- Email: [email protected]
WITHDRAWN: Economic lot-sizing game with price-dependent demand
Accepted Manuscript
Economic Lot-sizing Game with Price-dependent Demand Xiaojie Yan , Juliang Zhang , T.C.E. Cheng , Guowei Hua PII: DOI: Reference:
S0377-2217(17)30179-0 10.1016/j.ejor.2017.02.040 EOR 14277
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
28 January 2016 23 February 2017 27 February 2017
Please cite this article as: Xiaojie Yan , Juliang Zhang , T.C.E. Cheng , Guowei Hua , Economic Lotsizing Game with Price-dependent Demand, European Journal of Operational Research (2017), doi: 10.1016/j.ejor.2017.02.040
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Highlights A new cooperative game, economic lotsize game with dynamic pricing, is introduced; Show that the game is balanced and has a nonempty core under general conditions; Show that the game is convex for two special cases but not convex for general case; Numerical experiments are conducted to show the benefits of cooperation.
AC
CE
PT
ED
M
AN US
CR IP T
• • • •
1
ACCEPTED MANUSCRIPT Economic Lot-sizing Game with Price-dependent Demand Xiaojie Yan1, Juliang Zhang1*, T.C.E. Cheng2, Guowei Hua1 1
Department of Logistics Management School of Economics and Management Beijing Jiaotong University, Beijing, 100044, China 2
CR IP T
Department of Logistics and Maritime Studies The Hong Kong Polytechnic University Hung Hom, Kowloon, Hong Kong SAR
Abstract
AN US
We study the economic lot-sizing (ELS) game with price-dependent demand, whereby several retailers form a coalition to jointly place orders to a single supplier and determine their sale prices over a multi-period finite time horizon for profit maximization. We show that the game is balanced and has a non-empty core in general. We also show that the game is convex in two special cases where the retailers’ demand functions are the same and where the time horizon consists of two periods. We give a counter-example to demonstrate that the game
M
is non-convex in general. We perform numerical experiments to show the benefits of cooperating on inventory replenishment.
AC
CE
PT
ED
Key words: logistics; economic lot-sizing game; pricing; balancedness; core
*
Corresponding author, email: [email protected] 2
ACCEPTED MANUSCRIPT
1.
Introduction Increasing numbers of firms in a variety of industries have deployed innovative pricing strategies to
manage demand and improve profit (see, e.g., Elmaghraby and Keskinocak, 2003; Chen and Simchi-Levi, 2012). In addition, many companies have explored innovative collaboration strategies in efforts to improve
CR IP T
their core competence and bottom lines. For example, many retailers in Zhongguancun Electronic City in Beijing, China cooperate to place orders and stock their products in a centralized warehouse while only displaying sample products in their shops. In addition, Good Neighbor Pharmacy and Affiliated Foods Midwest employ collaborative strategies in the form of long-term alliance and collaborative logistics,
AN US
respectively, with a view to reducing costs. The former is a cooperative network of 2,700 independently owned and operated pharmacies, while the latter supplies more than 850 independent retailers in the 12 Midwest states with a full line of grocery products. These pharmacies and retailers form retailers’ cooperative groups which implement unified management such as inventory management, pricing assistance, logistics, warehousing and so on (Chen, 2009). In order to keep the long-term relationship, a very important issue faced
M
by the partnering firms is how to optimally allocate the profit or cost among them to ensure that the alliance is
ED
stable (van den Heuvel et al., 2007; Chen, 2009). Seeking to address this problem, we adopt the economic lot-sizing (ELS) game with price-dependent demand for modelling and analysis in this paper.
PT
There are two related steams of research. The first one is on inventory centralization games, for which Nagarajan and Sošić (2008), Drechsel (2010), and Montrucchio et al. (2012) gave comprehensive reviews. We
CE
briefly review the literature most closely related to our work. Hartman et al. (2000), Müller et al. (2002), Slikker et al. (2005), Chen and Zhang (2009), Özen et al. (2008), and Uhan (2015) studied the newsvendor
AC
game, while Chen (2009) studied the newsvendor game with price-dependent demand. They showed that the core is non-empty. Lu et al. (2014) studied how different firms cooperate in their recycling and pricing decisions using cooperative game theory. All these studies considered one-period problem and assumed that the retailers face random demands. Dror and Hartman (2007), Anily and Haviv (2007), and Zhang (2009) studied the inventory centralization game based on the economic order quantity (EOQ) model. They assumed that the demand is stationary. Karsten et al. (2009) studied the continuous-time inventory centralization game where demand occurs 3
ACCEPTED MANUSCRIPT
according to a Poisson process with a constant demand rate. Many researchers have studied the multi-period inventory game based on the ELS problem, which is called the ELS game. Van den Heuvel et al. (2007) showed that the ELS game is balanced and has a non-empty core under the assumption that backorders are not allowed. Guardiola et al. (2006) studied the case where backorders are allowed and showed how to find a core allocation. Chen and Zhang (2016) studied the game with a general ordering cost and allowing backorders. They developed a duality approach to compute a core allocation in
CR IP T
polynomial time. They pointed out that it remains a challenge to incorporate pricing decisions into the ELS game. Gopaladesikan and Uhan (2011) studied the cooperative cost sharing game based on the ELS problem considering re-manufacturing options. They showed that the game has an empty core in general and a non-empty core for two special cases. Drechsel (2010) extended the ELS games to include uncertain demand and capacity constraints with
AN US
transshipments, and studied a multi-level lot sizing game with restricted cooperation. Toriello and Uhan (2013) generalized the classical linear production game to the multi-period model with stochastic parameters. Guardiola et al. (2007), Esmaeili et al. (2009), Esmaeili and Zeephongsekul (2010), Anily and Haviv (2010), and Özener et al. (2013) studied applications of cooperative game theory to the management of centralized inventory systems. Hall
M
and Liu (2010), Chen and Chen (2013), and Yu et al. (2015) studied the inventory centralization game with capacity sharing using cooperative game theory. The above research assumed that the price is exogenous.
ED
The second research stream related to this study is on the joint decision on pricing and inventory control, on which Elmaghraby and Keskinocak (2003), Yano and Gilbert (2004), and Chen and Simchi-Levi (2012)
PT
gave extensive reviews. We only review some literature based on the ELS model. Wagner and Whitin (1958) first studied the ELS problem with price-dependent demand without capacity constraints. They established the
CE
zero inventory ordering (ZIO) property and proposed a cyclic algorithm with polynomial-time complexity. Thomas (1970) demonstrated explicitly that the optimal price in one period is consistently independent of the
AC
prices in the other periods. Florian and Morton (1971), and Florian et al. (1980) showed that the joint pricing and inventory management problem with capacity limits is NP-hard. Deng and Yano (2006) characterized properties of the optimal solution and gave an algorithm to solve the problem. Adeinat and Ventura (2015) generalized the problem to the case with multiple capacitated suppliers. All these studies considered only one retailer. In this paper we study the ELS game with price-dependent demand. In this game, there are several retailers that make ordering and pricing decisions over a multi-period finite time horizon. They may cooperate 4
ACCEPTED MANUSCRIPT
to place joint orders and keep inventory in a central warehouse. The demand of each retailer depends on its own sale price. At the beginning of each period, the retailers first review the initial inventory in the central warehouse, and then jointly decide on the order quantity and their sale prices. Associated with each order there are a fixed ordering cost and a purchase cost, which depends on the order quantity and the unit purchase cost. We assume that the lead time is zero so that the order is delivered immediately to the central warehouse. After the order delivery, the retailers’ demands are realized and the inventory in the central warehouse is allocated to
CR IP T
satisfy the demands; no shortages are allowed. The unsold inventory in any period is carried forward to the next period, which incurs a holding cost. We further assume that the inventory at the beginning of the time horizon is zero, and that the demand, ordering cost, and holding cost are independent, and the transportation cost from the central warehouse to each retailer’s warehouse is negligible. The objective is to make optimal
AN US
ordering and pricing decisions to maximize the total profit of all the retailers over all the periods in the time horizon. Also, we study whether there exists an allocation policy that ensures that the retailers can form a stable alliance, i.e., whether the core of the cooperative game is non-empty.
We show that the ZIO property holds and propose a polynomial-time algorithm to compute the optimal
M
total profits of the coalitions. We prove that the ELS game with price-dependent demand is balanced and has a non-empty core. We show that the game is convex in two special cases where the retailers’ demand functions
ED
are the same and where there are only two periods in the time horizon. We provide a counter-example to demonstrate that the game is non-convex in general.
PT
We organize the remainder of the paper as follows: In Section 2 we introduce the ELS game with price-dependent demand and show how to find the characteristic functions of coalitions. In Section 3 we prove
CE
that the game is balanced and has a non-empty core. In Section 4 we address the convexity issues of the game.
AC
We conclude the paper in the last section.
2. ELS game with price-dependent demand In this section we introduce the ELS game with price-dependent demand and show how to compute the
maximum payoff of the coalitions. Consider a set of retailers N 1,2
, n that sell the same product, and make pricing and ordering
decisions over a finite time horizon consisting of T periods. Retailer i ’s demand d ti in period t is 5
ACCEPTED MANUSCRIPT
deterministic and depends on its set price pti in that period. Assume that the demand function is dti Dti pti (i 1,2,..., n, t 1,2,..., T ) , which is a decreasing and concave function of the current price
(Talluri and Van Ryzin, 2004; Chen and Simchi-Levi, 2012). The retailers may cooperate to buy the product from the same manufacturer. When the retailers in a set (coalition) S N decide to cooperate, they place joint orders, which are delivered to a central warehouse. At the beginning of period t , they first review the
CR IP T
inventory I t 1 in the central warehouse, and then decide their sale prices pti (i S ) and the order quantity
xt . When an order is placed, a fixed ordering cost K t and a purchase cost ct xt are incurred, where ct is the unit purchase cost. Let us assume that the lead time is zero, so orders is delivered immediately. After the order delivery, the demands are realized and the inventory in the central warehouse is allocated to satisfy the
AN US
demands; no shortages are allowed. The unsold inventory is carried forward to the next period, which incurs a unit holding cost ht . Assume that the inventory at the beginning of period 1 is zero and that the transportation cost from the central warehouse to the retailers is zero.
The retailers can place orders individually or opt to cooperate to keep joint inventory in the central
M
warehouse. Following the arguments in van den Heuvel et al. (2007), we can show that it is profitable for the
ED
retailers to cooperate. The problem is whether there exists an allocation policy that ensures that the grand coalition is stable, i.e., all the retailers form a coalition.
PT
For each coalition S N , let F S be the characteristic profit function of S , i.e., the total maximum profit obtained by the retailers in the set S that cooperate in their pricing and ordering decisions. The game
CE
( N , F ) is called the ELS game with price-dependent demand.
Noting that there is a one-to-one correspondence relationship between the demand and price of each
AC
retailer, we adopt demand d ti (i 1,..., n) the analysis. It follows that
as the decision variable rather than price pti (i 1,..., n) to simplify
pti pti (dti ) Dti
1
d i t
(i 1,..., n) . Since dti Dti pti
(i 1,..., n) is a
decreasing and concave function by assumption, pti pti (dti ) (i 1,..., n) is a decreasing and concave function. i i i So the profit function dt pt dt (i 1,..., n) is also concave. The notation used in this paper is listed as
follows: 6
ACCEPTED MANUSCRIPT
d ti : demand of retailer i in period t, (a decision variable);
xt : order quantity in period t (a decision variable; we call t an ordering period if xt 0 ); I t : ending inventory in period t;
K t : fixed ordering cost in period t;
CR IP T
ct : unit purchase cost in period t; ht : unit holding cost in period t;
pti dti : the sale price of retailer i in period t when the demand is d ti ;
cst : defined as cst cs l s hl .
