A Stackelberg Game Model between Manufacturer and Wholesaler in a
Food Supply Chain
Javiera A. Bustos
1
, Sebastián H. Olavarría
1
, Víctor M. Albornoz
1
,
Sara V. Rodríguez
2
and Manuel Jiménez-Lizárraga
3
1
Departamento de Industrias, Campus Vitacura, UTFSM, Santiago, Chile
2
Facultad de Ing. Mecánica y Eléctrica, UANL, Monterrey, Mexico
3
Facultad de Ciencias Físicas y Matemáticas, UANL, Monterrey, Mexico
Keywords:
Stackelberg Game, Food Supply Chain, Bilevel Programming, Pork Supply Chain.
Abstract:
This paper describes an application of the Stackerlberg game model for the food supply chain. Specifically,
the focus of this work is on the pork industry and considers a production game. Such game includes two
players, manufacturer and wholesaler, who both aim to maximize profit. The role of leader is played by the
manufacturer, and follower by the wholesaler. Decisions involved in the game are the level of production,
quantity to be sold by the leader, and level of purchased products by the follower at each time period. This
paper presents a case study, and results show that coordination between these players is seen in cost savings
and improved service level.
1 INTRODUCTION
In developing countries, food demand continues
growing, given rising incomes and population growth.
In the context of countries with a strong and continu-
ous economic development, it is forecasted that pro-
tein consumption will rise, along with meat consump-
tion, resulting in an active industry. Meat production
will increase by 17% in developing countries and 2%
in developed countries from 2014 to 2024. Pork is
the most produced and consumed red meat worldwide
(FAO, 2016). In this context, several complex prob-
lems are faced by chain managers, who need to in-
tegrate stakeholder operations in order to coordinate
product flow along the chain. One of the most chal-
lenging problems is related to planning and schedul-
ing of operations for processing the carcasses (body
of the animal gutted and bloodless) into pork and by-
products, later to be sold to wholesalers to satisfy re-
quired demand. In this framework, the coordination
and integration between two agents is critical to im-
prove efficiency and increase supply chain productiv-
ity.
Operations Research (OR) is one of the most im-
portant disciplines that deal with advanced analytical
methods for decision making. OR is applied to a wide
range of problems arising in different areas, and their
fields of application involve the operations manage-
ment of the agriculture and food industry. There are
several works related to these topics, see (Ahumada
and Villalobos, 2009) for a review of agricultural sup-
ply chains; see (Bjørndal et al., 2012) for a review of
operations research applications in agriculture, fish-
eries, forestry and mining; see (Higgins et al., 2010)
for an application of agricultural value chains using
network analysis, agent-based modeling and dynam-
ical systems modeling; (Plà et al., 2014), draw out
insights for new opportunities regarding OR for the
agricultural industry. Specifically, (Rodríguez et al.,
2014) presents a key description of opportunities fo-
cused on the pork supply chain.
Furthermore, game theory has been deeply used
to analyze the interactions between different agents
in the supply chain (Hennet and Arda, 2008). Game
theory is a suitable tool to support decision making
where there is more than one participant (or player)
(Marulanda and Delgado, 2012). The Stackeberg
model was originally introduced in the context of
static competition games in 1934 by the economist
H. von Stackelberg (Von Stackelberg, 1952), with an
important impact on economic sciences. Such a prob-
lem can be also seen as a bilevel optimization problem
(Dempe, 2002). In this model, the leader announces
his strategy first. Next, the follower observes the
A. Bustos J., H. Olavarria S., M. Albornoz V., V. Rodrà guez S. and JimÃl’nez-Lizà ˛arraga M.
A Stackelberg Game Model between Manufacturer and Wholesaler in a Food Supply Chain.
DOI: 10.5220/0006201504090415
In Proceedings of the 6th International Conference on Operations Research and Enterprise Systems (ICORES 2017), pages 409-415
ISBN: 978-989-758-218-9
Copyright
c
 2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
409

leader’s actions and reacts to them, so as to maximize
profits. Interactions within the food supply chain are
captured through a game between a leader and a fol-
lower. In this model, here the leader presents advan-
tage, and consequently is who decides first about his
operational decisions: location, technology, amount
to processing, raw materials, and prices (Yue and You,
2014).
In many decision making processes there is a hi-
erarchical structure among agents and decisions are
taken at different levels of the hierarchy. Interactions
among participants in the supply chain are captured
through a game between a leader with a follower who
follows the structure of a Stackelberg game. When
we see this model as a two level decision problem,
it is called bilevel optimization program. In this
case, we have a leader (associated with the top level)
and a follower (associated with the lower level). If
both the leader’s and follower’s constraints are linear,
it is a bilevel linear programming problem (BLPP)
(Migdalas et al., 2013). Here, decisions of a player
on one level can affect decision-making behavior on
other levels, even though the leader does not com-
pletely control the actions of the followers.
