I.

• Ti = [ti, ti]: is the time window interval within which item i ∈ I is availableand has to be delivered. With ti being the time at which item i ∈ I becomesavailable and ti is the latest delivery time.

• γi: is the earliest delivery time of item i ∈ I, while δi: is its delivery due date,with ti ≤ γi ≤ δi ≤ ti.

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2. Notation related to the bins:

• J : is the set of all the bins available over the full time horizon. Each bin willbe indexed by j.

• Vj: is the volume capacity of the bin j ∈ J .

• Γj = [Γj,Γj]: is the availability time window of the bin j ∈ J .

• αj: is the time needed by bin j ∈ J to travel from the origin to the destination.

3. In addition, Tij: will denote the time window within which it could be possible toassign an item i ∈ I to a bin j ∈ J . i.e. Tij = [ti, ti−αj]∩Γj. Indeed an item canonly be assigned to a bin if the bin is actually available with the insurance that theitem will be delivered by its latest delivery time.

4. The cost parameters:

• fj: is the fixed cost of using bin j ∈ J .• pit: is the cost shipping item i ∈ I through the spot market at time periodt ∈ Ti.• atij: is the cost of assigning item i ∈ I to bin j ∈ J at time period t ∈ Tij.

5. The decision variables:

• ytj =

{1 if bin j ∈ J is selected at time period t ∈ Γj,0 otherwise.

• xtij =

{1 if item i ∈ I is assigned to bin j ∈ J at time period t ∈ Tij,0 otherwise.

• uti =

{1 if item i ∈ I is shipped via the spot market at time period t ∈ Ti,0 otherwise.

3.2 Formulation of the MVCSBPP-AC

The MVCSBPP-AC model can formally be written as follows.

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miny,x,u

∑j∈J

∑t∈Γj

fjytj +∑i∈I

∑j∈J

∑t∈Tij

atijxtij +

∑i∈I

∑t∈Ti

pituti (1a)

s.t.∑i∈I

vixtij ≤ Vjy

tj, ∀j ∈ J , t ∈ Γj, (1b)∑

j∈J

∑t∈Tij

xtij +∑t∈Ti

uti = 1, ∀i ∈ I, (1c)

∑t∈Γj

ytj ≤ 1, ∀j ∈ J , (1d)

ytj ∈ {0, 1}, ∀j ∈ J , t ∈ Tj, (1e)

xtij ∈ {0, 1}, ∀i ∈ I, j ∈ J , t ∈ Tij, (1f)

uti ∈ {0, 1}, ∀i ∈ I, t ∈ Ti. (1g)

The three terms of the objective function (1a) are respectively the total cost of se-lecting the bins, the total cost of assigning the items to the bins over the considered timeperiods and the total cost of using the spot market. The costs of selecting the bins andthe cost of using the spot market are straightforward to understand. However, the costof assigning an item to a bin at a specific time period takes into account the physicalcharacteristics and the time components of both the item and the bin. The constraints(1b) ensure that the volume capacity of each bin cannot be exceeded. The constraints(1c) ensure that each item is either assigned to a bin or shipped via the spot market overits active time interval. The constraints (1d) ensure that each bin can be selected atmost once during its availability time window. Finally, the constraints (1e)–(1g) enforcethe binary requirements of the variables.

3.3 Formulation of the VCSBPP-AC

The VCSBPP-AC is a sub-variant of the MVCSBPP-AC, described above, but yet, itcontains novel features with respect to existing variants of BPP. Indeed, it can be seenas a straightforward generalization of the VCSBPP (Crainic et al., 2011), that explicitlyaccounts for the specific costs of assigning items to bins. These costs depend on thevarious characteristics of both the items and the bins, and not only on the volumes of theitems as assumed in the VCSBPP. The mathematical formulation of the VCSBPP-ACis similar to that of (1), with the difference that the time component is dropped. It canformally be written as follows:

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miny,x,u

∑j∈J

fjyj +∑i∈I

∑j∈J

aijxij +∑i∈I

piui (2a)

s.t.∑i∈I

vixij ≤ Vjyj, ∀j ∈ J , (2b)∑j∈J

xij + ui = 1, ∀i ∈ I, (2c)

yj ∈ {0, 1}, ∀j ∈ J , (2d)

xij ∈ {0, 1}, ∀i ∈ I, j ∈ J , (2e)

ui ∈ {0, 1}, ∀i ∈ I. (2f)

The objective function (2a) minimizes the total cost of selecting the bins, the totalcost of assigning items to the bins and the total cost of using the spot market. Theconstraints (2b) are included to ensure that the volume capacity of each bin cannot beexceeded. The constraints (2c) ensure that each item is either assigned to a bin or shippedvia the spot market. Finally, the constraints (2d)–(2f) enforce the binary requirementsof the variables.

It should be noted that a solution to the MVCSBPP-AC can be found by solvingT instances of the VCSBPP-AC, i.e., one VCSBPP-AC instance for each time perioddefined by using the set of items and the set of bins that are made available for thetime period. Obviously, such an approach would solve the problem heuristically. Thisapproach will be explored in the computational experiments presented in Section 6 andwill serve as a benchmark to assess the value of the MVCSBPP-AC model.

4 Heuristic algorithms for the VCSBPP-AC model

In this section we will propose three constructive heuristic algorithms to efficiently findfeasible solutions to the VCSBPP-AC. For simplicity of notation we will also refer tothe VCSBPP-AC problem as the single period problem, and the corresponding heuristicalgorithm will be denoted HS (i.e., heuristic for the single period problem).

1. First heuristic for the VCSBPP-AC (HS1):The main idea of this heuristic algorithm is to find, for each item, the most suitablebin in which it can be loaded if such a bin has already been selected. If no suchbin has been selected yet, then it will simultaneously find the best suitable binfor this item as well as the best subset of items that can be packed into the binalong with the given item. The algorithm starts by initializing the set of selected

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bins and the set of assigned items to the empty sets. Then, the unassigned itemsare sequentially considered to find the most suitable bin in which they can beloaded. Indeed, for each item, the algorithm first looks for the selected bin withenough space for the item, and the smallest ratio of assignment cost over left-overcapacity once the item is loaded onto it. If such a bin exists, then the item will beloaded onto it. Otherwise, a new bin is possibly selected. Specifically, a new binis only selected if it is cost efficient to do so. The evaluation for selecting a newbin is performed by Algorithm 2 (the Profitable procedure), which measures thecontribution of selecting a new bin with respect to the overall cost. In particular,a new bin will be selected and the item loaded onto it, if there exists a subset ofunassigned items that can be loaded into the bin with overall cost smaller thanthe sum of individual costs of using the spot market for each of these items. Onceall the items have been investigated, the next step of the algorithm consists oftopping-up the left-over spaces in the selected bins where possible with unassigneditems. For each selected bin with left-over space, the unassigned items are sortedin the non-decreasing order of the ration of assignment cost over residual capacity,

i.e.,aij

V ∗j − vi. Then following this order, the items are loaded onto the bin until

it is full. The final step of the algorithm consists of looking for possible swapsbetween unassigned items and assigned ones. More precisely, the algorithm swapsan unassigned item with an assigned one if it is more profitable to do so. In theend, any item not assigned to a bin will then be shipped via the spot market. Thedetailed pseudo-code of this algorithm is described in Algorithm 1.

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Algorithm 1 : HS1

Input: Set of items I and set of bins J with their corresponding costs and volumes.Output: Set A of items assigned to bins and set of bins selected S.Initialize S = ∅Initialize A = ∅for all i ∈ I do

Identify the bin b ∈ S into which i can be loaded at smallest values ofaij

V ∗j − viand

aij < pi, with V ∗j being the residual capacity of bin j.if b exists then

Load i into b.A := A ∪ {i}.

elseIdentify the bin b ∈ J \ S such that Profitable(i, b) returns TRUE.if b exists then

Load i into bA := A ∪ {i}S := S ∪ {b}

elseLeave i as unassigned.

end ifend if

end for# Top-up the selected bins if possible.for all bin j ∈ S with left-over capacity V ∗j do

Sort the items in I \ A in the non-decreasing order ofaij

V ∗j − vi. Let us call the list

SILj.for i ∈ SILj do

if V ∗j − vi ≥ 0 thenLoad i into jV ∗j := V ∗j − vi

end ifend for

end for# Look for possible swaps between assigned items and unassigned ones.for i ∈ A assigned to a bin j0 do

for i′ ∈ I \ A doif (V ∗j0 − vi′ + vi ≥ 0) and (aij0 > ai′j0) then

Replace the item i with item i′ in the bin j0.V ∗j0 := V ∗j0 − vi′ + vi

end ifend for

end for

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Algorithm 2 : Profitable procedure for new bin selection for an item i

Input: Set of unassigned items I and set of unselected bins J with their correspondingcosts and volumes.Output: A bin index b with the corresponding packing.for all j ∈ J do

if aij < pi thenInitialize the total cost of the bin j to cj = fj + aijUpdate the left-over capacity Vj := Vj − viSort the items list I \ {i} in the non-decreasing order of

fj + ai′jVj − vi′

and call it

SILi

for all i′ ∈ SILi doif i′ can be loaded into j and ai′j < pi′ then

Load i′ into jcj := cj + ai′jVj := Vj − vi′

end ifend for

end ifend forSet b to be the bin such that cb = min {cj : j ∈ J }.Return b.