AN US
t 1
We now consider the computation of the characteristic function, F S . Evidently, F S is the optimal value of the following optimization problem: F S : max
d p d K x c x iS 1 t T
s.t.
i t
i t
i t
1 t T
I t xt I t 1 dti ,
t
t
t 1,
t t
ht I t
,T
M
iS
(2.1)
I 0 0，
,T
PT
1, xt 0, where xt 0, otherwise.
t 1,
ED
xt 0, I t 0，
The following proposition characterizes the optimal inventory policy for problem (2.1).
CE
Proposition 1. The ZIO property holds for problem (2.1), i.e., no order is placed whenever the initial inventory is positive.
AC
Proof. Suppose that the result does not hold, i.e., there exists an optimal ordering plan x x1 , x2 ,
, xT for
problem (2.1) and an ordering period t such that I t 1 0 and xt 0 . Since the inventory of each retailer at the beginning of period 1 is zero, the order quantity in period 1 is positive and period t is not period 1. Let period s be the ordering period immediately before period t . We keep the pricing decision unchanged and either shift the order quantity in period t to period s or shift an order quantity of I t 1 units from period s to period t . In the former case, no order is placed in period t 7
ACCEPTED MANUSCRIPT
and the resulting cost change is
cst ct xt , while in the latter case, the initial inventory level in period
t is
zero and the resulting cost change is cst ct It 1 , where cst cs l s hl is the marginal cost of t 1
satisfying the demand in period t by an order placed in period s . Clearly either case will lead to an ordering plan with a cost no more than that of the original plan. Hence the profit of new plan is larger than the profit of the original plan, contradicting x x1 , x2 ,
, xT is an optimal ordering plan, and the result follows.
CR IP T
□
From Proposition 1, we can convert problem (2.1) into the longest-path problem, which can be solved by a forward-type algorithm in O(T 2 ) time (see, e.g., Chen and Simchi-Levi, 2012), and design an algorithm
AN US
with polynomial-time complexity to solve it.
Algorithm 1: Step 1.
If periods a and b (a < b) are two consecutive ordering periods in an ordering plan, the ZIO property implies that the demand in period t a t b is satisfied by the order placed in period
M
a only, so the marginal cost of satisfying the demand in period t is cat . The associated optimal
ED
demand in period t a t b can then be derived by solving the following optimization problem: vatS max d i 0
p d c d . iS
i t
i t
at
i t
(2.2)
Construct an equivalent acyclic network G = (V, E) with the node set V 1,2,
PT
Step 2.
t
CE
set V a, b :1 a b T 1 . The arc
a, b
means that the demands from periods a through
b 1 are satisfied by the order in period a . The weight of the arc a, b is
AC
, T 1 and arc
b 1 S t a at
v Ka , which
is the total profit from periods a through b 1 with the demands determined by solving problem (2.2) for t a,
Step 3.
, b 1 .
Find the longest path from node 1 to node T+1 in the network G = (V, E). The longest path gives an optimal ordering plan with the ZIO property whose ordering periods exactly correspond to the nodes on the longest path, and the associated optimal demands are derived from solving problem (2.2). The length of the longest path is the maximum profit F S . The order quantity in an 8
ACCEPTED MANUSCRIPT
ordering period equals the sum of the demands from the current period to the period before the next ordering period. The associated optimal ordering decision is decision of retailer i is
i* 1
d
i* 1
,
, xT* . The optimal demand S
, dTi * . S
,
We obtain the optimal pricing decisions of retailer i demand decision
* 1
p d , i 1
i* 1
, pTi dTi*
S
by using the optimal
, dTi * . S
CR IP T
Step 4.
d
x ,
Remark 1. Note that the objective function in problem (2.2) is by assumption concave, implying (2.2) is an unconstrained concave programming and can be solved by a polynomial algorithm. This follows that Algorithm 1 is a polynomial-time algorithm.
Remark 2. Proposition 1 states that the alliance places an order only if the initial inventory is zero. Algorithm
AN US
1 helps the retailers to properly set prices, place orders, and allocate the inventory when the demands realized and the orders are delivered when they decide to cooperate to form an alliance.
3. The balancedness of the game
M
In Section 2 we show that it is profitable for retailers to cooperate. In this section we study whether there exists an allocation policy that ensures that the grand coalition is stable, i.e., whether the core of the
ED
cooperative game is non-empty. Noting that the objection function in (2.1) is non-linear and non-continuous, there exists a duality gap between the primal and dual problems. Chen (2009) and Lu et al (2014) considered
PT
the single-period pricing problem with a duality gap and showed that the coalition’s optimal order quantity increases as more players join the coalition. Based on this, they located a core allocation. Given that the ELS
CE
game is multi-period, the duality gap property no longer holds, so the duality approach developed by Chen and Zhang (2016) cannot be used to find a core allocation. Here we show that the game is balanced. The core
AC
of the game is non-empty by the Bondareva-Shapley theorem (Bondareva, 1963; Shapley, 1971). The proof is obtained by merging the ordering plans (of balanced collections) of coalitions to construct some ordering plans for the grand coalition in such a way that we can show the balancedness conditions of the Bondareva-Shapley theorem are satisfied. Let 2 N be the set consisting of all the subsets of N . Then B 2N is called a balanced collection of
9
ACCEPTED MANUSCRIPT
2 N if there exist weights S ( S B ) such that
e
SB
S
S
e N , where e S is the vector in R n with e S 1 i
F S F N
if i S and eiS 0 otherwise. A game is balanced if
S
SB
for all the balanced collections.
As noted before, ordering plans can be completely described by setting the ordering periods. Let X be an ordering plan and X t 1 if period t is an ordering period and X t 0 otherwise ( t 1, , T ). Let t 1
j max l t X l 1 and wti X max dti pti dti ct X dti . So the total
with
CR IP T
ct X c j l j hl
S profit F X of the ordering plan X is
T F S X wti X X t Kt . t 1 iS
AN US
For convenience, a plan X with X t 0 ( t 1, , T ) is called an empty plan, and we define ct X and wti X 0 under an empty plan X .
The following result states that the profit that retailer i obtains in period t under ordering plan X decreases in the unit marginal ordering cost of plan X in period t .
Suppose
ct X ct X
that
ED
Proof:
M
i Lemma 1. For any i S , wt X is strictly decreasing in ct X .
wti X max d i pti dti ct X dti
,
t
,
wti X max d i pti dti ct X dti , and d ti is an optimal solution for wti X . For any i S , we have t
PT
wti X max d i
p d c d p d c X d i t
t
i t
jt
i t
i t
i t
t
i t
CE
pti dti ct X d ti
max d i pti dti ct X dti t
wti X . □
AC
The lemma holds.
Theorem 1. The ELS game with price-dependent demand is balanced. Proof: Let B be a balanced collection of 2 N and S ( S B ) be the corresponding weights. It suffices to prove that
F S F N .
SB
S
(3.1)
Obviously, (3.1) holds for B N . Let B N and qB be the smallest integer for which qB S is 10
ACCEPTED MANUSCRIPT integral for all S B . Then (3.1) is equivalent to
q F S q F N .
SB
B S
(3.2)
B
The term qB S F S on the left hand side of (3.2) is qB S times of the profit of the optimal ordering plan for coalition S . So the left hand side of (3.2) is the total profit of nB qB S ordering plans. Let SB
, nB be the index set of coalitions (in an arbitrary order) and let S m be the coalition
CR IP T
N B 1,
corresponding to index m . So we have qB S indices for each coalition S B . Moreover, let X m be the optimal ordering plan corresponding to coalition S m , so the profit of plan X m for coalition S m is
F S m X m . If we can construct ordering plans Y j j 1, , qB for the grand coalition from the ordering , nB and show that the cumulative profit of these plans equals at least the total profit of
m plans X m 1,
, nB , i.e., nB
F
AN US
m plans X m 1,
S m
X F Y , N
then we have
j
(3.3)
j 1
M
m 1
qB
m
nB
qB
ED
qB S F S F S m X m F N Y j qB F N .
SB
S
j 1
1 for B N , nB qB S qB . Then we can construct q plans B
, qB from nB plans X m m 1,
SB
, nB .
CE
Y j j 1,
S B
PT
and (3.2) follows. Since
m 1
j In the remainder of the proof, we show how to construct plans Y j 1,
, qB for the grand coalition
AC
N based on the plans X m m 1, , nB and why the cumulative profit of plans Y j j 1, , qB equals at m least the cumulative profit of plans X m 1,
, nB . The construction procedure is given as follows:
Procedure A: Yt j : 0 j 1,
, qB , t 1,
,T
cmax : 11
ACCEPTED MANUSCRIPT
For t 1, , T For i 1,
, nB
i If X ti 1 and ct X cmax , then
k : arg max j 1,
c Y j
, qB
t
Yt k : 1
c Y
CR IP T
cmax : max j 1,
j
, qB
t
End If Next i
AN US
Next t
j In words, the procedure works as follows: We start with empty plans Y j 1, m periods t (t 1,2,..., T ) and over all the plans X m 1,
, qB and iterate over all the
, nB . Once we encounter an ordering period in
an ordering plan X m , if the unit marginal ordering cost is strictly smaller than the maximum of the current unit
, qB , we assign it to the plan Y j with the maximum current unit
M
j marginal ordering cost of plans Y j 1,
j assigned to plans Y j 1,
ED
j marginal ordering cost ct Y . This means that the qB smallest unit marginal ordering costs have been
, qB in period t .
PT
j Next we show why this procedure works, i.e., the cumulative profit of plans Y j 1,
X m m 1,
, nB . Note that we use every ordering period in each plan
CE
m least the cumulative profit of plans X m 1,
, nB at most once in constructing plans Y j j 1,
AC
j of plans Y j 1,
, qB equals at
, qB , so the cumulative fixed ordering cost
, qB equals at most the cumulative fixed ordering cost of plans X m m 1, qB
T
nB
T
Yt j Kt X tm Kt . j 1 t 1
, nB , i.e.,
(3.4)
m 1 t 1
j Because an ordering period is assigned to plan Y j with the maximum value ct Y
in period t and no
j ordering period is assigned if the unit marginal ordering costs is greater than the maximum value ct Y , the
12
ACCEPTED MANUSCRIPT
set
c Y j
t
plans
consists of the qB smallest unit marginal ordering costs of the ordering periods in the
j 1, , qB
X km m 1,
, nB
for
k 1,
periods
,t .
Assume
without
ct Y qB . Let A be an arbitrary subset of N B with
ct Y 1 ct Y 2
loss
of
generality
that
A qB . We choose qB
m arbitrary coalitions out of N B with the corresponding ordering plans X m m A . Let ct X
mA
be the
set of the corresponding unit marginal ordering costs. We may assume without loss of generality
ct X qB . Then, we have ct Y j ct X j for all j 1,
, qB and ct X 1 ct X 2
c Y
i j i j From Lemma 1, we have wt Y wt X . Otherwise,
CR IP T
that A 1,
j
t
, qB .
is not the set consisting of the qB
j 1, , qB
smallest unit marginal ordering costs. Then, for any i Sm ( m A ) and A N B with A qB , we have qB
i t
j 1
j
AN US
w Y w X . mA
i t
m
(3.5)
Now we compare the cumulative profit that retailer i ( i N ) obtains in period t ( t 1,2,..., T ) under m plans X m 1,
, nB and that under plans Y j j 1,
, qB . Let I i N B be the index set of coalitions in
c X
which player i is contained. Note that we have I i qB . Let
M
m
t
mIi
be the set of the corresponding
w X . On the other hand, the cumulative profit that retailer
mIi
j under plans Y j 1,
i t
m
, qB is
i obtains in period t
qB
w Y .
PT
X m m I i is
ED
unit marginal ordering costs. Then the total profit that retailer i obtains in period t under plans
j 1
i t
j
CE
From (3.5), we have
qB
w Y w X .
AC
j 1
i t
j
mI i
i t
m
(3.6)
So the cumulative profit that each retailer obtains in each period t under plans Y j is larger than that under plans X m .
Note that qB
qB
w Y w Y n
i 1 j 1
i t
j
j 1 iN
13
i t
j
(3.7)
ACCEPTED MANUSCRIPT
and nB
wti X j
wti X m .
