Operations Research Letters 42 (2014) 82–84

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Operations Research Letters

journal homepage: www.elsevier.com/locate/orl

Dynamic cost allocation for economic lot sizing games

Alejandro Toriello a,∗, Nelson A. Uhan b

a H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, United Statesb Mathematics Department, United States Naval Academy, Annapolis, MD 21402, United States

a r t i c l e i n f o

Article history:Received 6 November 2013Received in revised form11 December 2013Accepted 11 December 2013Available online 18 December 2013

Keywords:Cooperative gameEconomic lot sizingDynamic allocation

a b s t r a c t

We consider a cooperative game defined by an economic lot sizing problem with concave ordering costsover a finite time horizon, in which each player faces demand for a single product in each period andcoalitions can pool orders. We show how to compute a dynamic cost allocation in the strong sequentialcore of this game, i.e. an allocation over time that exactly distributes costs and is stable against coalitionaldefections at every period of the time horizon.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction and motivation

Production and inventory management are areas in which co-operation among independent agents has an intuitive appeal. Forexample, independent retailers sharing a warehouse may want tocombine orders from a common supplier to enjoy economies ofscale derived from larger order quantities. Viewed from a system-wide perspective, combining orders lowers the total cost, but theorder consolidation cannot be achieved unless the independent re-tailers can agree on a ‘‘fair’’ way to split this cost. Over the lastdecade or more, cooperative game theory research in production,distribution and inventory models has endeavored to determinefair cost allocations under a variety of settings.

The economic lot sizing problem is one of the canonical pro-duction and inventory models studied in operations research andmanagement science, almost since the field’s inception [10]. In acooperative setting, this model would capture the situation de-scribed above, with each retailer facing its own demand over theplanning horizon that must be satisfied with product orders or in-ventory, but with coalitions of retailers being able to order producttogether. The cooperative game derived from this model was pro-posed in [8] and subsequently studied in [2,3]. However, all of thesepapers approach the problem from a static perspective; that is, theauthors assume that an optimal solution is implemented over theentire horizon, and seek an up-front static cost allocation in thecore of this cooperative game.

∗ Corresponding author.E-mail addresses: atoriello3@isye.gatech.edu, atoriello@isye.gatech.edu

(A. Toriello), uhan@usna.edu (N.A. Uhan).

0167-6377/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.orl.2013.12.005

Unfortunately, a static cost allocation – even one in the core –suffers from a number of significant drawbacks. First, it assumesthat each retailer would be able to cover its portion of the entireplanning horizon’s cost up front, a significant financial burden thatis unrealistic in many settings. Second, once the allocated costs arecollected from the participating retailers, there may be incentivesfor individual retailers or coalitions to defect later on in the plan-ning horizon if they find themselves in an advantageous position.Finally, a static allocation ignores the typical rolling horizon ap-proach to thesemodels, in which solutions for one or a few periodsare implemented, and then a new model is formulated with up-dated parameters. Within a rolling horizon approach, it is unclearhow a static allocation could be implemented.

Our goal in this paper is to develop dynamic cost allocations forthe economic lot sizing problem. Unlike their static counterparts,dynamic allocations can be constrained to allocate costs as theyare incurred. They can also be designed so that no coalition everhas an incentive to defect throughout the entire planning horizon.Furthermore, because the costs are allocated as they are incurred,dynamic allocations can also be incorporated into a rolling horizonframework. Our allocation depends conceptually on extending thecore concept to dynamic settings (e.g. see [4]), and continues ourwork in [7] for multi-period models with linear costs.

Cooperative game theory has been successfully applied inmanyrelated models, and we cannot hope to adequately review themhere. Instead, we refer the interested reader to [1,5]. Two in-depthreferences on lot sizing andmore general production planning andinventory management are [6,12]. This paper has two remainingsections. In Section 2, we introduce the model and review somerelevant results. Then, in Section 3, we define the strong sequentialcore, a dynamic version of the core, and showhow to compute sucha dynamic allocation.

A. Toriello, N.A. Uhan / Operations Research Letters 42 (2014) 82–84 83

2. The economic lot sizing game

We consider the following deterministic economic lot sizingsetting with multiple retailers. A set of retailers N := {1, . . . , n}faces deterministic demand for a single product over a finite dis-crete time horizon. In particular, each retailer i ∈ N needs to sat-isfy demand dti in periods t = r, . . . , T ; we start in period r insteadof period 1 because we will eventually vary the starting period. Inevery period t = r, . . . , T , each retailer can order x units at a to-tal cost of ct(x); the function ct is concave and nondecreasing withct(0) = 0. Each retailer can also hold one unit of product in in-ventory in period t = r, . . . , T at a cost of ht . Finally, each retaileri ∈ N has an initial inventory of sr−1

i . For notational convenience,we define

d[j,k]R :=

i∈R

kt=j

dti for R ⊆ N, r ≤ j ≤ k ≤ T ,

sr−1R :=

i∈R

sr−1i .