2. Second heuristic for VCSBPP-AC (HS2):While the algorithm HS1 focuses of packing each item into the best possible binand complete the packing of the bin, the second heuristic, HS2, will simultaneouslyfocus on selecting the bins and the accompanying packing which makes the bestcontribution to the total cost. This algorithm proceeds as follows. Given the setof selected bins (initially empty), the algorithm starts by sorting the bins in thenon-decreasing order of their potentials for contribution to the total cost. The po-tential for contribution to the cost for a bin is measured in terms the ratio betweenthe cost of the bin and the capacity reduction, which is defined by how much ofthe available items can be loaded into the bin. In other word, the bins are sorted

in the non-decreasing order of the valuefj∑

i∈Ivi − Vj

. Note that in case this ratio is

equal for two different bins, the one with the largest capacity is placed first. Next,the algorithm picks a bin following this order and finds the best subset of itemsto be packed into it. The best packing is determined by sorting the unassigneditems in the non-decreasing order of the ratio between the cost of assigning theitem to the bin and the left-over capacity in the bin once the item is loaded. Then,following this order, the items are loaded into the bin if 1) there is enough spaceavailable and 2) the cost of assigning the item to that bin is smaller than the costof sending the item to the spot market. The final part of the algorithm also con-

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sists of topping-up the selected bins and swapping unassigned items with assignedones where possible. The pseudo-code of this algorithm is detailed in Algorithm 3.

Algorithm 3 : HS2

Input: Set of items I and set of bins J with their corresponding costs and volumes.Output: Set A of items assigned to bins and set of bins selected S.Initialize S = ∅Initialize A = ∅Sort the bins in J in the non-decreasing order of

fj∑i∈I

vi − Vj.

Let SBL be the sorted bin listfor all j ∈ SBL do

Sort the items in I in the non-decreasing order ofaij

Vj − viLet SIL be the sorted item listfor all i ∈ SIL do

if vi ≤ Vj with aij < p∗i , where p∗i is the lowest spot market cost of item i thenload i into jA := A ∪ {i}Vj := Vj − vi

end ifend forif j is non-empty thenS := S ∪ {j}

end ifend for# Top-up the selected bins if possible.for all bin j ∈ S with left-over capacity V ∗j do


V ∗j − vi. Let us call the list SILj.

for i ∈ SILj doif V ∗j − vi ≥ 0 then

Load i into jV ∗j := V ∗j − vi

end ifend for




end ifend for

end for

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3. Third heuristic for VCSBPP-AC (HS3):The main idea of this algorithm is very similar to that of HS2 and it follows thesame steps as defined in HS2. This means that HS3 also focuses simultaneously onselecting the bins and the accompanying packing which makes the best contributionto the total cost. However, the main difference with HS2 lies in how the potentialfor a bin to contribute to the total cost is calculated. Here, for a given bin, thepotential for contribution to the total cost is calculated by considering the cost ofthe bin plus the assignment cost of the best packing with all the items available.In other words, each bin is considered with how best it can be packed with theavailable items and the total cost of such a packing is calculated. Then all thebins are sorted in the non-increasing order of these values. Next, the algorithmpicks a bin following this order and determines the best packing for the bin. Notethat the step of determining the best packing here is needed again since once a binhas been selected and packed at this stage, the previous packing of the next binmay include items that are already assigned. At either stages, the packing is doneby sorting the items in the non-decreasing order of the ratio between the cost ofassigning the item to the bin and the left-over capacity in the bin once the item isloaded. The pseudo-code of this algorithm is given in Algorithm 6, which appearsin Appendix 1.

5 Heuristic algorithms for the MVCSBPP-AC model

This section describes the four constructive heuristic algorithms that we have developedfor finding feasible solutions for the MVCSBPP-AC. All four MVSCBPP-AC heuristicalgorithms follow the same framework, which will be detailed in Section 5.1. The maindifference between these algorithms lies in how the framework is implemented, as willbe discussed in Section 5.2. For the sake of clarity, we will refer to the heuristic for theMVCSBPP-AC as the heuristic for the multi-period problem, which will be denoted HM.

5.1 The main framework for the heuristic algorithms (HM)

The proposed main framework for solving the MVCSBPP-AC consists of the followinggeneral steps:

1. redefine the model by duplicating the bin and item selection variables for each timeperiod over the considered horizon,

2. decompose the MVCSBPP by time period by relaxing the constraints enforcingthe bins and items to be selected at most and exactly once, respectively, over their

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associated availability time window,

3. construct a feasible solution to the MVCSBPP by iteratively solving the VCSBPP-AC subproblems independently and, based on the current solutions obtained, iden-tify at which period specific bins can be selected in the most cost-efficient way,assign the items based on this selection and update the set of unassigned bins anditems accordingly.

In much detail, the framework starts by duplicating both the items and the bins, one foreach time period within their respective availability time windows. For example, if theavailability time widow of an item contains five time periods, it will then create five copiesof this item, one for each time period. The bins are duplicated similarly. The duplicationof the items and the bins allows us to decompose the problem by time period over theconsidered time horizon. The decomposed VCSBPP-AC instances can therefore be solvedusing either of the three heuristics presented in Section 4. At this point, the combinedsolution of all the independent VCSBPP-AC instances is most certainly infeasible for theMVCSBPP-AC, since some bins may be selected at different time periods or some itemsmay also be assigned to bins at different time periods.

Given the combined solution of these VCSBPP-AC instances, which recommendsa selection of bins and assignment of items for each time period in the time horizonconsidered, the next step of the framework determines the most efficient time period forselecting and packing each bin. As already remarked, since instances of VCSBPP-ACare solved for each time period with no regard to the other time periods, the solutionobtained may prescribe that a bin be selected and packed for more than one time period.Therefore, the procedure described in Algorithm 5 (the MostEfficient procedure) willallow to determine the time period at which selecting a bin could have the most significantimpact on the overall total cost. Once, such a time period is determined for a bin, itsselection for any other time period is cancelled and the items initially packed in it areset as unassigned. The process is repeated until there is no more unassigned item left orit becomes more costly to assign left-over items to a bin than to ship them via the spotmarket. For any unassigned item in the above procedure, the post-optimization part ofthe algorithm will try to look for a selected bin that can accommodate the item. If sucha bin exists, then the item is assigned to it provided it is less costly to do so than to usethe spot market. Finally, any item that cannot be assigned this way, is then shipped viathe spot market at the time period with the smallest spot cost.

Note that the construction of a feasible solution to MVCSBPP-AC from the solutionsof VCSBPP-AC is carried out in the sequence of the MostEfficient routine. Thisroutine allows to determine the time period for selecting and packing each bin as shownin Algorithm 5. Indeed, it considers all the time periods in the availability time windowof the bin and evaluates the bin’s loss-cost contribution. The loss-cost contribution for atime period is calculated as the total cost of the VCSBPP-AC corresponding to the time

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period minus the cost of selecting and packing the bin as given by the solution to theperiod’s VCSBPP-AC. The MostEfficient time period for selecting and packing thebin is then chosen to be the time period with the smallest loss-cost contribution value.

It can be seen in Algorithm 4 that the MostEfficient routine is sequentially calledin order to fix the time period for selecting and packing each bin. One can note that inAlgorithm 4, the bins are considered for selection without any specific ordering. However,it may be more efficient to consider the bins in a specific order. For example, we proposeto first look for the MostEfficient time period for selecting and packing each bin,then start with the bin that has the largest cost contribution among all the unassignedbins. This will result into the two approaches for constructing a feasible MVCSBPP-ACsolution from the combined solutions of the VCSBPP-AC instances, i.e., 1) no orderingof the bins, and 2) order the bins by cost contribution. Finally, the combination of thesetwo approaches with the three heuristics for solving the VCSBPP-AC has enabled us topropose four upper bound procedures for the MVCSBPP-AC as summarized in Section5.2.