(3.8)
w Y
wti X m .
(3.9)
n
m 1 iS m
i 1 jI i
qB
j 1 iN
i t
nB
j
m 1 iS m
Then we have F Y w Y Y qB
qB
N
T
j
j 1
j 1 t 1 qB
i t
iN
j
j
t
qB
CR IP T
It follows from (3.6)-(3.8) that
Kt
wti Y j Yt j K t T
j 1 t 1 iN nB
T
j 1 t 1 nB
wti X m X tm K t , T
AN US
T
m 1 t 1 iS m
m 1 t 1
nB T wti X m X tm K t m 1 t 1 iS m nB
F S m X m
M
m 1
where the inequality follows from (3.4) and (3.9). Then we prove (3.3) and the theorem.
□
ED
From Theorem 1 and the Bondareva-Shapley theorem (Bondareva, 1963; Shapley, 1967), we obtain the following result.
PT
Theorem 2. The ELS game with price-dependent demand has a non-empty core. Recall that the core of a profit game (N, F) is defined as follows:
AC
CE
n z R
z iN
i
F N and
z iS
i
F S for all S N , S .
An allocation l is in the core of the game (N, F) if
l F N and
jN j
l F S for any subset
jS j
SN.
Remark 3. It follows from Theorem 2 that it is not only profitable for the retailers to cooperate to form an alliance, but also there exists a pricing and inventory allocation policy that ensures that the grand alliance is stable. How to find the policy is a challenge since the game is not convex in general (please see Section 4). In the following section we show that the game is convex for two special cases. If a cooperative game is convex, 14
ACCEPTED MANUSCRIPT
then the Shapley value is an allocation in the core (Shapley, 1971).
4. The convexity issues of the game In this section we show that the game is convex in two special cases where the retailers face the same demand function and where the time horizon consists of two periods. We also give an example to demonstrate that the game is non-convex in general.
N, F
is convex if for each i N , we have
CR IP T
A profit game
F S i F S F R i F R for any S R N \ i .
(4.1)
First, we consider the case where the retailers face the same demand function.
AN US
Theorem 3. The ELS game with price-dependent demand is convex when the retailers face the same demand function.
Proof. Suppose the inverse demand function faced by all the retailers in period t is pt dt , i.e., ptn dt pt dt ( t 1,2,..., T ). Let s be the number of retailers in coalition S . We
prove that F S is convex in s .
M
pt1 dt pt2 dt
ED
Let P be an ordering plan represented by its ordering periods (in increasing order) and P T be the set of all the possible ordering plans. Let r t be the first ordering period after period t in the ordering plan
r j 1 F P S s max jP t j s
i t
t
i t
jt
r j 1
max p d c d t
jP t j
p d c d K i t
jt
i t
j
K j. jP
AC
CE
PT
P and r t T 1 if period t has no successor. Then the maximum profit of plan P P T is
P F P S and So under a given ordering plan P , F S is a linear function of s . Since F S Pmax P T
P T is finite, F S is piecewise linear convex. To complete the proof, let i N and S R N \ i . Using the convexity of function F S and S R , we have F S i F S F R i F R . The conclusion holds.
15
□
ACCEPTED MANUSCRIPT
When the retailers face the same demand function, it is easy to see that the average allocation is the Shapley value of the game and we have the following conclusion. Corollary 1. The policy that the retailers set the same prices and the inventory is allocated equally can ensure that the allocation is in the core when the retailers face the same demand function. Now we consider the case where the time horizon consists of two periods. In this case, we have two
• Ordering plan X 1 1,0
CR IP T
possible ordering plans, namely X 1 1,0 and X 2 1,1 .
In this case, we order all the demands d1 d 2 in period 1. The marginal costs in the two periods
1 1 are c1 X c1 and c2 X c1 h . The profit of coalition S N under plan X 1 is
AN US
F S X 1 max p1i d1i c1 d1i max p2i d 2i c1 h d 2i K1 . iS iS 2 •Ordering plan X 1,1 :
(4.2)
We order d1 in period 1 and order d 2 in period 2. The marginal costs in the two periods
M
2 2 are c1 X c1 and c2 X c2 . The profit of coalition S N under plan X 2 is
ED
F S X 2 max p1i d1i c1 d1i max p2i d 2i c2 d 2i K1 K 2 . iS iS
(4.3)
PT
S 1 S 2 The maximum profit of coalition S N is F S max F X , F X . It is easy to show the
following result from (4.2) and (4.3).
if max iS
CE
Lemma 2. We prefer ordering plan
p d c h d i 2
i 2
1
i 2
max iS
X 1 1,0
p d c d i 2
i 2
2
to ordering plan i 2
X 2 1,1
if and only
K2 .
AC
The following result states that the two-period ELS game with price dependent demand is convex.
Theorem 4. The two-period ELS game with price-dependent demand is convex. Proof. We prove that (4.1) holds for any coalitions S R . Obviously, (4.1) holds for S . Let
S R N \ j
and
S . Let d1i , d 2i and
d 2i i S
be the optimal solutions
i i i i i i i i i of v11 max p1 d1 c1 d1 , v12 max p2 d 2 c1 h d 2 , and v22 max p2 d 2 c2 d 2 iS iS iS
16
ACCEPTED MANUSCRIPT
respectively.
w11i p1i d1i c1 d1i i S
Let
,
w12i p2i d2i c1 h d 2i i S
,
and
i w22 p2i d 2i c2 d 2i i S . We consider the following two cases: c1 h c2 and c1 h c2 .
Case (i). c1 h c2 i In this case, Lemma 1 implies that w12i w22 i S for any i S . From Lemma 2, we know that
CR IP T
ordering plan X 1 1,0 is optimal for all the coalitions. Then it is easy to show that (4.1) holds. Case (ii). c1 h c2
2 In this case, we first show that if the optimal plan for some coalition S is X 1,1 , then the optimal
Since the optimal plan for coalition
iS j
w12i
iS j
AN US
2 i plan of coalition S j is also X 1,1 . It follows from c1 h c2 and Lemma 1 that w12i w22 i N .
S
is
X 2 1,1 , we have
w w iS
i 12
iS
i 22
K 2 . Hence
i w22 K 2 , so the assertion is true. From this, we know that there are six possible
combinations of the ordering plans in this case, which are listed in Table 3.
M
Table 3: The possible combinations of the ordering plans when T 2 and c1 h c2 . S j
R
R j
(1,0)
(1,0)
(1,0)
(1,0)
(1,0)
(1,1)
(1,0)
(1,0)
(1,1)
(1,1)
(1,0)
(1,1)
(1,0)
(1,1)
(1,0)
(1,1)
(1,1)
(1,1)
(1,1)
(1,1)
(1,1)
(1,1)
S
Case 1
(1,0)
Case 2
(1,0)
PT
ED
Coalition
Case 4 Case 5
AC
Case 6
CE
Case 3
In the following, we check (4.1) for each case. Case 1.
Obviously, we have
F S j F S w11j w12j F R j F R . 2 Case 2. Note that the optimal plan of coalition R j is X 1,1 , Lemma 2 implies that
17
ACCEPTED MANUSCRIPT
iR j
w12i
iR j
i w22 K2 .
So we have F S j F S w11j w12j
w11j
iR j
iR j
i w22
iR j
w12i K 2
i w22 w12i K 2 iR
CR IP T
w11j w12j
F R j F R .
Case 3. We have
F S j F S w11j w12j w11j w22j F R j F R .
w
iR S
i 22
F S j F S w11j
w12i 0 . So
AN US
Case 4. It is easy to check that
iS j
w w j 11
j 12
j 12
i w22
w
w j 11
i 22
iS j
iR j
w12i K 2
w12i K 2
iS j
iS j
w
iR S
i 22
w12i .
w w K2
M
iS
iS j
w w j 11
i w22 w12i K 2
i 22
iR
i 12
ED
F R j F R .
1 Case 5. Note that the optimal plan of coalition S is X 1,0 . From Lemma 2, we have
iS
i 12
iS
i 22
K 2 . Hence
PT
w w
AC
CE
F S j F S w11j
iS j
i w22 w12i K 2 iS
w w w w12i K 2 j 11
j 22
iS
i 22
iS
w w j 11
j 22
F R j F R .
Case 6. Obviously, we have
F S j F S w11j w22j F R j F R . Summarizing the above analysis, we establish the theorem.
□
In the sequel, we provide an example to illustrate that the ELS game with price-dependent demand 18
ACCEPTED MANUSCRIPT where T 3 and the demand functions are not the same is non-convex. Let N 1,2,3 , T 3 , K1 K2 K3 9 , c1 2 , c2 1 , c3 0 , and ht 0 . The demand functions
i i i i of the retailers are pt dt t dt , where 11 4, 12 4, 13 4 , 21 12, 22 5.5, 23 4.5 , 31 11 ,
32 4 , 33 4 . In this setting, the profit F S X of coalition S N under plan X is
For this game, we have four possible ordering plans as follows: 1 • Ordering plan X 1,0,0
Under
this
plan,
we
order
d1 d2 d3
in
period
CR IP T
3 2 1 F S X ti ct X X t Kt . t 1 4 iS
1.
The
marginal
costs
are
AN US
c1 X 1 c2 X 1 c3 X 1 c1 . The profit of coalition S N under plan X 1 is 2 1 3 ti c1 K1 4 t 1 iS 2 1 3 ti 2 9. 4 t 1 iS
M
FS X1
ED
2 •Ordering plan X 1,1,0
2 Under this plan, we order d1 in period 1 and d 2 d3 in period 2. The marginal costs are c1 X c1 ,
PT
2 2 and c2 X c3 X c2 . The profit of coalition S N under plan X 2 is
CE
FS X 2
2 2 2 1 1 1 1i c1 2i c2 3i c2 K1 K 2 4 iS 4 iS 4 iS 2 2 2 1 1 1 1i 2 2i 1 3i 1 18. 4 iS 4 iS 4 iS
AC
3 •Ordering plan X 1,0,1
Under this plan, we order d1 d 2 in period 1 and d 3 in period 3. The marginal costs are
c1 X 3 c2 X 3 c1 , and c3 X 3 c3 . The profit of coalition S N under plan X 3 is FS X3
2 2 2 1 1 1 1i c1 2i c1 3i c3 K1 K 3 4 iS 4 iS 4 iS 2 2 2 1 1 1 1i 2 2i 2 3i 18. 4 iS 4 iS 4 iS
19
ACCEPTED MANUSCRIPT
• Ordering plan X 4 1,1,1 Under this plan, we order d1 in period 1, d 2 in period 2, and d 3 in period 3. The marginal costs are
c1 X 4 c1 , c2 X 4 c2 , and c3 X 4 c3 . The profit of coalition S N under plan X 4 is 2 2 2 1 1 1 1i c1 2i c2 3i c3 K1 K 2 K3 4 iS 4 iS 4 iS 2 2 2 1 1 1 1i 2 2i 1 3i 27. 4 iS 4 iS 4 iS
CR IP T
FS X 4
Thus, the maximum profit of coalition S N is
F S max F S X 1 , F S X 2 , F S X 3 , F S X 4 .
From this, we have
AN US
F 1 max F 1 X 1 , F 1 X 2 , F 1 X 3 , F 1 X 4 max 37.25,38.25,38.25,34.5 38.25，
F 1, 2 max F 1,2 X 1 , F 1,2 X 2 , F 1,2 X 3 , F 1,2 X 4 max 42.3125, 46.5625, 46.3125, 44.5625
M
46.5625，
F 1,3 max F 1,3 X 1 , F 1,3 X 2 , F 1,3 X 3 , F 1,3 X 4
ED
max 40.8125, 44.5625, 44.8125, 42.5625 44.8125,
PT
F 1, 2,3 max F 1,2,3 X 1 , F 1,2,3 X 2 , F 1,2,3 X 3 , F 1,2,3 X 4 max 45.875,52.875,52.875,52.625
AC
Then
CE
52.875.