Here and throughout, we take a summation in which the initial in-dex is greater than the ending index to be vacuous and equal tozero; for example, d[j,j−1]

R = 0.The economic lot sizing problem for a subset of retailers R ⊆ N

seeks to satisfy the demands drR, . . . , dTR with initial inventory sr−1

Rin a way that minimizes the total ordering and inventory cost. Theeconomic lot sizing game is a transferable utility cooperative game(N, fr(·, sr−1)) in which each retailer corresponds to a player, andthe cost fr(R, sr−1) to a subset of players R ⊆ N is the optimal valueof its economic lot sizing problem. Following standard terminologyin the cooperative game theory literature, we refer to a subset ofplayers as a coalition, and the set of all players as the grand coalition.

It is well-known that any instance of the economic lot sizingproblem can be transformed into an equivalent instance with zeroinventory by using the initial inventory to greedily satisfy demandin the beginning time periods [11]. It is also well-known that whenthe initial inventory is zero, there exists an optimal solution thatsatisfies the zero-inventory property in which orders only occur inperiods when the inventory level is zero [9,10]. Therefore, wheninitial inventories are of the form

sr−1i = d[r,τ−1]

i for i ∈ N

for some τ ≥ r , we can model the economic lot sizing problem forcoalition R ⊆ N with the following linear program [2]:

fr(R, sr−1) =

τ−1t=r

htd[t+1,τ−1]R + min

x

Tj=τ

Tk=j

cjd[j,k]R

+

kt=j

htd[t+1,k]R

xjk (1a)

s.t.t

j=r

Tk=t

xjk = 1 for t = τ , . . . , T , (1b)

xjk ≥ 0 for j, k : τ ≤ j ≤ k ≤ T , (1c)

where xjk is a decision variable that indicates an order in period j tomeet demands in periods j, . . . , k, for all j, k such that τ ≤ j ≤ k ≤

T . Although the decision variables are continuous, this interpreta-tion is well-defined: Chen and Zhang [2] showed that there alwaysexists an optimal solution to (1) such that xjk ∈ {0, 1} for all j, k.The first term of the objective (1a) is the cost of greedily using theinitial inventory sr−1

R to satisfy demand in periods r, . . . , τ −1, andthe coefficient of xjk in the second term of the objective is the costof satisfying demand in periods j, . . . , k with an order in period j.The constraints (1b) ensure that demand is satisfied in each period

τ , . . . , T . We slightly abuse notation and use fr(R, sr−1) to refer tothe model (1) as well as its optimal value.

The dual of fr(R, sr−1) is

fr(R, sr−1) =

τ−1t=r

htd[t+1,τ−1]R + max

α

Tt=τ

dtRαt (2a)

s.t.k

t=j

dtRαt ≤ cjd[j,k]R

+

kt=j

htd[t+1,k]R

for j, k : τ ≤ j ≤ k ≤ T . (2b)Gopaladesikan et al. [3] proposed a polynomial-time combinato-rial algorithm that solves fr(R, sr−1) and its dual (actually, an affinetransformation of its dual), and used the primal and dual optimalsolutions in conjunction with some structural properties estab-lished byChen andZhang [2] to compute an allocation in the core ofthe economic lot sizing game (N, fr(·, sr−1)). We summarize theseresults in the lemma below.

Lemma 1 ([2,3]). Let x∗ and α∗ respectively be optimal solutions tofr(R, sr−1) and its dual, obtained using the algorithm in [3]. Then, x∗

and α∗ satisfy the following properties:(a) α∗

t + ht+1 ≥ α∗

t+1 for t = τ , . . . , T .(b) For all dR = (d r

R , . . . , dTR ) such that d t

R ≤ dtR for t = r, . . . , T ,k

t=j

d tRα

∗

t ≤ cjd [j,k]R

+

kt=j

ht d[t+1,k]R

for j, k : τ ≤ j ≤ k ≤ T .

(c) There exist replenishment intervals {(a1, b1), . . . , (am, bm)}such that a1 = τ , bm = T , aℓ ≤ bℓ and bℓ +1 = aℓ+1 for all ℓ =

1, . . . ,m, and

x∗

aℓbℓ= 1 for ℓ = 1, . . . ,m,

bℓt=aℓ

dtRα∗

t = caℓd[aℓ,bℓ]

R

+

bℓt=aℓ

htd[t+1,bℓ]

R

for ℓ = 1, . . . ,m.

3. Dynamic cost allocation

Now suppose that the time horizon starts in period r = 1, witheach player having zero initial inventory, and the players agreeto implement an optimal solution x∗ to f1(N, 0) with correspond-ing dual optimal solutionα∗ and replenishment intervals {(a1, b1),. . . , (am, bm)} with a1 = 1 and bm = T , obtained by the algorithmin [3].