5.2 Summary and description of the four heuristic algorithms

In Section 5.1 we presented a main framework for solving the problem MVCSBPP-AC.However, there are different possibilities of implementing this framework that will poten-tially result in different solutions. The main variations in implementing this frameworklie in 1) the heuristic that is applied to solve the VCSBPP-AC subproblems, and 2) theorder by which the bins are selected. The combination of these components has leadus to adopt four upper bounding approaches for the MVCSBPP-AC problem, which aresummarized as follows:

1. First heuristic for MVCSBPP-AC (HM1):For each time period, the VCSBPP-AC instance is solved using Algorithm 1. ThenAlgorithm 5 is used to determine the most efficient time period for selecting andpacking a bin, without any ranking of the bins.

2. Second heuristic for MVCSBPP-AC (HM2):For each time period, the VCSBPP-AC instance is solved using Algorithm 3. ThenAlgorithm 5 is used to determine the most efficient time period for selecting andpacking a bin, without any ranking of the bins.

3. Third heuristic for MVCSBPP-AC (HM3):For each time period, the VCSBPP-AC instance is solved using Algorithm 6. ThenAlgorithm 5 is used to determine the most efficient time period for selecting andpacking a bin, without any ranking of the bins.

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Algorithm 4 : The main framework of the the MVCSBPP-AC heuristic algorithms

Input: Set of all the items I and set of all the bins J with their corresponding cost,time and volume parameters.Initialize the set of selected bins for each time period St = ∅ for all t = 1, . . . , TInitialize the set of assigned items for each time period At = ∅ for all t = 1, . . . , Twhile (I 6= ∅) and (one item packing is made) do

for t = 1, . . . , T doSolve the VCSBPP-AC with items set It and bins set Jt.

end forfor all j ∈ J do

Identify the time period t∗ for which MostEfficient(j, t∗) returns TRUEif t∗ exists then

Add j to the list of bins selected at time t∗ i.e. St∗ := St∗ ∪ {j}Delete all the items loaded into j from the set of items that are yet to be

assignedfor period t∗ i.e. It∗ = It∗ \ (It∗(j)), where It∗(j) is the set of items loaded

into bin j.Add the items packed into bin j to the set of items already assigned at time

t∗

i.e. At∗ = At∗ ∪ It∗(j).for all t ∈ Γj \ {t∗} do

Delete all the items loaded into j from the set of items that are yet to beassigned for period t i.e. It = It \ (At∗ ∩ It(j)).

end forend if

end forUpdate the set I i.e. I = I \ ∪t=1,...,TAt

Update the set J i.e. J = J \ ∪t=1,...,TStend whileLook for a selected bin which can accommodate any unassigned item.Look for an assigned item that can be replaced by an unassigned one and swap.Assign any left-over item to spot market at the period with the smallest spot cost.

4. Fourth heuristic for MVCSBPP-AC (HM4):For each time period, the VCSBPP-AC instance is solved using Algorithm 6. Thenin order to determine the most efficient time period for selecting and packing a bin,the values qtj =

∑i∈A(j)

q∗i representing the alternative cost of shipping the items in j

at time period t via the spot market are calculated for all the bins. The bin j∗ isselected at time period t∗ for which qt

∗j∗ = max

{qt∗j = max

{qtj : t ∈ Γj

}: j ∈ J

}.

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Algorithm 5 : MostEfficient procedure for time period to select a bin j

for all t ∈ Γj doDelete any item that has already been assigned somewhere else from jTop-up the bin j with unassigned items if it is non-emptyLet qtj be the alternative cost of shipping the items in j at time period t via the

spot marketi.e. qtj =

∑i∈A(j)

q∗i , where A(j) is the set of items loaded into j and q∗i = min{qti :

∀t ∈ Ti}end forSet t∗ to be the time period such that qt

∗j = max

{qtj : t ∈ Γj

}.

Return t∗.

6 Computational Results

In this section, we present the outcome of the computational experiments that we carriedout for the two mathematical models and the proposed heuristic algorithms. The goalof these experiments are threefold. Firstly, we evaluate the quality of the upper boundsprovided by the proposed heuristic algorithms. Here the quality is mainly measured interms of computational time and in terms of the optimality gap. Secondly, we assessthe ability of the heuristics solutions to improve the performance of commercial solvers ifused as warm-start solutions. In other words, we set the objective function value obtainedby the heuristic algorithm as the initial upper bound within the commercial solver, sothat every branch with upper bound larger than the pre-set value will automaticallybe pruned. It is noteworthy to mention that most literatures report the inefficiency ofcommercial solvers in solving bin packing models (Crainic et al., 2011; Baldi et al., 2012;Correia et al., 2008). Finally, we measure the impact of solving the MVCSBPP-AC modelin terms of the total cost reduction gain by comparison to a myopic approach which solvesthe same instances without considering the multi-period setting. These experiments areconducted with four sets of randomly generated data (two sets for testing the VCSBPP-AC algorithms and two sets for testing the MVCSBPP-AC algorithms) for a total of 1300test instances. These sets of instances are generated so as to represent situations wherethe items sizes are relatively small compared to the bins capacities, as well as situationswhere there is a mix of large and medium size items compared the capacities of the bins.The details of the procedure applied to generate the data will be described in Section6.1.

All the algorithms were coded in the C programming language and were compiledwith gcc version 4.8.1 (GNU, 2018). In order to compute the optimality gap of theinstances, they were solved using the MIP solver of CPLEX version 12.6.3 (IBM, 2016).The computational platform used for running all the heuristic algorithms was a machinewith 8GB of memory and a processor at 1.8 GHz, while the computational platform used

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for solving the instances to optimality was the supercomputer “CEDAR” managed by“Calcul Quebec” and “Compute Canada”. We used a single node on this cluster witha 2.26 GHz processor. The memory requested to solve the problems were 8 GB for allthe instances with a time limit of 1 hour. Furthermore, we set the internal parametersof CPLEX to only use one thread (i.e. no parallel computing) and to generate as manycutting planes as needed. The rest of this section is divided as follows: we first presenthow the test instances were generated (Section 6.1) and then we discuss the results ofthe experiments conducted.

6.1 Description of the test instances

In this subsection, we will describe the test instances that we used to assess the efficiencyof our algorithms for both the VCSBPP-AC and the MVCSBPP-AC. For each of theseproblems, we have used two sets of test instances that have been generated randomly.The two sets of instances used for the MVCSBPP-AC have some similarities with the twosets used for the VCSBPP-AC, with the major difference being that the MVCSBPP-ACinstances are generated over multiple periods. Therefore, for both the VCSBPP-AC andthe MVCSBPP-AC, we will designate the two sets of test instances simply as Set-1 andSet-2. The instances of Set-1, for both problems, represent the situation where the sizesof the items are very small compared to the sizes of the bins. While, instances of Set-2represent the situation where the items are of a mix of large and medium sizes. We shouldnote that these instances are similar in spirit with the instances used by (Baldi et al.,2012; Crainic et al., 2011). However, instead of considering bins of specific capacities,we assume that the bins capacities are completely random in order better to reflect thefact that, in practice, the bins in our models refer to residual capacities of standardcontainers.

1. Set-1 for MVCSBPP-AC:Ten instances are randomly created for each combination of the following parame-ters:

• The number of items in the set {100, 200, 300, 500, 750, 1000}.• The volume of items in the set {1, 2, . . . , 20}.• The bins’ capacities are also randomly generated in the interval [50, 250].

• Five combinations of the number of bins and the number of time periods, whichdefine the five types of instances that we have in this set, have been used asfollows: (1) the number of bins for each instance is randomly generated in theinterval [10, 100] and there are 10 time periods; (2) the number of bins foreach instance is randomly generated in the interval [20, 100] and there are 20time periods; (3) the number of bins for each instance is randomly generated

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in the interval [50, 100] and there are 30 time periods; (4) the number of binsfor each instance is randomly generated in the interval [50, 100] and there are10 time periods, with the bin capacities restricted to the interval [50, 150];(5)the number of bins for each instance is randomly generated in the interval[50, 100] and there are 15 time periods, with all bin capacities set to be equalto 150.

• Item volumes randomly generated in the set {1, 2, . . . , 20}• The cost of selecting each bin bin j is set to 100

√Vj.

• the availability time windows for both the items and the bins are randomlychosen time intervals between 0 and T and with lengths randomly varyingfrom 3 to 6 time periods.