F 1, 2 F 1,3 46.5625 44.8125 91.375, F 1 F 1, 2,3 38.25 52.875 91.125.
It follows that
F 1 F 1,2,3 F 1,2 F 1,3 .
This means that the game is not convex.
20
ACCEPTED MANUSCRIPT
5. Numerical Experiments In this section we perform numerical experiments to show that the cooperation benefits can be significant. Consider an ELS game with price-dependent demand. There are three retailers N 1,2,3 over 6 periods with fixed ordering costs K1 K3 K5 12 , K2 K4 K6 6 , unit purchase costs ct c 3 and unit holding costs ht h 1 . The retailers’ inverse demand functions are given in Table 1. And the
CR IP T
characteristic functions are summarized in Table 2.
Table 1. The inverse demand functions 1
2
3
4
5
6
Retailer 1
10 d11
9 d 21
11 d31
10 d 41
9 d51
10 d61
Retailer 2
9 d12
8 d 22
9 d32
8 d 42
10 d52
8 d62
Retailer 3
8 d13
11 d 23
8 d33
8 d 43
9 d53
9 d63
AN US
Period
Table 2. The characteristic functions {1}
{2}
F S
34.25
18
{3}
M
S
24.5
{1, 2}
{1, 3}
{2, 3}
{1, 2, 3}
77.25
82
60.75
125
ED
When the retailers cooperate to jointly place their orders and keep inventory in the central warehouse, they can get the total profit F 1, 2,3 125 , which is larger than the sum of individual profits
PT
F 1 F 2 F 3 77 . Note that the total fixed ordering cost is 28, the total purchase cost is 130.5,
CE
the total unit holding cost is 48, and the total income is 283.25 when they place orders individually, while the total fixed ordering cost is 156, the total purchase cost is 123, the total unit holding cost is 30, and the total
AC
income is 327 when they cooperate to place orders. So the retailers’ cooperation can not only save costs, but also increase their total income. Next, we study the impacts of the unit purchase cost ct c on the profit increases achieved by
cooperation.
The
F1,2,3( F {1} F {2} F {3}) F {1} F {2} F {3} F1,2,3 ( F {1,3} F {2}) F {1,3} F {3}
results
are
depicted
in
Figure
1,
where
with respect to c ; l2 depicts the changes
and l4 depicts the changes
F1,2,3 ( F {1} F {2,3}) F {1} F {2,3}
21
.
curve
l1
F1,2,3 ( F {1,2} F {3}) F {1,2} F {3}
depicts
the
changes
; l3 depicts the changes
CR IP T
ACCEPTED MANUSCRIPT
Figure 1. Profit increase ratio with respect to c
From Figure 1, we find that the total profit increases as more players join the coalition. Moreover, the
AN US
larger the unit purchase cost c is, the more the cooperation profits are.
We then study the impact of the holding cost ht h on the profit increases. The results is depicted in Figure 2 and l1 , l2 , l3 and l4 have the similar meanings as those in Figure 1. Figure 2 shows that as the unit holding cost h increases, cooperation can also bring more profit. The more the players join the coalition,
AC
CE
PT
ED
M
the larger the profit is.
Figure 2. Profit increase ratio with respect to h
6. Conclusions In this paper we introduce a new cooperative game, i.e., the ELS game with price-dependent demand, in which the demand depends on the sale price set by the retailers in each period. We first show that the ZIO 22
ACCEPTED MANUSCRIPT
property holds and give a polynomial-time algorithm to make optimal ordering and pricing decisions, and compute the characteristic profit function based on ZIO. We then show that the game is balanced and has a non-empty core in general, i.e., there exists an allocation policy that ensures that the retailers can form a stable alliance. Finally, we show that the game is convex for two special cases where the retailers face the same demand function and where the time horizon consists of two periods. We provide a counter-example to demonstrate that the game is not convex in general.
CR IP T
In reality, there are many situations in which retailers set their prices by themselves and cooperate to replenish their inventories, e.g., chain stores, retailers in a special region etc. Also, in real practice of e-commerce, many retailers sell the same products on the same e-commerce platform, and cooperate to manage inventory and delivery. Our results show that cooperation on inventory replenishment can bring
AN US
significant profits to retailers and there exists an allocation policy that can ensure that cooperation among retailers is stable. Also, retailers can use our algorithm to derive their optimal pricing and ordering policies. Future research could address some of our limiting assumptions. In this study we assume that shortage is not allowed. However, in some practical situations, backlog is allowed. Extending our model to the case
M
where backlog is allowed is both interesting and challenging.
Another interesting point of departure is to drop the assumption that the adjustment of price incurs no
ED
cost and the demand only depends on the price in the current period. In reality, the demand in a period may depend on the price in the current period and those in the past periods. Also the adjustment of price may incur
PT
costs. Extending our model to consider these two issues are interesting topics. It is also relevant to study dependent retailers where the demands faced by retailers are dependent. In
CE
reality, there are situations where the demand faced by a retailer depends on its own price as well as the other retailers’ prices. The ELS game with inter-dependent demand functions is a topic that needs further research.
AC
Finally, extending our model to the case where the supplier is capacitated is an interesting and challenging topic.
Acknowledgements We thank the editor and reviewers for their helpful comments, which help us improve the quality of the paper. This work was supported by the National Natural Science Foundation of China (grant numbers: 23
ACCEPTED MANUSCRIPT
71390334 and 71661167009), the Program for New Century Excellent Talents in University (NCET-13-0660) and National Social Science Foundation of China (grant numbers: 15ZDA022).
References
CR IP T
Adeinat, H.，Ventura, J.A. (2015) Determining the retailer's replenishment policy considering multiple capacitated suppliers and price-sensitive demand. European Journal of Operational Research, 247(1), 83-92. Anily, S., Haviv, M. (2007) The cost allocation problem for the first order interaction joint replenishment model. Operations Research, 55(2), 292-302. Anily, S., Haviv, M. (2010) Cooperation in service systems. Operations Research, 58(3), 660-673.
Bondareva, O.N. (1963) Some applications of linear programming methods to the theory of cooperative games. Problemy Kibernetiki, 10, 119-139.
AN US
Chen, X. (2009) Inventory centralization games with price-dependent demand and quantity discount. Operations Research, 57(6), 1394-1406. Chen, X., Chen, Z. (2013) Cost allocation of capacity investment games. Naval Research Logistics, 60(6), 512-523.
M
Chen, X., Simchi-Levi D. (2012) Pricing and inventory management. In Özer, Ö., Phillips, R. (Eds.), Oxford Handbook of Pricing Management, (pp. 784-822), Oxford University Press. Chen, X., Zhang, J. (2009) A stochastic programming duality approach to inventory centralization games. Operations Research, 57(4), 840-851.
ED
Chen, X., Zhang, J. (2016). Duality approaches to economic lot‐sizing games. Production and Operations Management, 25(7), 1203-1215.
PT
Deng, S., Yano, C.A. (2006) Joint production and pricing decisions with setup costs and capacity constraints. Management Science, 52(5), 741-756.
CE
Drechsel, J. (2010) Cooperative lot sizing games in supply chains. Lecture Notes in Economics and Mathematical Systems, Volume 644, Springer Berlin Heidelberg.
AC
Dror, M., Hartman, B.C. (2007) Shipment consolidation: who pays for it and how much? Management Science, 53(1), 78-87. Elmaghraby, W., Keskinocak, P. (2003) Dynamic pricing in the presence of inventory considerations: Research overview, current practices, and future directions. Management Science, 49(10), 1287-1309. Esmaeili, M., Aryanezhad, M.B., Zeephongsekul, P. (2009) A game theory approach in seller–buyer supply chain. European Journal of Operational Research, 195(2), 442-448. Esmaeili, M., Zeephongsekul, P. (2010) Seller-buyer models of supply chain management with an asymmetric information structure. International Journal of Production Economics, 123(1), 146-154. Florian, M., Klein, M. (1971) Deterministic production planning with concave costs and capacity 24
ACCEPTED MANUSCRIPT
constraints. Management Science, 18(1), 12-20. Florian, M., Lenstra, J.K., Rinnooy Kan, A.H.G. (1980) Deterministic production planning: Algorithms and complexity. Management Science, 26(7), 669-679. Guardiola, L.A., Meca, A., Timmer, J. (2007) Cooperation and profit allocation in distribution chains. Decision Support Systems, 44(1), 17-27. Gopaladesikan, M., Uhan, N.A. (2011) Cost sharing for the economic lot-sizing problem with remanufacturing options, http://www.optimization-online.org/DB_FILE/2010/09/2733.pdf, 2011.
CR IP T
Guardiola, L.A., Meca, A., Puerto, J. (2006) Coordination in periodic review inventory situations. Working Paper, Centro de Investigación Operativa, Universidad Miguel Hernández de Elche. Hall, N.G., Liu, Z. (2010) Capacity allocation and scheduling in supply chains. Operations Research, 58(6), 1711-1725.
AN US
Hartman, B.C., Dror, M., Shaked, M. (2000) Cores of inventory centralization games. Games and Economic Behavior, 31(1), 26-49. Karsten, F.J.P., Slikker, M., van Houtum, G.J. (2009) Spare parts inventory pooling games. Working Paper, School of Industrial Engineering, Eindhoven University of Technology. Lu, L., Qi, X., Liu, Z. (2014) On the cooperation of recycling operations. European Journal of Operational Research, 233(2), 349-358.
M
Müller, A., Scarsini, M., Shaked, M. (2002) The newsvendor game has a nonempty core. Games and Economic Behavior, 38(1), 118-126.
ED
Montrucchio, L., Norde, H., Özen, U., Scarsini, M., Slikker, M. (2012) Cooperative newsvendor games: A review. In Choi, T.M. (Ed.), Handbook of Newsvendor Problems (pp. 137-162), Springer, New York.
PT
Nagarajan, M., Sošić, G. (2008) Game-theoretic analysis of cooperation among supply chain agents: Review and extensions. European Journal of Operational Research, 187(3), 719-745.
CE
Özen, U., Fransoo, J., Norde, H., Slikker, M. (2008) Cooperation between multiple newsvendors with warehouses. Manufacturing & Service Operations Management, 10(2), 311-324.
AC
Özener, O.Ö., Ergun, Ö., Savelsbergh, M. (2013) Allocating cost of service to customers in inventory routing. Operations Research, 61(1), 112-125.
Shapley, L.S. (1971) Cores of convex games. International Journal of Game Theory, 1(1), 11-26. Slikker, M., Fransoo, J., Wouters, M. (2005) Cooperation between transshipments. European Journal of Operational Research, 167(2), 370-380.
multiple
newsvendors
with
Talluri, K.T., Van Ryzin, G.J. (2004) The Theory and Practice of Revenue Management. Springer. Thomas, J. (1970) Price-production decisions with deterministic demand. Management Science, 16(11), 747-750. Toriello, A., Uhan, N.A. (2013) Dynamic linear production games under uncertainty. Working Paper, University of Chicago, Booth School of Business Operations/Management Science Workshop.
25
ACCEPTED MANUSCRIPT
Uhan, N.A. (2015) Stochastic linear programming games with concave preferences. European Journal of Operational Research, 243(2), 637-646. van den Heuvel, W., Borm, P., Hamers, H. (2007) Economic lot-sizing games. European Journal of Operational Research, 176(2), 1117-1130. Wagner, H.M., Whitin, T.M. (1958) Dynamic problems in the theory of the firm. Naval Research Logistics Quarterly, 5(1), 53-74.
CR IP T
Yano, C.A., Gilbert, S.M. (2004) Coordinated pricing and production/procurement decisions: A review. In Chakravarty, A.K., Eliashberg, J. (Eds.), Managing Business Interfaces (pp. 65-103), Springer USA. Yu, Y., Benjaafar, S., Gerchak, Y. (2015) Capacity sharing and cost allocation among independent firms with congestion. Production and Operations Management, 24(8), 1285-1310.
AC
CE
PT
ED
M
AN US
Zhang, J. (2009) Cost allocation for joint replenishment models. Operations Research, 57(1), 146-156.