In previous approaches to economic lot sizing games [2,3,8],the authors assume that the entire cost of the optimal solution isallocated up front at the start of the horizon. Instead, we make themore realistic assumption that ordering costs must be allocated inthe same period in which an order takes place, and holding costsfor each period must be allocated in that same period. Finally, weassume that the players agree that at any point in time, each playerowns the inventory that was ordered to meet its own demand; inotherwords, for each ℓ = 1, . . . ,m, the inventory of player i ∈ N is

saℓ−1i = 0, (3a)

sr−1i = d[r,bℓ]

i for r = aℓ + 1, . . . , bℓ. (3b)

Definition 2 ([4,7]). In a dynamic allocation χ = (χ ti )i∈N,t=1,...,T ,

player i ∈ N is allocated a cost of χ ti in period t . The strong sequen-

tial core of the economic lot sizing game (N, (fr)r=1,...,T ) is the setof dynamic allocations χ that satisfy the following conditions forsome optimal solution x∗ to f1(N, 0):

84 A. Toriello, N.A. Uhan / Operations Research Letters 42 (2014) 82–84

(a) Stage-wise efficiency: For each ℓ = 1, . . . ,m,i∈N

χaℓi = caℓ

d[aℓ,bℓ]

N

+ haℓd

[aℓ+1,bℓ]

N , (4a)

i∈N

χ ti = htd

[t+1,bℓ]

N for t = aℓ + 1, . . . , bℓ. (4b)

In other words, the costs incurred in each period must be fullydistributed among the grand coalition.

(b) Time-consistent stability: For each period r = aℓ, . . . , bℓ for allℓ = 1, . . . ,m,i∈R

Tt=r

χ ti ≤ fr(R, sr−1) for R ⊆ N, (4c)

where the initial inventories sr−1 are as defined in (3). In otherwords, at any point in the time horizon, the cost allocated toany coalition from that point forward does not exceed its costif it abandons the grand coalition and continues on its own.

We define the dynamic allocation χ as follows:

χaℓi =

bℓt=aℓ

dti

α∗

t −

t−1u=aℓ

hu

+ haℓd

[aℓ+1,bℓ]

i

χ ti = htd

[t+1,bℓ]

i for t = aℓ + 1, . . . , bℓ

for ℓ = 1, . . . ,m. (5)

Theorem 3. The dynamic allocation χ defined in (5) is in the strongsequential core of the economic lot sizing game (N, (fr)r=1,...,T ).

Proof. First, we show that χ is stage-wise efficient. For any ℓ =

1, . . . ,m,i∈N

χaℓi =

i∈N

bℓ

t=aℓ

dti

α∗

t −

t−1u=aℓ

hu

+ haℓd

[aℓ+1,bℓ]

i

=

bℓt=aℓ

dtN

α∗

t −

t−1u=aℓ

hu

+ haℓd

[aℓ+1,bℓ]

N

=

bℓt=aℓ

dtNα∗

t −

bℓt=aℓ

t−1u=aℓ

hudtN + haℓd[aℓ+1,bℓ]

N

=

bℓt=aℓ

dtNα∗

t −

bℓt=aℓ

bℓu=t+1

htduN + haℓd[aℓ+1,bℓ]

N

=

bℓt=aℓ

dtNα∗

t −

bℓt=aℓ

htd[t+1,bℓ]

N + haℓd[aℓ+1,bℓ]

N

(i)= caℓ

dt

[aℓ,bℓ]

+ haℓd

[aℓ+1,bℓ]

N ,

where (i) follows from Lemma 1(c). In addition, we have, for all ℓ =

1, . . . ,m,i∈N

χ ti =

i∈N

htd[t+1,bℓ]

i = htd[t+1,bℓ]

N for t = aℓ + 1, . . . , bℓ.

Next,we show that χ is time-consistently stable. Fix r ∈ {ap+1,. . . , bp, ap+1}, for some p ∈ {0, 1, . . . ,m} (ignoring a0 + 1, . . . , b0

and am+1). For any R ⊆ N , we have thati∈R

Tt=r

χ ti =

bpt=r

htd[t+1,bp]R +

mℓ=p+1

bℓ

t=aℓ

dtR

α∗

t −

t−1u=aℓ

hu

+

bℓt=aℓ

htd[t+1,bℓ]

R

=

bpt=r

htd[t+1,bp]R +

mℓ=p+1

bℓ

t=aℓ

dtRα∗

t

−

bℓt=aℓ

t−1u=aℓ

hudtR +

bℓt=aℓ

htd[t+1,bℓ]

R

=

bpt=r

htd[t+1,bp]R +

mℓ=p+1

bℓ

t=aℓ

dtRα∗

t

−

bℓt=aℓ

bℓu=t+1

htduR +

bℓt=aℓ

bℓu=t+1

htduR

=

bpt=r

htd[t+1,bp]R +

mℓ=p+1

bℓt=aℓ

dtRα∗

t

(ii)≤ fr(R, sr−1)

where (ii) holds because Lemma 1(b) implies that α∗ is a dual fea-sible solution for fr(R, sr−1). �

Acknowledgment

The authors’ work was partially supported by the US NationalScience Foundation via grant CMMI 1265616.

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