• The cost of assigning an item to a bin are generated as follows: atij = β1 +β2×(t− ti) + β3 ×max{(γi − t), 0}+ β4 ×max{(t− δi), 0}, where βk, k = 1, 2, 3, 4are random numbers in the intervals [20, 30], [3, 5], [1, 2] and [3, 5], respectively.This formula represents in practice the fix item-to-bin assignment costs as wellas the different penalty costs related to late shipping, early delivery and latedelivery, respectively.

• The cost of using the spot market is chosen as uti = u1 +β2× (t− ti), where u1

is a random number between [100, 200] and β2 is the same as in the assignmentcost.

2. Set-2 for MVCSBPP-AC:Ten instances are randomly created for each combination of the following parame-ters:

• Number of items in the set {100, 200, 300, 500, 750, 1000}.• The bins’ capacities for all the instances are randomly generated in the interval

[20, 250].

• Six combinations of the number of bins, the volumes of the items and thenumber of time periods which define the six types of instances that we havein this set: (1) the number of bins are in the interval [75, 100], there are 10time periods and the items volumes are in the interval [1, 100]; (2) the numberof bins are in the interval [100, 150], there are 10 time periods and the itemsvolumes are in the interval [20, 100]; (3) the number of bins are in the interval[150, 200], there are 10 time periods and the items volumes are in the interval[50, 100]; (4) the number of bins are in the interval [75, 100], there are 20 timeperiods and the items volumes are in the interval [1, 100]; (5) the number ofbins are in the interval [100, 150], there are 20 time periods and the itemsvolumes are in the interval [20, 100]; (6) the number of bins are in the interval[150, 200], there are 20 time periods and the items volumes are in the interval[50, 100].

• All the remaining parameters are generated as in Set-1.

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3. Set-1 for VCSBPP-AC:Ten instances are randomly created for each combination of the following parame-ters:

• Number of items in the set {100, 200, 300, 400, 500, 600, 700, 800, 900, 1000}.• Item volumes randomly generated in the set {1, 2, . . . , 20}• Two combinations of the number of bins and the bins capacities which define

the two types of instances that we have in this set: (1) the number of binsfor each instance is randomly generated in the interval [10, 100] and bins ca-pacities are in the interval [50, 250]; (2) the number of bins for each instanceis randomly generated in the interval [10, 100] and bins capacities are in theinterval [200, 250];

• All the remaining parameters are generated as in Set-1 for MVCSBPP-AC.

4. Set-2 for VCSBPP-AC:Ten instances are randomly created for each combination of the following parame-ters:

• Number of items in the set {100, 200, 300, 500, 750, 1000}.• Six combinations of the number of bins, the volumes of the items and the

capacities of the bins which define the six types of instances that we have inthis set: (1) the number of bins are in the interval [75, 100], the bins capacitiesare in the interval [50, 250] and the items volumes are in the interval [1, 100];(2) the number of bins are in the interval [100, 150], the bins capacities arein the interval [50, 250] and the items volumes are in the interval [20, 100];(3) the number of bins are in the interval [150, 200], the bins capacities arein the interval [50, 250] and the items volumes are in the interval [50, 100];(4) the number of bins are in the interval [50, 100], the bins capacities are inthe interval [100, 200] and the items volumes are in the interval [1, 100]; (5)the number of bins are in the interval [100, 150], the bins capacities are in theinterval [100, 200] and the items volumes are in the interval [20, 100]; (6) thenumber of bins are in the interval [150, 200], the bins capacities are all equalto 150 and the items volumes are in the interval [50, 100];

• All the remaining parameters are generated as in for MVCSBPP-AC.

Overall, the reader may notice that the various parameters, e.g., the number of itemsin the set, the item volumes, the bins’ capacities, etc., associated with each set aresometimes quite different from one another. The idea behind this is to ensure that ourinstances reflect a higher diversity in problem characteristics as well as the generality ofthe models, which do not not deal with specific types of bins or items, etc.

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6.2 Quality of the heuristic solutions

The objective of this section is to present the efficiency of the proposed heuristic algo-rithms described in Sections 4 and 5. Indeed, these algorithms are constructive heuristicalgorithms, which aim to produce a feasible solution in a very short amount of time.These heuristics can be directly applied to solve practical problems, or, they can producewarm-start solutions for a commercial solver, as it will be shown in Section 6.3, or moresophisticated metaheuristic algorithms.

6.2.1 Quality assessment for the VCSBPP-AC heuristic solutions

The results obtained for assessing the quality of the heuristic solutions provided by thethree VCSBPP-AC algorithms HS1, HS2 and HS3 are shown in Table 1 for the test in-stances in Set-1 and in Table 2 for the test instances in Set-2. In each of these tables,the first column indicates the type group of the instances being solved and the secondcolumns ‘|I|’ indicates the number of items. For the next eight columns, we have theaverage (out of ten instances) percentage optimality gap and the average computationaltime in seconds for the three heuristic algorithms indicated as ‘HS1’, ‘HS2’ and ‘HS3’,respectively, as well as the optimality of the branch-and-cut algorithm of CPLEX, indi-cated as ‘BnC’. In fact, since we have set a time limit of 3600 seconds into CPLEX, ithas often been the case that CPLEX could not find an optimal solution within this timelimit. Therefore, the gap reported here refers to the gap between the best upper boundfound by CPLEX and best the lower bound found. Similarly, the gap for each of ourthree heuristics are calculated with respect to the best lower bound found by CPLEX asshown in Equation (3). In Appendix 1, we present the overall spread of the percentagegaps for these instances.

gap =

(Heuritic Solution− CPLEX lower bound

CPLEX lower bound

)× 100. (3)

From the results for the instances of Set-1 reported in Table 1, it appears that HS2performs very badly for these instances, with its average optimality gap varying between21% and 50%. The performance of HS3 seems not to be competitive for these instancesas well with the average optimality gap nearing 20% in some cases. However, HS1 has avery good performance for this set of instances as it can consistently find solutions within6% of optimality for most of the instances of Type-1 and within 3% of optimality for mostof the instances of Type-2. In terms of computational times, one can note that all threeheuristics (HS1, HS2 and HS3) only require few seconds to solve the problem. Indeed,the average computational times of HS1 for instances with more than 500 items are 5seconds or less. While, for instances with less than 500 items, it only requires less thanhalf a second. On the other hand, one can note that the Branch-and-Cut algorithm of

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Table 1: Output of VCSBPP-AC instances of Set-1

HS1 HS2 HS3 BnC|I| Gap Time Gap Time Gap Time Gap Time

100 10.99 0.01 40.06 0.01 7.69 0.01 0.00 32.80200 9.75 0.06 40.90 0.11 7.94 0.11 0.01 260.43300 8.41 0.17 37.99 0.30 13.63 0.33 0.01 1,328.06400 6.91 0.44 31.01 0.56 12.67 0.76 0.04 2,970.02

Type-1 500 6.66 0.62 24.94 0.79 19.41 1.05 0.05 2,345.45600 6.65 0.90 19.75 1.18 18.02 1.49 0.05 2,608.62700 6.49 1.72 21.20 1.89 16.96 2.96 0.10 2,732.17800 6.43 3.38 20.43 3.55 15.20 6.09 0.12 3,600.07900 6.56 3.04 17.42 3.08 16.55 5.65 0.11 3,240.27

1000 6.83 5.79 14.97 5.83 16.10 10.23 0.15 3,600.07100 6.25 0.01 51.77 0.02 5.87 0.01 0.00 1.58200 3.44 0.06 48.83 0.11 4.40 0.11 0.00 11.59300 3.15 0.17 42.02 0.32 5.67 0.32 0.01 13.12400 3.60 0.39 29.57 0.66 3.52 0.73 0.01 33.94500 3.12 0.54 34.10 0.88 18.25 1.02 0.02 535.34

Type-2 600 4.00 0.77 30.29 0.99 20.82 1.45 0.00 92.08700 3.20 1.55 27.15 1.86 16.14 2.90 0.00 122.21800 3.79 3.13 23.78 3.54 12.39 5.83 0.00 311.00900 4.34 2.91 23.76 2.97 19.66 5.45 0.00 502.09

1000 4.24 5.59 21.38 5.52 16.71 10.08 0.00 586.34

CPLEX is unable to optimally solve all the instances within the 3600 seconds time limit.But, within this time limit, it is able to find feasible solutions within 0.15% of optimalityfor all the instances. Most of these gaps are observed for the Type-1 instances, which arealso the instances for which the optimality gap of the best of our three algorithms (HS1)is above 6% of optimality. This suggests that the instances in this group may have somecharacteristics such as symmetries which make them harder to solve for both CPLEX andour algorithms. Indeed, having items of small volumes against bins with larger capacitiesimplies that there will be a lot of similarities between the items in the sense that replacingan item by another will often have no significant effect on the objective function.