26
Economic Lot-sizing Game with Price-dependent Demand Xiaojie Yan , Juliang Zhang , T.C.E. Cheng , Guowei Hua PII: DOI: Reference:
S0377-2217(17)30179-0 10.1016/j.ejor.2017.02.040 EOR 14277
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
28 January 2016 23 February 2017 27 February 2017
Please cite this article as: Xiaojie Yan , Juliang Zhang , T.C.E. Cheng , Guowei Hua , Economic Lotsizing Game with Price-dependent Demand, European Journal of Operational Research (2017), doi: 10.1016/j.ejor.2017.02.040
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Highlights A new cooperative game, economic lotsize game with dynamic pricing, is introduced; Show that the game is balanced and has a nonempty core under general conditions; Show that the game is convex for two special cases but not convex for general case; Numerical experiments are conducted to show the benefits of cooperation.
AC
CE
PT
ED
M
AN US
CR IP T
• • • •
1
ACCEPTED MANUSCRIPT Economic Lot-sizing Game with Price-dependent Demand Xiaojie Yan1, Juliang Zhang1*, T.C.E. Cheng2, Guowei Hua1 1
Department of Logistics Management School of Economics and Management Beijing Jiaotong University, Beijing, 100044, China 2
CR IP T
Department of Logistics and Maritime Studies The Hong Kong Polytechnic University Hung Hom, Kowloon, Hong Kong SAR
Abstract
AN US
We study the economic lot-sizing (ELS) game with price-dependent demand, whereby several retailers form a coalition to jointly place orders to a single supplier and determine their sale prices over a multi-period finite time horizon for profit maximization. We show that the game is balanced and has a non-empty core in general. We also show that the game is convex in two special cases where the retailers’ demand functions are the same and where the time horizon consists of two periods. We give a counter-example to demonstrate that the game
M
is non-convex in general. We perform numerical experiments to show the benefits of cooperating on inventory replenishment.
AC
CE
PT
ED
Key words: logistics; economic lot-sizing game; pricing; balancedness; core
*
Corresponding author, email: [email protected] 2
ACCEPTED MANUSCRIPT
1.
Introduction Increasing numbers of firms in a variety of industries have deployed innovative pricing strategies to
manage demand and improve profit (see, e.g., Elmaghraby and Keskinocak, 2003; Chen and Simchi-Levi, 2012). In addition, many companies have explored innovative collaboration strategies in efforts to improve
CR IP T
their core competence and bottom lines. For example, many retailers in Zhongguancun Electronic City in Beijing, China cooperate to place orders and stock their products in a centralized warehouse while only displaying sample products in their shops. In addition, Good Neighbor Pharmacy and Affiliated Foods Midwest employ collaborative strategies in the form of long-term alliance and collaborative logistics,
AN US
respectively, with a view to reducing costs. The former is a cooperative network of 2,700 independently owned and operated pharmacies, while the latter supplies more than 850 independent retailers in the 12 Midwest states with a full line of grocery products. These pharmacies and retailers form retailers’ cooperative groups which implement unified management such as inventory management, pricing assistance, logistics, warehousing and so on (Chen, 2009). In order to keep the long-term relationship, a very important issue faced
M
by the partnering firms is how to optimally allocate the profit or cost among them to ensure that the alliance is
ED
stable (van den Heuvel et al., 2007; Chen, 2009). Seeking to address this problem, we adopt the economic lot-sizing (ELS) game with price-dependent demand for modelling and analysis in this paper.
PT
There are two related steams of research. The first one is on inventory centralization games, for which Nagarajan and Sošić (2008), Drechsel (2010), and Montrucchio et al. (2012) gave comprehensive reviews. We
CE
briefly review the literature most closely related to our work. Hartman et al. (2000), Müller et al. (2002), Slikker et al. (2005), Chen and Zhang (2009), Özen et al. (2008), and Uhan (2015) studied the newsvendor
AC
game, while Chen (2009) studied the newsvendor game with price-dependent demand. They showed that the core is non-empty. Lu et al. (2014) studied how different firms cooperate in their recycling and pricing decisions using cooperative game theory. All these studies considered one-period problem and assumed that the retailers face random demands. Dror and Hartman (2007), Anily and Haviv (2007), and Zhang (2009) studied the inventory centralization game based on the economic order quantity (EOQ) model. They assumed that the demand is stationary. Karsten et al. (2009) studied the continuous-time inventory centralization game where demand occurs 3
ACCEPTED MANUSCRIPT
according to a Poisson process with a constant demand rate. Many researchers have studied the multi-period inventory game based on the ELS problem, which is called the ELS game. Van den Heuvel et al. (2007) showed that the ELS game is balanced and has a non-empty core under the assumption that backorders are not allowed. Guardiola et al. (2006) studied the case where backorders are allowed and showed how to find a core allocation. Chen and Zhang (2016) studied the game with a general ordering cost and allowing backorders. They developed a duality approach to compute a core allocation in
CR IP T
polynomial time. They pointed out that it remains a challenge to incorporate pricing decisions into the ELS game. Gopaladesikan and Uhan (2011) studied the cooperative cost sharing game based on the ELS problem considering re-manufacturing options. They showed that the game has an empty core in general and a non-empty core for two special cases. Drechsel (2010) extended the ELS games to include uncertain demand and capacity constraints with
AN US
transshipments, and studied a multi-level lot sizing game with restricted cooperation. Toriello and Uhan (2013) generalized the classical linear production game to the multi-period model with stochastic parameters. Guardiola et al. (2007), Esmaeili et al. (2009), Esmaeili and Zeephongsekul (2010), Anily and Haviv (2010), and Özener et al. (2013) studied applications of cooperative game theory to the management of centralized inventory systems. Hall
M
and Liu (2010), Chen and Chen (2013), and Yu et al. (2015) studied the inventory centralization game with capacity sharing using cooperative game theory. The above research assumed that the price is exogenous.
ED
The second research stream related to this study is on the joint decision on pricing and inventory control, on which Elmaghraby and Keskinocak (2003), Yano and Gilbert (2004), and Chen and Simchi-Levi (2012)
PT
gave extensive reviews. We only review some literature based on the ELS model. Wagner and Whitin (1958) first studied the ELS problem with price-dependent demand without capacity constraints. They established the
CE
zero inventory ordering (ZIO) property and proposed a cyclic algorithm with polynomial-time complexity. Thomas (1970) demonstrated explicitly that the optimal price in one period is consistently independent of the
AC
prices in the other periods. Florian and Morton (1971), and Florian et al. (1980) showed that the joint pricing and inventory management problem with capacity limits is NP-hard. Deng and Yano (2006) characterized properties of the optimal solution and gave an algorithm to solve the problem. Adeinat and Ventura (2015) generalized the problem to the case with multiple capacitated suppliers. All these studies considered only one retailer. In this paper we study the ELS game with price-dependent demand. In this game, there are several retailers that make ordering and pricing decisions over a multi-period finite time horizon. They may cooperate 4
ACCEPTED MANUSCRIPT
to place joint orders and keep inventory in a central warehouse. The demand of each retailer depends on its own sale price. At the beginning of each period, the retailers first review the initial inventory in the central warehouse, and then jointly decide on the order quantity and their sale prices. Associated with each order there are a fixed ordering cost and a purchase cost, which depends on the order quantity and the unit purchase cost. We assume that the lead time is zero so that the order is delivered immediately to the central warehouse. After the order delivery, the retailers’ demands are realized and the inventory in the central warehouse is allocated to
CR IP T
satisfy the demands; no shortages are allowed. The unsold inventory in any period is carried forward to the next period, which incurs a holding cost. We further assume that the inventory at the beginning of the time horizon is zero, and that the demand, ordering cost, and holding cost are independent, and the transportation cost from the central warehouse to each retailer’s warehouse is negligible. The objective is to make optimal
AN US
ordering and pricing decisions to maximize the total profit of all the retailers over all the periods in the time horizon. Also, we study whether there exists an allocation policy that ensures that the retailers can form a stable alliance, i.e., whether the core of the cooperative game is non-empty.
We show that the ZIO property holds and propose a polynomial-time algorithm to compute the optimal
M
total profits of the coalitions. We prove that the ELS game with price-dependent demand is balanced and has a non-empty core. We show that the game is convex in two special cases where the retailers’ demand functions
ED
are the same and where there are only two periods in the time horizon. We provide a counter-example to demonstrate that the game is non-convex in general.
PT
We organize the remainder of the paper as follows: In Section 2 we introduce the ELS game with price-dependent demand and show how to find the characteristic functions of coalitions. In Section 3 we prove
CE
that the game is balanced and has a non-empty core. In Section 4 we address the convexity issues of the game.
AC
We conclude the paper in the last section.
2. ELS game with price-dependent demand In this section we introduce the ELS game with price-dependent demand and show how to compute the
maximum payoff of the coalitions. Consider a set of retailers N 1,2
, n that sell the same product, and make pricing and ordering
decisions over a finite time horizon consisting of T periods. Retailer i ’s demand d ti in period t is 5
ACCEPTED MANUSCRIPT
deterministic and depends on its set price pti in that period. Assume that the demand function is dti Dti pti (i 1,2,..., n, t 1,2,..., T ) , which is a decreasing and concave function of the current price
(Talluri and Van Ryzin, 2004; Chen and Simchi-Levi, 2012). The retailers may cooperate to buy the product from the same manufacturer. When the retailers in a set (coalition) S N decide to cooperate, they place joint orders, which are delivered to a central warehouse. At the beginning of period t , they first review the
CR IP T
inventory I t 1 in the central warehouse, and then decide their sale prices pti (i S ) and the order quantity
xt . When an order is placed, a fixed ordering cost K t and a purchase cost ct xt are incurred, where ct is the unit purchase cost. Let us assume that the lead time is zero, so orders is delivered immediately. After the order delivery, the demands are realized and the inventory in the central warehouse is allocated to satisfy the
AN US
demands; no shortages are allowed. The unsold inventory is carried forward to the next period, which incurs a unit holding cost ht . Assume that the inventory at the beginning of period 1 is zero and that the transportation cost from the central warehouse to the retailers is zero.
The retailers can place orders individually or opt to cooperate to keep joint inventory in the central
M
warehouse. Following the arguments in van den Heuvel et al. (2007), we can show that it is profitable for the
ED
retailers to cooperate. The problem is whether there exists an allocation policy that ensures that the grand coalition is stable, i.e., all the retailers form a coalition.
PT
For each coalition S N , let F S be the characteristic profit function of S , i.e., the total maximum profit obtained by the retailers in the set S that cooperate in their pricing and ordering decisions. The game
CE
( N , F ) is called the ELS game with price-dependent demand.
Noting that there is a one-to-one correspondence relationship between the demand and price of each
AC
retailer, we adopt demand d ti (i 1,..., n) the analysis. It follows that
as the decision variable rather than price pti (i 1,..., n) to simplify
pti pti (dti ) Dti
1
d i t
(i 1,..., n) . Since dti Dti pti
(i 1,..., n) is a
decreasing and concave function by assumption, pti pti (dti ) (i 1,..., n) is a decreasing and concave function. i i i So the profit function dt pt dt (i 1,..., n) is also concave. The notation used in this paper is listed as
follows: 6
ACCEPTED MANUSCRIPT
d ti : demand of retailer i in period t, (a decision variable);
xt : order quantity in period t (a decision variable; we call t an ordering period if xt 0 ); I t : ending inventory in period t;
K t : fixed ordering cost in period t;
CR IP T
ct : unit purchase cost in period t; ht : unit holding cost in period t;
pti dti : the sale price of retailer i in period t when the demand is d ti ;
cst : defined as cst cs l s hl .
AN US
t 1
We now consider the computation of the characteristic function, F S . Evidently, F S is the optimal value of the following optimization problem: F S : max
d p d K x c x iS 1 t T
s.t.
i t
i t
i t
1 t T
I t xt I t 1 dti ,
t
t
t 1,
t t
ht I t
,T
M
iS
(2.1)
I 0 0，
,T
PT
1, xt 0, where xt 0, otherwise.
t 1,
ED
xt 0, I t 0，
The following proposition characterizes the optimal inventory policy for problem (2.1).
CE
Proposition 1. The ZIO property holds for problem (2.1), i.e., no order is placed whenever the initial inventory is positive.
AC
Proof. Suppose that the result does not hold, i.e., there exists an optimal ordering plan x x1 , x2 ,
, xT for
problem (2.1) and an ordering period t such that I t 1 0 and xt 0 . Since the inventory of each retailer at the beginning of period 1 is zero, the order quantity in period 1 is positive and period t is not period 1. Let period s be the ordering period immediately before period t . We keep the pricing decision unchanged and either shift the order quantity in period t to period s or shift an order quantity of I t 1 units from period s to period t . In the former case, no order is placed in period t 7
ACCEPTED MANUSCRIPT
and the resulting cost change is
cst ct xt , while in the latter case, the initial inventory level in period
t is
zero and the resulting cost change is cst ct It 1 , where cst cs l s hl is the marginal cost of t 1
satisfying the demand in period t by an order placed in period s . Clearly either case will lead to an ordering plan with a cost no more than that of the original plan. Hence the profit of new plan is larger than the profit of the original plan, contradicting x x1 , x2 ,
, xT is an optimal ordering plan, and the result follows.
CR IP T
□
From Proposition 1, we can convert problem (2.1) into the longest-path problem, which can be solved by a forward-type algorithm in O(T 2 ) time (see, e.g., Chen and Simchi-Levi, 2012), and design an algorithm
AN US
with polynomial-time complexity to solve it.
Algorithm 1: Step 1.
If periods a and b (a < b) are two consecutive ordering periods in an ordering plan, the ZIO property implies that the demand in period t a t b is satisfied by the order placed in period
M
a only, so the marginal cost of satisfying the demand in period t is cat . The associated optimal
ED
demand in period t a t b can then be derived by solving the following optimization problem: vatS max d i 0
p d c d . iS
i t
i t
at
i t
(2.2)
Construct an equivalent acyclic network G = (V, E) with the node set V 1,2,
PT
Step 2.
t
CE
set V a, b :1 a b T 1 . The arc
a, b
means that the demands from periods a through
b 1 are satisfied by the order in period a . The weight of the arc a, b is
AC
, T 1 and arc
b 1 S t a at
v Ka , which
is the total profit from periods a through b 1 with the demands determined by solving problem (2.2) for t a,
Step 3.
, b 1 .
Find the longest path from node 1 to node T+1 in the network G = (V, E). The longest path gives an optimal ordering plan with the ZIO property whose ordering periods exactly correspond to the nodes on the longest path, and the associated optimal demands are derived from solving problem (2.2). The length of the longest path is the maximum profit F S . The order quantity in an 8
ACCEPTED MANUSCRIPT
ordering period equals the sum of the demands from the current period to the period before the next ordering period. The associated optimal ordering decision is decision of retailer i is
i* 1
d
i* 1
,
, xT* . The optimal demand S
, dTi * . S
,
We obtain the optimal pricing decisions of retailer i demand decision
* 1
p d , i 1
i* 1
, pTi dTi*
S
by using the optimal
, dTi * . S
CR IP T
Step 4.
d
x ,
Remark 1. Note that the objective function in problem (2.2) is by assumption concave, implying (2.2) is an unconstrained concave programming and can be solved by a polynomial algorithm. This follows that Algorithm 1 is a polynomial-time algorithm.
Remark 2. Proposition 1 states that the alliance places an order only if the initial inventory is zero. Algorithm
AN US
1 helps the retailers to properly set prices, place orders, and allocate the inventory when the demands realized and the orders are delivered when they decide to cooperate to form an alliance.
3. The balancedness of the game
M
In Section 2 we show that it is profitable for retailers to cooperate. In this section we study whether there exists an allocation policy that ensures that the grand coalition is stable, i.e., whether the core of the
ED
cooperative game is non-empty. Noting that the objection function in (2.1) is non-linear and non-continuous, there exists a duality gap between the primal and dual problems. Chen (2009) and Lu et al (2014) considered
PT
the single-period pricing problem with a duality gap and showed that the coalition’s optimal order quantity increases as more players join the coalition. Based on this, they located a core allocation. Given that the ELS
CE
game is multi-period, the duality gap property no longer holds, so the duality approach developed by Chen and Zhang (2016) cannot be used to find a core allocation. Here we show that the game is balanced. The core
AC
of the game is non-empty by the Bondareva-Shapley theorem (Bondareva, 1963; Shapley, 1971). The proof is obtained by merging the ordering plans (of balanced collections) of coalitions to construct some ordering plans for the grand coalition in such a way that we can show the balancedness conditions of the Bondareva-Shapley theorem are satisfied. Let 2 N be the set consisting of all the subsets of N . Then B 2N is called a balanced collection of
9
ACCEPTED MANUSCRIPT
2 N if there exist weights S ( S B ) such that
e
SB
S
S
e N , where e S is the vector in R n with e S 1 i
F S F N
if i S and eiS 0 otherwise. A game is balanced if
S
SB
for all the balanced collections.
As noted before, ordering plans can be completely described by setting the ordering periods. Let X be an ordering plan and X t 1 if period t is an ordering period and X t 0 otherwise ( t 1, , T ). Let t 1
j max l t X l 1 and wti X max dti pti dti ct X dti . So the total
with
CR IP T
ct X c j l j hl
S profit F X of the ordering plan X is
T F S X wti X X t Kt . t 1 iS
AN US
For convenience, a plan X with X t 0 ( t 1, , T ) is called an empty plan, and we define ct X and wti X 0 under an empty plan X .
The following result states that the profit that retailer i obtains in period t under ordering plan X decreases in the unit marginal ordering cost of plan X in period t .
Suppose
ct X ct X
that
ED
Proof:
M
i Lemma 1. For any i S , wt X is strictly decreasing in ct X .
wti X max d i pti dti ct X dti
,
t
,
wti X max d i pti dti ct X dti , and d ti is an optimal solution for wti X . For any i S , we have t
PT
wti X max d i
p d c d p d c X d i t
t
i t
jt
i t
i t
i t
t
i t
CE
pti dti ct X d ti
max d i pti dti ct X dti t
wti X . □
AC
The lemma holds.
Theorem 1. The ELS game with price-dependent demand is balanced. Proof: Let B be a balanced collection of 2 N and S ( S B ) be the corresponding weights. It suffices to prove that
F S F N .
SB
S
(3.1)
Obviously, (3.1) holds for B N . Let B N and qB be the smallest integer for which qB S is 10
ACCEPTED MANUSCRIPT integral for all S B . Then (3.1) is equivalent to
q F S q F N .
SB
B S
(3.2)
B
The term qB S F S on the left hand side of (3.2) is qB S times of the profit of the optimal ordering plan for coalition S . So the left hand side of (3.2) is the total profit of nB qB S ordering plans. Let SB
, nB be the index set of coalitions (in an arbitrary order) and let S m be the coalition
CR IP T
N B 1,
corresponding to index m . So we have qB S indices for each coalition S B . Moreover, let X m be the optimal ordering plan corresponding to coalition S m , so the profit of plan X m for coalition S m is
F S m X m . If we can construct ordering plans Y j j 1, , qB for the grand coalition from the ordering , nB and show that the cumulative profit of these plans equals at least the total profit of
m plans X m 1,
, nB , i.e., nB
F
AN US
m plans X m 1,
S m
X F Y , N
then we have
j
(3.3)
j 1
M
m 1
qB
m
nB
qB
ED
qB S F S F S m X m F N Y j qB F N .
SB
S
j 1
1 for B N , nB qB S qB . Then we can construct q plans B
, qB from nB plans X m m 1,
SB
, nB .
CE
Y j j 1,
S B
PT
and (3.2) follows. Since
m 1
j In the remainder of the proof, we show how to construct plans Y j 1,
, qB for the grand coalition
AC
N based on the plans X m m 1, , nB and why the cumulative profit of plans Y j j 1, , qB equals at m least the cumulative profit of plans X m 1,
, nB . The construction procedure is given as follows:
Procedure A: Yt j : 0 j 1,
, qB , t 1,
,T
cmax : 11
ACCEPTED MANUSCRIPT
For t 1, , T For i 1,
, nB
i If X ti 1 and ct X cmax , then
k : arg max j 1,
c Y j
, qB
t
Yt k : 1
c Y
CR IP T
cmax : max j 1,
j
, qB
t
End If Next i
AN US
Next t
j In words, the procedure works as follows: We start with empty plans Y j 1, m periods t (t 1,2,..., T ) and over all the plans X m 1,
, qB and iterate over all the
, nB . Once we encounter an ordering period in
an ordering plan X m , if the unit marginal ordering cost is strictly smaller than the maximum of the current unit
, qB , we assign it to the plan Y j with the maximum current unit
M
j marginal ordering cost of plans Y j 1,
j assigned to plans Y j 1,
ED
j marginal ordering cost ct Y . This means that the qB smallest unit marginal ordering costs have been
, qB in period t .
PT
j Next we show why this procedure works, i.e., the cumulative profit of plans Y j 1,
X m m 1,
, nB . Note that we use every ordering period in each plan
CE
m least the cumulative profit of plans X m 1,
, nB at most once in constructing plans Y j j 1,
AC
j of plans Y j 1,
, qB equals at
, qB , so the cumulative fixed ordering cost
, qB equals at most the cumulative fixed ordering cost of plans X m m 1, qB
T
nB
T
Yt j Kt X tm Kt . j 1 t 1
, nB , i.e.,
(3.4)
m 1 t 1
j Because an ordering period is assigned to plan Y j with the maximum value ct Y
in period t and no
j ordering period is assigned if the unit marginal ordering costs is greater than the maximum value ct Y , the
12
ACCEPTED MANUSCRIPT
set
c Y j
t
plans
consists of the qB smallest unit marginal ordering costs of the ordering periods in the
j 1, , qB
X km m 1,
, nB
for
k 1,
periods
,t .
Assume
without
ct Y qB . Let A be an arbitrary subset of N B with
ct Y 1 ct Y 2
loss
of
generality
that
A qB . We choose qB
m arbitrary coalitions out of N B with the corresponding ordering plans X m m A . Let ct X
mA
be the
set of the corresponding unit marginal ordering costs. We may assume without loss of generality
ct X qB . Then, we have ct Y j ct X j for all j 1,
, qB and ct X 1 ct X 2
c Y
i j i j From Lemma 1, we have wt Y wt X . Otherwise,
CR IP T
that A 1,
j
t
, qB .
is not the set consisting of the qB
j 1, , qB
smallest unit marginal ordering costs. Then, for any i Sm ( m A ) and A N B with A qB , we have qB
i t
j 1
j
AN US
w Y w X . mA
i t
m
(3.5)
Now we compare the cumulative profit that retailer i ( i N ) obtains in period t ( t 1,2,..., T ) under m plans X m 1,
, nB and that under plans Y j j 1,
, qB . Let I i N B be the index set of coalitions in
c X
which player i is contained. Note that we have I i qB . Let
M
m
t
mIi
be the set of the corresponding
w X . On the other hand, the cumulative profit that retailer
mIi
j under plans Y j 1,
i t
m
, qB is
i obtains in period t
qB
w Y .
PT
X m m I i is
ED
unit marginal ordering costs. Then the total profit that retailer i obtains in period t under plans
j 1
i t
j
CE
From (3.5), we have
qB
w Y w X .
AC
j 1
i t
j
mI i
i t
m
(3.6)
So the cumulative profit that each retailer obtains in each period t under plans Y j is larger than that under plans X m .
Note that qB
qB
w Y w Y n
i 1 j 1
i t
j
j 1 iN
13
i t
j
(3.7)
ACCEPTED MANUSCRIPT
and nB
wti X j
wti X m .
(3.8)
w Y
wti X m .