Although CPLEX is able to find feasible solutions within 0.15% of optimality, thecomputational times required are rather unacceptable. For example, instances in theType-2 group require up to 500 of seconds, while most of those in the Type-1 grouprequire up to the 3600 seconds limit to get to this optimality gap. This may not beacceptable for some practical applications. Indeed, bin packing problems are often solvedat the operational level where the time available to make decision is often quite limited(Crainic et al., 2011). This is achievable by one of our proposed heuristic algorithm(HS1), which is able to produce good quality solutions within less than five seconds.Moreover, given that the computational times of HS1 are so small, it means that thisheuristic algorithm is a good candidate for generating starting solutions that may be used

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either by much sophisticated metaheuristic algorithms, or even by commercial solvers. Inthe latter case, we will show in Section 6.3 how these solutions are able to dramaticallyimprove the computational time of CPLEX for the difficult instances in Set-1.

Table 2: Output of VCSBPP-AC instances of Set-2

HS1 HS2 HS3 BnC|I| Gap Time Gap Time Gap Time Gap Time

100 2.52 0.11 7.11 0.01 7.04 0.01 0.00 2.92200 4.44 0.70 7.43 0.05 9.15 0.06 0.01 23.58

Type-1 300 4.11 1.56 7.77 0.13 9.58 0.16 0.01 51.05500 4.51 4.43 7.89 0.37 10.16 0.45 0.01 169.04750 3.69 10.23 7.67 0.94 10.11 1.05 0.01 356.58

1000 3.10 17.20 7.33 1.57 9.59 2.16 0.01 908.36100 0.01 0.10 0.01 0.01 0.01 0.01 0.00 0.21200 0.29 0.67 0.29 0.03 0.29 0.04 0.01 0.96

Type-2 300 0.22 2.05 0.19 0.06 0.19 0.09 0.01 2.69500 0.35 5.67 0.38 0.16 0.38 0.23 0.01 53.53750 0.41 12.45 0.41 0.35 0.41 0.49 0.01 66.68

1000 0.39 20.99 0.38 0.57 0.38 0.86 0.01 47.63100 0.00 0.18 0.00 0.01 0.00 0.01 0.00 0.20200 0.00 1.08 0.00 0.03 0.00 0.05 0.00 0.32

Type-3 300 0.00 3.44 0.00 0.08 0.00 0.12 0.00 0.46500 0.00 9.77 0.00 0.21 0.00 0.32 0.00 0.81750 0.00 20.95 0.00 0.46 0.00 0.68 0.00 1.20

1000 0.00 36.22 0.00 0.80 0.00 1.21 0.00 1.66100 3.17 0.11 13.24 0.01 13.41 0.01 0.00 0.85200 4.22 0.75 14.30 0.04 14.78 0.04 0.00 1.43

Type-4 300 2.99 1.60 14.50 0.10 14.87 0.12 0.01 1.36500 3.52 4.52 14.57 0.24 14.68 0.44 0.01 2.86750 3.07 9.29 14.28 0.57 14.43 0.70 0.01 5.43

1000 2.68 14.95 13.62 0.57 13.61 1.07 0.01 8.66100 0.61 0.10 0.70 0.01 0.70 0.01 0.00 0.25200 1.14 0.95 1.39 0.02 1.39 0.03 0.01 0.92

Type-5 300 1.02 3.34 1.21 0.06 1.21 0.08 0.01 1.27500 1.04 8.73 1.31 0.15 1.31 0.22 0.01 3.26750 1.04 20.65 1.24 0.32 1.24 0.48 0.00 6.88

1000 0.97 35.76 1.09 0.57 1.09 0.84 0.01 11.65100 0.00 0.17 0.00 0.01 0.00 0.01 0.00 0.19200 0.00 1.07 0.00 0.03 0.00 0.05 0.00 0.31

Type-6 300 0.00 3.24 0.00 0.08 0.00 0.12 0.00 0.48500 0.00 8.92 0.00 0.22 0.00 0.31 0.00 0.82750 0.00 19.92 0.00 0.46 0.00 0.66 0.00 1.33

1000 0.00 34.85 0.00 0.80 0.00 1.17 0.00 1.65

It appears in Table 2 that all our three upper bounding algorithms perform very wellfor all the instances in Set-2. Indeed, optimality gaps larger than 1% are only foundfor the Type-1 instances (where there are items with very small volumes). This againsuggests that instances that include items of small size are difficult to solve. Nevertheless,in Set-2, HS1 seems to be the best performing of the three algorithms again, with itsoptimality gap varying between 0.00% and 4%. Unlike Set-1, CPLEX efficiently solvesthe instances of Set-2. It can consistently find the optimal solutions for these instances,with computational times exceeding 50 seconds only for a handful of instances. Whereas,

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the computational times for the three heuristic algorithms never exceed 36 seconds.

In summary, the computational experiments conducted in this section shows thatthe newly proposed VCSBPP-AC model may sometimes be solved to optimality usingcommercial solvers. This may mainly be the case for instances with large size items or amix of large and medium size items. In the case of small size items, the CPLEX solverfinds it difficult to solve the instances. However, one of our proposed upper boundingtechnique (HS1) has proved to be able to provide a great trade-off between computationaltime and optimality gap regardless of the type of the problem at hand. HS1 seems tohave some useful traits for practical problems. First, it has the ability of producinga good quality solution in a very small amount of time, which is extremely importantin practical applications. Secondly, it is also a good candidate for generating startingsolutions either for more sophisticated metaheuristic algorithms, or, as a mean to speed-up the performance of commercial solvers.

6.2.2 Quality assessment for the MVCSBPP-AC heuristic solutions

In this section we discuss the results obtained for assessing the quality of the solutionsprovided by the proposed four MVCSBPP-AC heuristic algorithms: HM1, HM2, HM3and HM4 (presented in Section 5.2). These results are presented in Table 3 for thetest instances in Set-1 and in Table 4 for the test instances in Set-2. The first twelvecolumns of each of these tables can be understood in the same way as in Tables 1 and2. In these experimentations, there have been numerous instances that could not besolved to optimality (i.e., within 0.05% of optimality) by CPLEX, and sometimes, thereare instances for which CPLEX could not find a feasible solution within 3600 seconds.Therefore, we report in each of the tables, the number of instances out of ten thatwere solved by CPLEX (Column “#Sol.”) as well as the number of instances out often for which CPLEX was able to find at least one feasible solution (Column “#Feas.”)within the time limit. Note that the average optimality gap of CPLEX here refers to theaverage optimality gap for the instances for which CPLEX was able to find both upperand lower bounds. We also provide an overall view of the spread of the percentage gapsin Appendix 2.

From the results for the instances of Set-1 reported in Table 3, it appears that theaverage percentage gaps vary between 0.17% and 17.04% for HM1, between 2.41% and22.30% for HM2, between 2.67% and 17.17% for HM3, and between 0.94% and 11.38.%for HM4. It can also be seen that the optimality gap of CPLEX varies between 0.00%and 3.51%. A quick comparison between our four upper bounds shows that HM4 is ableto provide the best quality feasible solutions among the four heuristics. The results inTable 3 also show that all four proposed heuristics procedures for MVCSBPP-AC are veryfast. For example HM4 is consistently able to provide solutions within 6% of optimalityin less than 15 seconds. In contrast to this, CPLEX proved to be very computationally

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Table 3: Output of MVCSBPP-AC instances of Set-1

HM1 HM2 HM3 HM4 BnC|I| Gap Time Gap Time Gap Time Gap Time Gap Time #Sol. #Feas.100 15.82 0.01 17.55 0.03 12.90 0.03 7.45 0.05 0.00 16.12 10 10200 16.24 0.06 21.64 0.16 16.05 0.23 7.07 0.23 0.06 1,493.74 8 10

Type-1 300 12.32 0.13 18.73 0.42 14.47 0.69 8.07 0.51 0.09 3,035.26 8 10500 7.89 0.39 14.07 1.15 14.72 2.18 8.84 1.21 0.12 2,650.34 6 10750 4.27 1.05 8.70 3.40 9.68 7.58 4.36 2.96 0.15 2,917.40 6 101000 1.06 2.50 5.98 7.89 7.31 15.84 0.94 6.90 0.11 3,600.21 6 10100 16.81 0.02 19.71 0.06 14.76 0.07 8.76 0.11 0.01 103.48 10 10200 14.56 0.06 17.28 0.18 15.25 0.26 7.23 0.24 0.02 1,072.54 9 10