(3.9)
n
m 1 iS m
i 1 jI i
qB
j 1 iN
i t
nB
j
m 1 iS m
Then we have F Y w Y Y qB
qB
N
T
j
j 1
j 1 t 1 qB
i t
iN
j
j
t
qB
CR IP T
It follows from (3.6)-(3.8) that
Kt
wti Y j Yt j K t T
j 1 t 1 iN nB
T
j 1 t 1 nB
wti X m X tm K t , T
AN US
T
m 1 t 1 iS m
m 1 t 1
nB T wti X m X tm K t m 1 t 1 iS m nB
F S m X m
M
m 1
where the inequality follows from (3.4) and (3.9). Then we prove (3.3) and the theorem.
□
ED
From Theorem 1 and the Bondareva-Shapley theorem (Bondareva, 1963; Shapley, 1967), we obtain the following result.
PT
Theorem 2. The ELS game with price-dependent demand has a non-empty core. Recall that the core of a profit game (N, F) is defined as follows:
AC
CE
n z R
z iN
i
F N and
z iS
i
F S for all S N , S .
An allocation l is in the core of the game (N, F) if
l F N and
jN j
l F S for any subset
jS j
SN.
Remark 3. It follows from Theorem 2 that it is not only profitable for the retailers to cooperate to form an alliance, but also there exists a pricing and inventory allocation policy that ensures that the grand alliance is stable. How to find the policy is a challenge since the game is not convex in general (please see Section 4). In the following section we show that the game is convex for two special cases. If a cooperative game is convex, 14
ACCEPTED MANUSCRIPT
then the Shapley value is an allocation in the core (Shapley, 1971).
4. The convexity issues of the game In this section we show that the game is convex in two special cases where the retailers face the same demand function and where the time horizon consists of two periods. We also give an example to demonstrate that the game is non-convex in general.
N, F
is convex if for each i N , we have
CR IP T
A profit game
F S i F S F R i F R for any S R N \ i .
(4.1)
First, we consider the case where the retailers face the same demand function.
AN US
Theorem 3. The ELS game with price-dependent demand is convex when the retailers face the same demand function.
Proof. Suppose the inverse demand function faced by all the retailers in period t is pt dt , i.e., ptn dt pt dt ( t 1,2,..., T ). Let s be the number of retailers in coalition S . We
prove that F S is convex in s .
M
pt1 dt pt2 dt
ED
Let P be an ordering plan represented by its ordering periods (in increasing order) and P T be the set of all the possible ordering plans. Let r t be the first ordering period after period t in the ordering plan
r j 1 F P S s max jP t j s
i t
t
i t
jt
r j 1
max p d c d t
jP t j
p d c d K i t
jt
i t
j
K j. jP
AC
CE
PT
P and r t T 1 if period t has no successor. Then the maximum profit of plan P P T is
P F P S and So under a given ordering plan P , F S is a linear function of s . Since F S Pmax P T
P T is finite, F S is piecewise linear convex. To complete the proof, let i N and S R N \ i . Using the convexity of function F S and S R , we have F S i F S F R i F R . The conclusion holds.
15
□
ACCEPTED MANUSCRIPT
When the retailers face the same demand function, it is easy to see that the average allocation is the Shapley value of the game and we have the following conclusion. Corollary 1. The policy that the retailers set the same prices and the inventory is allocated equally can ensure that the allocation is in the core when the retailers face the same demand function. Now we consider the case where the time horizon consists of two periods. In this case, we have two
• Ordering plan X 1 1,0
CR IP T
possible ordering plans, namely X 1 1,0 and X 2 1,1 .
In this case, we order all the demands d1 d 2 in period 1. The marginal costs in the two periods
1 1 are c1 X c1 and c2 X c1 h . The profit of coalition S N under plan X 1 is
AN US
F S X 1 max p1i d1i c1 d1i max p2i d 2i c1 h d 2i K1 . iS iS 2 •Ordering plan X 1,1 :
(4.2)
We order d1 in period 1 and order d 2 in period 2. The marginal costs in the two periods
M
2 2 are c1 X c1 and c2 X c2 . The profit of coalition S N under plan X 2 is
ED
F S X 2 max p1i d1i c1 d1i max p2i d 2i c2 d 2i K1 K 2 . iS iS
(4.3)
PT
S 1 S 2 The maximum profit of coalition S N is F S max F X , F X . It is easy to show the
following result from (4.2) and (4.3).
if max iS
CE
Lemma 2. We prefer ordering plan
p d c h d i 2
i 2
1
i 2
max iS
X 1 1,0
p d c d i 2
i 2
2
to ordering plan i 2
X 2 1,1
if and only
K2 .
AC
The following result states that the two-period ELS game with price dependent demand is convex.
Theorem 4. The two-period ELS game with price-dependent demand is convex. Proof. We prove that (4.1) holds for any coalitions S R . Obviously, (4.1) holds for S . Let
S R N \ j
and
S . Let d1i , d 2i and
d 2i i S
be the optimal solutions
i i i i i i i i i of v11 max p1 d1 c1 d1 , v12 max p2 d 2 c1 h d 2 , and v22 max p2 d 2 c2 d 2 iS iS iS
16
ACCEPTED MANUSCRIPT
respectively.
w11i p1i d1i c1 d1i i S
Let
,
w12i p2i d2i c1 h d 2i i S
,
and
i w22 p2i d 2i c2 d 2i i S . We consider the following two cases: c1 h c2 and c1 h c2 .
Case (i). c1 h c2 i In this case, Lemma 1 implies that w12i w22 i S for any i S . From Lemma 2, we know that
CR IP T
ordering plan X 1 1,0 is optimal for all the coalitions. Then it is easy to show that (4.1) holds. Case (ii). c1 h c2
2 In this case, we first show that if the optimal plan for some coalition S is X 1,1 , then the optimal
Since the optimal plan for coalition
iS j
w12i
iS j
AN US
2 i plan of coalition S j is also X 1,1 . It follows from c1 h c2 and Lemma 1 that w12i w22 i N .
S
is
X 2 1,1 , we have
w w iS
i 12
iS
i 22
K 2 . Hence
i w22 K 2 , so the assertion is true. From this, we know that there are six possible
combinations of the ordering plans in this case, which are listed in Table 3.
M
Table 3: The possible combinations of the ordering plans when T 2 and c1 h c2 . S j
R
R j
(1,0)
(1,0)
(1,0)
(1,0)
(1,0)
(1,1)
(1,0)
(1,0)
(1,1)
(1,1)
(1,0)
(1,1)
(1,0)
(1,1)
(1,0)
(1,1)
(1,1)
(1,1)
(1,1)
(1,1)
(1,1)
(1,1)
S
Case 1
(1,0)
Case 2
(1,0)
PT
ED
Coalition
Case 4 Case 5
AC
Case 6
CE
Case 3
In the following, we check (4.1) for each case. Case 1.
Obviously, we have
F S j F S w11j w12j F R j F R . 2 Case 2. Note that the optimal plan of coalition R j is X 1,1 , Lemma 2 implies that
17
ACCEPTED MANUSCRIPT
iR j
w12i
iR j
i w22 K2 .
So we have F S j F S w11j w12j
w11j
iR j
iR j
i w22
iR j
w12i K 2
i w22 w12i K 2 iR
CR IP T
w11j w12j
F R j F R .
Case 3. We have
F S j F S w11j w12j w11j w22j F R j F R .
w
iR S
i 22
F S j F S w11j
w12i 0 . So
AN US
Case 4. It is easy to check that
iS j
w w j 11
j 12
j 12
i w22
w
w j 11
i 22
iS j
iR j
w12i K 2
w12i K 2
iS j
iS j
w
iR S
i 22
w12i .
w w K2
M
iS
iS j
w w j 11
i w22 w12i K 2
i 22
iR
i 12
ED
F R j F R .
1 Case 5. Note that the optimal plan of coalition S is X 1,0 . From Lemma 2, we have
iS
i 12
iS
i 22
K 2 . Hence
PT
w w
AC
CE
F S j F S w11j
iS j
i w22 w12i K 2 iS
w w w w12i K 2 j 11
j 22
iS
i 22
iS
w w j 11
j 22
F R j F R .
Case 6. Obviously, we have
F S j F S w11j w22j F R j F R . Summarizing the above analysis, we establish the theorem.
□
In the sequel, we provide an example to illustrate that the ELS game with price-dependent demand 18
ACCEPTED MANUSCRIPT where T 3 and the demand functions are not the same is non-convex. Let N 1,2,3 , T 3 , K1 K2 K3 9 , c1 2 , c2 1 , c3 0 , and ht 0 . The demand functions
i i i i of the retailers are pt dt t dt , where 11 4, 12 4, 13 4 , 21 12, 22 5.5, 23 4.5 , 31 11 ,
32 4 , 33 4 . In this setting, the profit F S X of coalition S N under plan X is
For this game, we have four possible ordering plans as follows: 1 • Ordering plan X 1,0,0
Under
this
plan,
we
order
d1 d2 d3
in
period
CR IP T
3 2 1 F S X ti ct X X t Kt . t 1 4 iS
1.
The
marginal
costs
are
AN US
c1 X 1 c2 X 1 c3 X 1 c1 . The profit of coalition S N under plan X 1 is 2 1 3 ti c1 K1 4 t 1 iS 2 1 3 ti 2 9. 4 t 1 iS
M
FS X1
ED
2 •Ordering plan X 1,1,0
2 Under this plan, we order d1 in period 1 and d 2 d3 in period 2. The marginal costs are c1 X c1 ,
PT
2 2 and c2 X c3 X c2 . The profit of coalition S N under plan X 2 is
CE
FS X 2
2 2 2 1 1 1 1i c1 2i c2 3i c2 K1 K 2 4 iS 4 iS 4 iS 2 2 2 1 1 1 1i 2 2i 1 3i 1 18. 4 iS 4 iS 4 iS
AC
3 •Ordering plan X 1,0,1
Under this plan, we order d1 d 2 in period 1 and d 3 in period 3. The marginal costs are
c1 X 3 c2 X 3 c1 , and c3 X 3 c3 . The profit of coalition S N under plan X 3 is FS X3
2 2 2 1 1 1 1i c1 2i c1 3i c3 K1 K 3 4 iS 4 iS 4 iS 2 2 2 1 1 1 1i 2 2i 2 3i 18. 4 iS 4 iS 4 iS
19
ACCEPTED MANUSCRIPT
• Ordering plan X 4 1,1,1 Under this plan, we order d1 in period 1, d 2 in period 2, and d 3 in period 3. The marginal costs are
c1 X 4 c1 , c2 X 4 c2 , and c3 X 4 c3 . The profit of coalition S N under plan X 4 is 2 2 2 1 1 1 1i c1 2i c2 3i c3 K1 K 2 K3 4 iS 4 iS 4 iS 2 2 2 1 1 1 1i 2 2i 1 3i 27. 4 iS 4 iS 4 iS
CR IP T
FS X 4
Thus, the maximum profit of coalition S N is
F S max F S X 1 , F S X 2 , F S X 3 , F S X 4 .
From this, we have
AN US
F 1 max F 1 X 1 , F 1 X 2 , F 1 X 3 , F 1 X 4 max 37.25,38.25,38.25,34.5 38.25，
F 1, 2 max F 1,2 X 1 , F 1,2 X 2 , F 1,2 X 3 , F 1,2 X 4 max 42.3125, 46.5625, 46.3125, 44.5625
M
46.5625，
F 1,3 max F 1,3 X 1 , F 1,3 X 2 , F 1,3 X 3 , F 1,3 X 4
ED
max 40.8125, 44.5625, 44.8125, 42.5625 44.8125,
PT
F 1, 2,3 max F 1,2,3 X 1 , F 1,2,3 X 2 , F 1,2,3 X 3 , F 1,2,3 X 4 max 45.875,52.875,52.875,52.625
AC
Then
CE
52.875.
F 1, 2 F 1,3 46.5625 44.8125 91.375, F 1 F 1, 2,3 38.25 52.875 91.125.