Type-2 300 10.77 0.13 14.33 0.59 11.36 0.85 5.71 0.81 0.47 3,600.05 2 9500 11.47 0.54 14.58 3.38 12.89 4.98 5.76 3.18 0.73 3,600.06 0 9750 5.58 1.34 8.17 8.53 7.76 13.83 1.90 6.97 0.33 3,600.19 0 61000 1.45 2.67 3.72 15.50 3.98 25.81 1.16 13.33 0.52 3,600.33 0 3100 14.40 0.03 16.35 0.09 16.12 0.11 7.94 0.18 0.01 124.89 10 10200 17.04 0.09 19.21 0.41 17.17 0.56 8.31 0.64 0.20 1,991.09 7 10

Type-3 300 14.12 0.19 17.98 1.04 13.63 1.41 7.05 1.38 0.56 3,600.01 0 10500 11.47 0.54 14.58 3.31 12.89 4.89 5.76 3.18 0.73 3,600.02 0 9750 5.44 1.34 8.01 8.55 7.59 13.69 1.75 6.97 0.33 3,600.29 0 61000 0.17 2.68 2.41 16.16 2.67 25.98 1.41 13.33 0.47 3,600.38 0 2100 15.64 0.02 22.30 0.06 13.00 0.07 8.11 0.09 0.01 90.11 10 10200 14.12 0.07 20.68 0.26 14.70 0.38 7.70 0.37 0.04 2,770.56 8 10

Type-4 300 13.54 0.18 20.28 0.71 15.08 1.12 7.60 0.82 0.09 3,437.43 3 10500 10.03 0.59 14.54 2.39 12.52 4.07 6.27 2.43 0.24 3,600.03 0 9750 5.18 1.60 8.38 6.06 9.21 10.84 3.27 4.96 0.30 3,600.21 0 61000 4.34 3.12 7.61 11.13 10.02 21.36 3.69 8.45 0.38 3,600.13 0 6100 11.13 0.03 12.04 0.09 12.58 0.17 11.38 0.26 2.49 2,774.82 3 10200 6.32 0.14 5.80 0.48 7.51 0.94 5.89 1.08 3.51 3,600.02 0 10

Type-5 300 7.88 0.24 6.76 0.95 8.40 1.83 7.39 1.63 2.16 3,600.06 0 10500 6.10 0.87 5.17 4.03 7.37 7.14 6.15 5.38 1.47 3,600.04 0 8750 5.73 2.38 4.74 12.28 7.00 23.47 6.13 13.40 1.86 3,600.06 0 81000 6.69 3.78 5.91 20.81 7.98 42.14 6.74 16.40 1.18 3,600.06 0 10

expensive when applied to solve these instances. In fact, the time limit of 3600 secondswas reached for a vast majority of the instances in Set-1. Moreover, the two last columnsof Table 3 suggest that out of 300 test instances in Set-1, CPLEX was unable to find afeasible solutions to 41 of them, and was also unable to find an optimal solution for about194 instances. This poor performance of CPLEX may also be linked to the structure ofthe problem. In fact, for these instances, the size of the items are significantly smallerthan the capacity of the bins. Therefore, branching on a variable, which in theory maymean removing an item from a bin, or adding an item to a bin, will often not be sodifferent from one item to another. Thus increasing the size of the branch-and-boundtree. The most important part of these results is that, the proposed heuristics are ableto find feasible solutions to all the 300 instances, even when CPLEX fails to do so withinthe time limit. This shows again the practical importance of these heuristic algorithms.

For the instances of Set-2, reported in Table 4, it appears that all our proposedheuristics perform very well in the sense that, they have all been able to find feasible

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Table 4: Output of MVCSBPP-AC instances of Set-2

HM1 HM2 HM3 HM4 BnC|I| Gap Time Gap Time Gap Time Gap Time Gap Time #Sol. #Feas.100 2.01 0.06 3.26 0.02 3.96 0.02 3.96 0.30 0.01 2.74 10 10200 3.42 0.20 5.13 0.08 6.20 0.13 5.57 1.74 0.01 13.69 10 10

Type-1 300 3.90 0.42 6.04 0.21 7.76 0.28 6.57 3.32 0.01 66.94 10 10500 3.16 1.42 5.89 0.76 7.53 1.11 6.05 9.92 0.01 441.73 10 10750 3.69 3.42 6.08 2.14 7.68 3.19 6.24 22.31 0.02 1,530.40 9 101000 2.31 7.61 4.88 4.34 6.60 6.66 5.32 36.61 0.02 2,849.11 8 8100 0.00 0.05 0.00 0.01 0.00 0.02 0.00 0.45 0.00 0.11 10 10200 0.00 0.17 0.00 0.04 0.00 0.06 0.00 1.72 0.00 0.30 10 10

Type-2 300 0.04 0.41 0.04 0.08 0.04 0.14 0.04 3.96 0.00 0.63 10 10500 0.14 1.20 0.14 0.22 0.14 0.36 0.14 11.27 0.01 4.67 10 10750 0.15 2.72 0.15 0.46 0.15 0.78 0.15 25.09 0.01 8.28 10 101000 0.18 5.11 0.18 0.76 0.18 1.25 0.18 42.57 0.01 225.17 10 10100 0.00 0.08 0.00 0.02 0.00 0.03 0.00 0.80 0.00 0.12 10 10200 0.00 0.31 0.00 0.06 0.00 0.08 0.00 3.36 0.00 0.19 10 10

Type-3 300 0.00 0.72 0.00 0.11 0.00 0.18 0.00 7.71 0.00 0.26 10 10500 0.00 1.90 0.00 0.29 0.00 0.54 0.00 18.66 0.00 0.40 10 10750 0.00 4.50 0.00 0.61 0.00 1.05 0.00 41.09 0.00 0.66 10 101000 0.00 8.17 0.00 1.06 0.00 1.73 0.00 71.41 0.00 0.86 10 10100 0.65 0.05 1.00 0.02 1.42 0.02 1.42 0.32 0.01 24.07 10 10200 2.32 0.20 3.67 0.12 4.38 0.10 4.41 1.76 0.01 148.40 10 10

Type-4 300 3.09 0.37 5.02 0.32 6.19 0.31 6.14 3.94 0.01 142.31 10 10500 3.02 1.27 5.15 1.04 6.75 1.04 5.88 10.66 0.02 1,728.06 10 10750 2.31 3.29 4.54 2.88 5.92 3.28 4.81 21.27 0.05 2,377.78 6 81000 2.50 6.03 4.72 5.63 5.88 8.44 5.01 41.56 0.07 3,462.01 4 8100 0.00 0.04 0.00 0.02 0.00 0.03 0.00 0.54 0.00 0.07 10 10200 0.00 0.14 0.00 0.05 0.00 0.08 0.00 2.00 0.00 0.14 10 10

Type-5 300 0.00 0.34 0.00 0.11 0.00 0.16 0.00 4.84 0.00 0.24 10 10500 0.13 1.19 0.13 0.22 0.13 0.35 0.13 11.51 0.01 4.15 10 10750 0.15 2.73 0.15 0.46 0.15 0.74 0.15 24.51 0.01 8.19 10 101000 0.18 5.11 0.18 0.76 0.18 1.22 0.18 40.39 0.01 222.81 10 10100 0.00 0.06 0.00 0.03 0.00 0.04 0.00 0.98 0.00 0.07 10 10200 0.00 0.25 0.00 0.08 0.00 0.11 0.00 3.63 0.00 0.13 10 10

Type-6 300 0.00 0.58 0.00 0.15 0.00 0.21 0.00 8.36 0.00 0.18 10 10500 0.00 1.55 0.00 0.34 0.00 0.50 0.00 20.76 0.00 0.26 10 10750 0.00 3.66 0.00 0.70 0.00 1.07 0.00 46.58 0.00 0.42 10 101000 0.03 6.06 0.03 1.14 0.03 1.77 0.03 74.97 0.00 4.00 10 10

solutions with gaps less than 5%. Indeed, apart from instances of Type-1 and Type-4,HM1 was able to solve all the instances with an optimality gap no larger than 0.18%. Evenfor the instances of Type-1 and Type-4, the optimality gaps of HM1 vary between 0.6%and 3%. Moreover, as it has been the case for all the other instances already discussed,the computational times for these instances of Set-2 are eight seconds or less for HM1,and still very fast for all the other upper bounding procedures. These instances seemindeed easier to solve, but CPLEX still fails on a number of cases. Furthermore, ourheuristics clearly dominate in terms of computational efficiency. A visual comparisonof the computational times for these instances can be seen in Figure 1b provided in

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Appendix 2.

In summary, the computational experiments conducted in this section also highlightthe practical usefulness of the proposed heuristic algorithms. More precisely, it hasbeen observed that on certain instances, CPLEX can be quite efficient. However, onall instance types, we can see that, as the size of the instances increases so does thecomputational times of CPLEX (i.e., CPLEX fails to even find a feasible solution oncertain instances). The increases in the computational times of the heuristics are notnearly as pronounced (completing the search under a minute in most cases) and theheuristics are quite robust (producing high-quality solutions on all instances solved). Inaddition, the consistently low computational times of these heuristic algorithms showthat they may also be used as good candidates for generating starting solutions eitherfor much sophisticated metaheuristic algorithms, or even to speed-up the performance ofcommercial solvers (as shown in Section 6.3).

6.3 Heuristic solutions as warm-start solutions

In this section, we assess the impact of using the solutions of the heuristic algorithmsdescribed in Section 4 and Section 5 as initial solutions for a commercial solver. The aimof these experiments is to show that the solutions provided by the heuristic algorithmscan be used as warm-start solutions to lower the computational times of CPLEX. For thispurpose we will focus the experiments on test instances for which the algorithms seemsto struggle i.e., instances in Set-1 for VCSBPP-Ac and in Set-1 for MVCSBPP-AC aswell. For each of these instances, we first run the corresponding heuristic algorithm (HS1for VCSBPP-AC instances and HM4 for MVCSBPP-AC instances). Then, the objectivefunction value thus obtained is set as the problem upper bound in CPLEX. This impliesthat CPLEX will automatically prune any branch that has an upper bound larger thanthis objective function value. All the other CPLEX setting are left the same as the onesfor the previous experiments.

The outcome of this experiment is shown in Figure 1. More precisely, the graphs inFigure 1 shows the comparison between the CPU times (s) of CPLEX when run with andwithout the warm-start heuristic solutions. The results for the VCSBPP-AC instancesof Set-1 are plotted in Figure 1a and those for the MVCSBPP-AC instances of Set-1are plotted in Figure 1b. One can clearly observe that the use of the heuristic solutionsas a warm-start enables the commercial solver to reduce its computational times by asignificant amount. Indeed, CPLEX is now able to solve all the VCSBPP-AC instancesof Set-1 within 40 seconds. While, most of the MVCSBPP-AC instances of Set-1 canbe solved within 20 minutes with the exception of 10 instances out of 300. Recallingthat the results in Table 3 suggested that there were 41 instances (out of 300) for whichCPLEX was unable to find a feasible solution within the 1 hour time limit. Now, withthe warm-start, it has been able to find a feasible solution for all the instances. The 1

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hour time limit has been reached for 9 of these instances. Nevertheless, the correspondingoptimality gap were 3% or less.

The results of these experiments therefore, emphasize the usefulness of our proposedheuristic algorithms. Indeed, these results clearly illustrate how our heuristic algorithmscan be employed within other solution strategies to clearly enhance their computationalperformance.

6.4 Comparisons with a myopic approach

We first recall that the main idea of our proposed MVCSBPP-AC model in this papersuggests that there is a benefit of solving VCSBPP-AC over multiple periods of timeinstead of just solving one VCSBPP-AC instance for each individual time period withthe available items and bins. This allows to take advantage of the fact that some itemsor bins could be delayed for a much better consolidation opportunity in a future timeperiod. However, a drawback of such delays may be the penalty costs involved as well asthe risks of missing the delivery due dates. In fact, the current practice in the literaturedoes not explicitly account for the time component. Although, some stochastic variantsof the bin packing problem try to regard into the future by modelling the uncertainavailability of items and bins (Crainic et al., 2016), they do not account for all the timecharacteristics that are accounted for in this paper (e.g., availability time windows of theitems and the bins, or delivery time windows of the items, etc.).

The purpose of this section is to compare the two planing approaches, mainly weassess the impact of solving MVCSBPP-AC models instead of just solving VCSBPP-ACfor each time period independently within the same time horizon. More precisely, givenan MVCSBPP-AC instance with T time periods, we will first solve the correspond modelusing our proposed heuristic algorithms. Then, we will also solve T VCSBPP-AC in-stances (one for each time period with the items and bins made available for that timeperiod) and assess the value of solving the problem instance as an MVCSBPP-AC modelinstead of solving it as independent VCSBPP-AC models. This value will be assessed interms of the improvement in the cost reduction. For example, if Cost(MVCSBPP-AC)is the total cost of the MVCSBPP-AC model for an instance with T time periods andCost(VCSBPP-AC) is the corresponding cost of solving the same instance as T inde-pendent VCSBPP-AC models, then the improvement in the cost reduction is calculatedas:

Improvement =

(Cost(MVCSBPP-AC)− Cost(VCSBPP-AC)

Cost(VCSBPP-AC)

)× 100.

The results of these comparisons are shown in Table 5 for both the test instances inSet-1 and in Set-2. In this table, the numbers in each of the columns (‘HM1’, ‘HM2’,‘HM3’ and ‘HM4’) represent the average (out of ten instances) percentage improvement

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(a) VCSBPP-AC instances of Set-1

(b) MVCSBPP-AC instances of Set-1

Figure 1: Impact of warm-starting with heuristic solutions: time comparisons

in the cost reduction when the corresponding heuristic procedure is used to solve theinstances and compared with the myopic approach. Note that in order to solve theVCSBPP-AC instances for the myopic approach, we have used Algorithm 1 since itproved to dominate both Algorithm 3 and Algorithm 6 as shown in Figure 2. Note also

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Table 5: Average Improvement of our heuristics with respect to the myopic approach

Set-1 Set-2|I| HM1 HM2 HM3 HM4 HM1 HM2 HM3 HM4100 14.98 13.7 17.07 21.09 73.84 73.5 73.32 73.32200 24.17 20.78 24.2 30.02 62.21 61.58 61.19 61.42

Type-1 300 22.82 18.56 21.19 25.47 51.78 50.81 50.01 50.56500 17.6 12.78 12.09 16.53 37.59 35.94 34.94 35.84750 14.68 10.94 10 14.39 27.55 25.88 24.76 25.771000 13.95 9.71 8.56 13.92 23.09 21.16 19.87 20.82100 41.52 39.99 42.52 45.48 77.82 77.82 77.82 77.82200 23.98 22.11 23.46 28.86 69.96 69.96 69.96 69.96

Type-2 300 20.19 17.65 19.74 23.79 61.81 61.81 61.81 61.81500 17.62 15.34 16.56 21.82 49.27 49.27 49.27 49.27750 19.95 17.97 18.31 22.75 38.24 38.24 38.24 38.241000 17.53 15.67 15.43 18.79 31.14 31.14 31.14 31.14100 55.9 55.17 55.27 58.38 81.64 81.64 81.64 81.64200 36.86 35.68 36.76 41.52 75.97 75.97 75.97 75.97

Type-3 300 25.38 22.84 25.69 29.99 69.44 69.44 69.44 69.44500 17.62 15.34 16.56 21.82 57.31 57.31 57.31 57.31750 19.95 17.97 18.31 22.75 47.55 47.55 47.55 47.551000 17.53 15.67 15.43 18.79 40.89 40.89 40.89 40.89100 16.89 12.07 18.79 22.27 72.4 72.3 72.18 72.18200 28.06 23.92 27.66 32.07 64.37 63.9 63.65 63.64

Type-4 300 25.25 20.82 24.26 29.16 55.09 54.24 53.73 53.75500 18.04 14.69 16.17 20.79 41.03 39.8 38.88 39.38750 14.35 11.73 11.03 15.83 30.58 29.07 28.14 28.91000 13.61 10.86 8.85 14.1 24 22.36 21.52 22.16100 37.56 37.06 36.78 37.42 78.16 78.16 78.16 78.16200 27.6 27.96 26.78 27.91 71.9 71.9 71.9 71.9

Type-5 300 20.13 20.96 19.74 20.49 64.81 64.81 64.81 64.81500 14.53 15.28 13.51 14.48 49.75 49.75 49.75 49.75750 12.58 13.39 11.52 12.25 38.24 38.24 38.24 38.241000 11.74 12.39 10.67 11.71 31.14 31.14 31.14 31.14100 82.06 82.06 82.06 82.06200 77.58 77.58 77.58 77.58

Type-6 300 72.5 72.5 72.5 72.5500 60.75 60.75 60.75 60.75750 50.8 50.8 50.8 50.81000 41.82 41.82 41.82 41.82

that the blank space in Table 5 for Type-6 of Set-1 is simply because Set-1 only hasinstances of Type-1 to Type-5.