It follows that
F 1 F 1,2,3 F 1,2 F 1,3 .
This means that the game is not convex.
20
ACCEPTED MANUSCRIPT
5. Numerical Experiments In this section we perform numerical experiments to show that the cooperation benefits can be significant. Consider an ELS game with price-dependent demand. There are three retailers N 1,2,3 over 6 periods with fixed ordering costs K1 K3 K5 12 , K2 K4 K6 6 , unit purchase costs ct c 3 and unit holding costs ht h 1 . The retailers’ inverse demand functions are given in Table 1. And the
CR IP T
characteristic functions are summarized in Table 2.
Table 1. The inverse demand functions 1
2
3
4
5
6
Retailer 1
10 d11
9 d 21
11 d31
10 d 41
9 d51
10 d61
Retailer 2
9 d12
8 d 22
9 d32
8 d 42
10 d52
8 d62
Retailer 3
8 d13
11 d 23
8 d33
8 d 43
9 d53
9 d63
AN US
Period
Table 2. The characteristic functions {1}
{2}
F S
34.25
18
{3}
M
S
24.5
{1, 2}
{1, 3}
{2, 3}
{1, 2, 3}
77.25
82
60.75
125
ED
When the retailers cooperate to jointly place their orders and keep inventory in the central warehouse, they can get the total profit F 1, 2,3 125 , which is larger than the sum of individual profits
PT
F 1 F 2 F 3 77 . Note that the total fixed ordering cost is 28, the total purchase cost is 130.5,
CE
the total unit holding cost is 48, and the total income is 283.25 when they place orders individually, while the total fixed ordering cost is 156, the total purchase cost is 123, the total unit holding cost is 30, and the total
AC
income is 327 when they cooperate to place orders. So the retailers’ cooperation can not only save costs, but also increase their total income. Next, we study the impacts of the unit purchase cost ct c on the profit increases achieved by
cooperation.
The
F1,2,3( F {1} F {2} F {3}) F {1} F {2} F {3} F1,2,3 ( F {1,3} F {2}) F {1,3} F {3}
results
are
depicted
in
Figure
1,
where
with respect to c ; l2 depicts the changes
and l4 depicts the changes
F1,2,3 ( F {1} F {2,3}) F {1} F {2,3}
21
.
curve
l1
F1,2,3 ( F {1,2} F {3}) F {1,2} F {3}
depicts
the
changes
; l3 depicts the changes
CR IP T
ACCEPTED MANUSCRIPT
Figure 1. Profit increase ratio with respect to c
From Figure 1, we find that the total profit increases as more players join the coalition. Moreover, the
AN US
larger the unit purchase cost c is, the more the cooperation profits are.
We then study the impact of the holding cost ht h on the profit increases. The results is depicted in Figure 2 and l1 , l2 , l3 and l4 have the similar meanings as those in Figure 1. Figure 2 shows that as the unit holding cost h increases, cooperation can also bring more profit. The more the players join the coalition,
AC
CE
PT
ED
M
the larger the profit is.
Figure 2. Profit increase ratio with respect to h
6. Conclusions In this paper we introduce a new cooperative game, i.e., the ELS game with price-dependent demand, in which the demand depends on the sale price set by the retailers in each period. We first show that the ZIO 22
ACCEPTED MANUSCRIPT
property holds and give a polynomial-time algorithm to make optimal ordering and pricing decisions, and compute the characteristic profit function based on ZIO. We then show that the game is balanced and has a non-empty core in general, i.e., there exists an allocation policy that ensures that the retailers can form a stable alliance. Finally, we show that the game is convex for two special cases where the retailers face the same demand function and where the time horizon consists of two periods. We provide a counter-example to demonstrate that the game is not convex in general.
CR IP T
In reality, there are many situations in which retailers set their prices by themselves and cooperate to replenish their inventories, e.g., chain stores, retailers in a special region etc. Also, in real practice of e-commerce, many retailers sell the same products on the same e-commerce platform, and cooperate to manage inventory and delivery. Our results show that cooperation on inventory replenishment can bring
AN US
significant profits to retailers and there exists an allocation policy that can ensure that cooperation among retailers is stable. Also, retailers can use our algorithm to derive their optimal pricing and ordering policies. Future research could address some of our limiting assumptions. In this study we assume that shortage is not allowed. However, in some practical situations, backlog is allowed. Extending our model to the case
M
where backlog is allowed is both interesting and challenging.
Another interesting point of departure is to drop the assumption that the adjustment of price incurs no
ED
cost and the demand only depends on the price in the current period. In reality, the demand in a period may depend on the price in the current period and those in the past periods. Also the adjustment of price may incur
PT
costs. Extending our model to consider these two issues are interesting topics. It is also relevant to study dependent retailers where the demands faced by retailers are dependent. In
CE
reality, there are situations where the demand faced by a retailer depends on its own price as well as the other retailers’ prices. The ELS game with inter-dependent demand functions is a topic that needs further research.
AC
Finally, extending our model to the case where the supplier is capacitated is an interesting and challenging topic.
Acknowledgements We thank the editor and reviewers for their helpful comments, which help us improve the quality of the paper. This work was supported by the National Natural Science Foundation of China (grant numbers: 23
ACCEPTED MANUSCRIPT
71390334 and 71661167009), the Program for New Century Excellent Talents in University (NCET-13-0660) and National Social Science Foundation of China (grant numbers: 15ZDA022).
References
CR IP T
Adeinat, H.，Ventura, J.A. (2015) Determining the retailer's replenishment policy considering multiple capacitated suppliers and price-sensitive demand. European Journal of Operational Research, 247(1), 83-92. Anily, S., Haviv, M. (2007) The cost allocation problem for the first order interaction joint replenishment model. Operations Research, 55(2), 292-302. Anily, S., Haviv, M. (2010) Cooperation in service systems. Operations Research, 58(3), 660-673.
Bondareva, O.N. (1963) Some applications of linear programming methods to the theory of cooperative games. Problemy Kibernetiki, 10, 119-139.
AN US
Chen, X. (2009) Inventory centralization games with price-dependent demand and quantity discount. Operations Research, 57(6), 1394-1406. Chen, X., Chen, Z. (2013) Cost allocation of capacity investment games. Naval Research Logistics, 60(6), 512-523.
M
Chen, X., Simchi-Levi D. (2012) Pricing and inventory management. In Özer, Ö., Phillips, R. (Eds.), Oxford Handbook of Pricing Management, (pp. 784-822), Oxford University Press. Chen, X., Zhang, J. (2009) A stochastic programming duality approach to inventory centralization games. Operations Research, 57(4), 840-851.
ED
Chen, X., Zhang, J. (2016). Duality approaches to economic lot‐sizing games. Production and Operations Management, 25(7), 1203-1215.
PT
Deng, S., Yano, C.A. (2006) Joint production and pricing decisions with setup costs and capacity constraints. Management Science, 52(5), 741-756.
CE
Drechsel, J. (2010) Cooperative lot sizing games in supply chains. Lecture Notes in Economics and Mathematical Systems, Volume 644, Springer Berlin Heidelberg.
AC
Dror, M., Hartman, B.C. (2007) Shipment consolidation: who pays for it and how much? Management Science, 53(1), 78-87. Elmaghraby, W., Keskinocak, P. (2003) Dynamic pricing in the presence of inventory considerations: Research overview, current practices, and future directions. Management Science, 49(10), 1287-1309. Esmaeili, M., Aryanezhad, M.B., Zeephongsekul, P. (2009) A game theory approach in seller–buyer supply chain. European Journal of Operational Research, 195(2), 442-448. Esmaeili, M., Zeephongsekul, P. (2010) Seller-buyer models of supply chain management with an asymmetric information structure. International Journal of Production Economics, 123(1), 146-154. Florian, M., Klein, M. (1971) Deterministic production planning with concave costs and capacity 24
ACCEPTED MANUSCRIPT
constraints. Management Science, 18(1), 12-20. Florian, M., Lenstra, J.K., Rinnooy Kan, A.H.G. (1980) Deterministic production planning: Algorithms and complexity. Management Science, 26(7), 669-679. Guardiola, L.A., Meca, A., Timmer, J. (2007) Cooperation and profit allocation in distribution chains. Decision Support Systems, 44(1), 17-27. Gopaladesikan, M., Uhan, N.A. (2011) Cost sharing for the economic lot-sizing problem with remanufacturing options, http://www.optimization-online.org/DB_FILE/2010/09/2733.pdf, 2011.
CR IP T
Guardiola, L.A., Meca, A., Puerto, J. (2006) Coordination in periodic review inventory situations. Working Paper, Centro de Investigación Operativa, Universidad Miguel Hernández de Elche. Hall, N.G., Liu, Z. (2010) Capacity allocation and scheduling in supply chains. Operations Research, 58(6), 1711-1725.
AN US
Hartman, B.C., Dror, M., Shaked, M. (2000) Cores of inventory centralization games. Games and Economic Behavior, 31(1), 26-49. Karsten, F.J.P., Slikker, M., van Houtum, G.J. (2009) Spare parts inventory pooling games. Working Paper, School of Industrial Engineering, Eindhoven University of Technology. Lu, L., Qi, X., Liu, Z. (2014) On the cooperation of recycling operations. European Journal of Operational Research, 233(2), 349-358.
M
Müller, A., Scarsini, M., Shaked, M. (2002) The newsvendor game has a nonempty core. Games and Economic Behavior, 38(1), 118-126.
ED
Montrucchio, L., Norde, H., Özen, U., Scarsini, M., Slikker, M. (2012) Cooperative newsvendor games: A review. In Choi, T.M. (Ed.), Handbook of Newsvendor Problems (pp. 137-162), Springer, New York.
PT
Nagarajan, M., Sošić, G. (2008) Game-theoretic analysis of cooperation among supply chain agents: Review and extensions. European Journal of Operational Research, 187(3), 719-745.
CE
Özen, U., Fransoo, J., Norde, H., Slikker, M. (2008) Cooperation between multiple newsvendors with warehouses. Manufacturing & Service Operations Management, 10(2), 311-324.
AC
Özener, O.Ö., Ergun, Ö., Savelsbergh, M. (2013) Allocating cost of service to customers in inventory routing. Operations Research, 61(1), 112-125.
Shapley, L.S. (1971) Cores of convex games. International Journal of Game Theory, 1(1), 11-26. Slikker, M., Fransoo, J., Wouters, M. (2005) Cooperation between transshipments. European Journal of Operational Research, 167(2), 370-380.
multiple
newsvendors
with
Talluri, K.T., Van Ryzin, G.J. (2004) The Theory and Practice of Revenue Management. Springer. Thomas, J. (1970) Price-production decisions with deterministic demand. Management Science, 16(11), 747-750. Toriello, A., Uhan, N.A. (2013) Dynamic linear production games under uncertainty. Working Paper, University of Chicago, Booth School of Business Operations/Management Science Workshop.
25
ACCEPTED MANUSCRIPT
Uhan, N.A. (2015) Stochastic linear programming games with concave preferences. European Journal of Operational Research, 243(2), 637-646. van den Heuvel, W., Borm, P., Hamers, H. (2007) Economic lot-sizing games. European Journal of Operational Research, 176(2), 1117-1130. Wagner, H.M., Whitin, T.M. (1958) Dynamic problems in the theory of the firm. Naval Research Logistics Quarterly, 5(1), 53-74.
CR IP T
Yano, C.A., Gilbert, S.M. (2004) Coordinated pricing and production/procurement decisions: A review. In Chakravarty, A.K., Eliashberg, J. (Eds.), Managing Business Interfaces (pp. 65-103), Springer USA. Yu, Y., Benjaafar, S., Gerchak, Y. (2015) Capacity sharing and cost allocation among independent firms with congestion. Production and Operations Management, 24(8), 1285-1310.
AC
CE
PT
ED
M
AN US
Zhang, J. (2009) Cost allocation for joint replenishment models. Operations Research, 57(1), 146-156.
26