One of the first obvious observation in Table 5 is that there is no negative entry

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value. This means that for all the combined 660 test instances of both Set-1 and Set-2,solving the MVCSBPP-AC model always offers a much smaller total cost than solvingindependent VCSBPP-AC instances for each time period, regardless of the heuristicapproach used. Indeed, the average improvements vary between approximately 9% and60% for the instances in Set-1, and from about 21% up to 82% for the instances in Set-2. This shows that our proposed MVCSBPP-AC model is not only important from thetheoretical point of view, but it has the potential of significantly reducing the operationalcosts for practical applications.

6.5 Summary of the computational experiments

The computational experiments presented in this paper have targeted three main ob-jectives. First, we assessed the quality of the upper bounds provided by the proposedheuristic algorithms for both the VCSBPP-AC and the MVCSBPP-AC models. It re-sulted that CPLEX was quite efficient on certain instances. However, on all instancetypes, one could see that, as the size of the instances increases so does the computa-tional times of CPLEX. Indeed, CPLEX failed to even find a feasible solution on a largenumber of instances. The increases in the computational times of the heuristics wererather negligible and the heuristics proved to be quite robust in producing high-qualitysolutions on all the instances.

Given the rapid computational time of our algorithms, it is obvious on one handthat these algorithms can be used within more sophisticated metaheuristic algorithms forimproved search performance. What did not appear so obvious was whether the solutionsof the algorithms could be used as starting solutions for commercial solvers with the aimof speeding-up their computational times, especially for the ‘difficult’ instances. Thishas therefore been the objective of our second set experimentations. The outcome of theexperiments suggest that, indeed, using the solutions of the heuristic algorithms, one isable to reduce the computational time of CPLEX by a very large amount. This puts anadditional emphasis on the usefulness of the proposed heuristic algorithms.

The third set of experiments was aimed at measuring the impact of solving a MVCSBPP-AC as compared to solving individual VCSBPP-AC instances within the same time hori-zon. To this end, we measured the possible advantages of using the MVCSBPP-ACmodel in terms of the total cost reduction by comparison to a myopic approach thatsolves the same instances without considering the multi-period setting. It was observedthat the MVCSBPP-AC is able to reduce the operational costs by up to 82%. This mayrepresent a significant advantage in many practical applications.

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7 Conclusion

In this research, we proposed two new variants of the well known bin packing problem.Indeed, the bin packing problems already studied in the literature mainly focus on select-ing the bins at the lowest costs and loading them with any items. Thus assuming thatthe cost of loading items into the bins depends only on the items volumes and not thebin type. However, in some practical applications such as multi-modal logistics systems,the cost of assigning an item to a bin needs to account for both the physical characteris-tics of the bins and other economical attributes of the system. The first contribution ofthis paper was to bridge this gap in the literature and propose a new variant of the binpacking problem which accounts for the variability in the cost and size of the bins as wellas the cost of loading specific items into specific bins. The proposed model was referredto as the variable cost and size bin packing problem with assignment cost (VCSBPP-AC).

The second contributed mathematical model presented in the paper is the multi-periodvariable cost and size bin packing problem with assignment cost (MVCSBPP-AC). Thismodel formulates a logistics system wherein retailers and wholesalers need to ship goodsfrom one geographical region to another on a regular basis. In this setting, we haveshown that the cost to assign an item to a bin (transportation mode) needs to accountalso for the delivery time window as well as for the travel time of the bin, consideringthat a delay or an early delivery may hugely impact the operating costs.

In addition to the new variants of the bin packing problem, we also developed a totalof seven constructive heuristic algorithms for both models. Computational experimentshave been conducted on a set of about 1300 problem instances to assess the efficiency ofboth the proposed models and solution algorithms. The results showed that dependingon the structure of the problem, some instances of these two problems can be solvedby commercial solvers within an acceptable amount of time. While commercial solversstruggled to solve a large number of these instances, the proposed constructive heuristicscould provide great trade-offs between computational time and optimality gap regardlessof the structure of the problem at hand. The heuristics proved to have the abilityto quickly generate good quality feasible solutions to the problems that can either beused directly in practice, or, as a starting solution to warm-start commercial solvers.Furthermore, a comparison between MVCSBPP-AC solutions with those of a benchmarkapproach showed that MVCSBPP-AC can reduce the operating costs by up to about 82%.

As future research avenues, it would be interesting to develop more sophisticatedmetaheuristic algorithms for these problems. Such metaheuristics can naturally use theheuristic algorithms presented in this paper as starting points. On the other hand, theproposed MVCSBPP-AC models considers multiple period of time and assumes that theavailability of both the items and the bins are exactly known for the entire time horizon.This assumption may be too optimistic. Therefore, an extension of this research mayfocus on investigating the various sources of uncertainty that may be related to the items

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and the bins and propose a stochastic optimization method for the problem.

Acknowledgements

While working on the project, the first author was Adjunct Professor, Department ofComputer Science and Operations Research, Universite de Montreal. We gratefully ac-knowledge the financial support provided by the Natural Sciences and Engineering Coun-cil of Canada (NSERC), through its Discovery, Acceleration Grant and Engage programs.We also gratefully acknowledge the support of Fonds de recherche Nature et Technologiesdu Quebec through their team research project program. Thanks are also due to CalculQuebec and Compute Canada for providing the authors access to their high-performancecomputing infrastructure.

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Appendix

Appendix 1: Pseudo-code of Algorithm HS3

Algorithm 6 : Procedure 3 for solving a VCSBPP-AC

Input: Set of items I and set of bins J with their corresponding costs and volumes.Output: Set A of items assigned to bins and set of bins selected S.Initialize S = ∅Initialize A = ∅for all j ∈ J do

Let cj to be the cost of the bin j



if vi ≤ Vj with aij < p∗i , where p∗i is the lowest spot market cost of item i thenload i into jcj := cj + aijVj := Vj − vi

end ifend for

end forSort the bins in J in the non-decreasing order of cj.Let SBL be the sorted bin listRe-initialize Vj to the capacity of the bin, for all j ∈ Jfor all j ∈ SBL do



if vi ≤ Vj with aij < p∗i , where p∗i is the lowest spot market cost of item i thenload i into jA := A ∪ {i}Vj := Vj − vi

end ifend forif j is non-empty thenS := S ∪ {j}

end ifend for# Top-up the selected bins if possible.for all bin j ∈ S with left-over capacity V ∗j do


V ∗j − vi. Let us call the list SILj.

for i ∈ SILj doif V ∗j − vi ≥ 0 then

Load i into jV ∗j := V ∗j − vi

end ifend for




end ifend for

end for

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Appendix 2: VCSBPP-AC

(a) Overview for Set-1

(b) Overview for Set-2

Figure 2: Overview of VCSBPP-AC results Gap and CPU time

In addition to the results in Table 1 and Table 2, we provide a somehow detailedsummary of the results for all the instances. This summary is shown in Figure 2a forinstances in Set-1 and Figure 2b. In each of these figures, the left-hand graph summarizes

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the optimality gaps (%) for all the instances, while the right-hand graph summarizes thecomputational times (in seconds) for all the instances. We have chosen boxplots forthis purpose, which are able to provide a good visual overview of the distribution of theresults and can easily enable comparisons. From both Figure 2a and Figure 2b, it canbe observed that UB1 (the upper bounding procedure described in Algorithm 1) is thebest performing of our three proposed algorithms in terms of optimality gap. It canconsistently find solutions around 6% of optimality and around 2% of optimality for themajority of instances in Set-1 and in Set-2 respectively, and all this with about 15 secondsor less. Although CPLEX has been able to find at least one feasible solution for all theinstances in both Set-1 and Set-2, it proves to be very time consuming for the instancesSet-1 and some instances in Set-2.

Appendix 3: VCSBPP-AC

In addition to the results in Table 3 and Table 4, we also provide an overview of theresults for all the instances. This summary is shown in Figure 3a for instances in Set-1and Figure 3b for instances in Set-2. It is obvious from Figure 3a that HM4 providesthe best upper bound (as compared to HM1, HM2, and HM3) for instances in Set-1. Itis also apparent that CPLEX struggles quite a lot for these instances both in terms ofoptimality gap and in terms of computational time. On the other hand, Figure 3b showsthat all our four proposed upper bounding procedure can solve the majority of instancesin Set-2 within about 0.1% of optimality (based on the median lines of the boxplots). Italso shows that HM1 dominates the other three procedure on the instances in Set-2 interms of optimality gap. The overall observation from both figures could therefore bethat our proposed heuristic provides a good trade-off between computational time andoptimality gap.

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(a) Overview for Set-1

(b) Overview for Set-2

Figure 3: Overview of MVCSBPP-AC results Gap and CPU time

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