A vast majority of research on bilevel program-
ming has centered on the linear version of this prob-
lem, alternatively known as the linear Stackelberg
game. The BLPP can be written as follows (Bard,
2013):
For x ∈ X ⊂ R
n
, y ∈ Y ⊂ R
m
, F : X × Y → R
1
, and
f : X × Y → R
1
.
min
x∈X
F(x,y) = c
1
x + d
1
y (1)
s.t A
1
x + B
1
y 5 b
1
(2)
min
y∈Y
f(x,y) = c
2
x + d
2
y (3)
s.t A
2
x + B
2
y 5 b
2
(4)
where c
1
, c
2
∈ R
n
, d
1
, d
2
∈ R
m
, b
1
∈ R
p
, b
2
∈ R
q
,
A
1
∈ R
p×n
, B
1
∈ R
p×m
, A
2
∈ R
q×n
, B
2
∈ R
q×m
. Sets
X and Y place additional constraints on the variables,
such as upper and lower bounds. In this model, once
the Ieader selects an x, the first term in the follower’s
objective function becomes a constant; and the same
is valid for the follower’s constraints.
Bard (2013) also presents a necessary condition
that (x
*
, y
*
) solves the linear BLPP (1)-(4) if there ex-
ist (row) vectors u
*
and v
*
such that (x
*
, y
*
, u
*
, v
*
)
solves:
minc
1
x + d
1
y (5)
s.t A
1
x + B
1
y 5 b
1
(6)
uB
2
− v = −d
2
(7)
u(b
2
− A
2
x − B
2
) + vy = 0 (8)
A
2
x + B
2
y 5 b
2
(9)
x = 0, y = 0, u = 0, v = 0 (10)
Thus, a new nonlinear constraint (8) is generated,
to represent the optimization model for the follower.
However, this formulation has played a key role in
the development of algorithms. One advantage that
it offers is that it allows for a more robust model to
be solved without introducing any new computational
difficulties. There are several algorithms proposed
for solving the linear BLPP since the field caught the
attention of researchers in the mid-1970s. Many of
these are of academic interest only because they are
either impractical to implement or grossly inefficient.
The most popular method for solving the linear BLPP
is known as the "Kuhn-Tucker" approach and concen-
trates on (5)-(10). The fundamental idea is to use a
branch and bound strategy to deal with the comple-
mentary constraint (8). Omitting or relaxing this con-
straint leaves a standard linear program which is easy
to solve. Various methods proposed employ different
techniques for assuring that the complementary con-
straint is ultimately satisfied (Bard, 2013). (Fortuny-
Amat and McCarl, 1981), proposed a reformulation
of the non-linear constraints; and (Bard and Moore,
1990) developed an algorithm that ensured global op-
timum with high efficiency.
In what follows, we propose a Stackelberg game
between two food supply chain players involved in
processing and selling activities. This is carried out
assuming that there is a leader-follower relationship
among the players. The primary challenge in this
model is to support in the coordination and integra-
tion of activities and information among two supply
chain agents. In Section 2, we detail the proposed
Stackelberg game and model to solve this problem.
After this, Section 3 presents a case study and pro-
vides the obtained results. Finally, in Section 4 main
conclusions and future research are presented.
2 MATERIALS AND METHODS
To have a good relationship between agents, coor-
dinate activities and share information , the chain
should be aligned, improving efficiency and produc-
tivity. This section presents a detailed description
of the Stackelberg game between two supply chain
agents under the leader and follower scheme; and an
optimization problem that models the interaction be-
tween the players.
ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems
410

2.1 Games Description
In the meat supply chain, a manufacturer is in charge
of processing the raw material, from carcasses to meat
products. This player decides based on demand and
the yielding rate of each cutting pattern, the level of
production and inventory for each product, and hence
the number of carcasses required. The processing
plant aims to maximize profits through the sale of
meat products; and the wholesaler is in charge of dis-
tribution and marketing of the products. Meat prod-
ucts are purchased from the manufacturer, and then
sold to end customers, with the aim of maximizing
level of service.
Interactions between both players in the supply
chain are captured through a game with a leader and
a follower who follows the structure of Stackelberg
game. The problem arises when the optimal quan-
tity produced and sold by the manufacturer is not
enough to supply the needs of wholesaler. Thus, the
wholesaler imposes a penalty cost on the manufac-
turer for the unsatisfied demand. This penalty cost
can be reduced through coordination and integration
of activities, and information exchange between the
two supply chain agents. By sharing information
about consumer preferences and demand, the manu-
facturer avoids expired products and cooperates with
the wholesaler to maximize their level of service, im-
proving performance of the whole supply chain.
The proposed game considers the manufacturer as
the leader and the wholesaler as the follower. Given
the characteristics of this Stackelberg game, static and
not cooperative, this problem can be modeled by a
bilevel linear programming problem.
2.2 Bilevel Linear Programming Model
This model supports and assists in the coordination
and integration of two food supply chain agents, un-
der the leader and follower structure given by the
Stackelberg game. The main assumptions and consid-
erations for the formulation of this bilevel program-
ming model are based on(Albornoz et al., 2015).
Sets, indexes and variables used in the model are
described below:
Sets and Indexes:
T : Number of periods of the planning
horizon.
J : Number of cutting patterns.
k ∈ K : Set of sections per carcass.
j ∈ J
k
: Set of cutting patterns per section k.
r ∈ R : Set to represent the different types
of carcasses.
i ∈ P : Set of Products.
Parameters:
H : Carcasses available to process
during the whole planning horizon.
α
r
: Proportion of carcasses of type r .
ψ
i jr
: Yield of product i using cutting
pattern j on carcasses of type r .
p
i
: Selling price per product i .
c
j
: Operational cost of pattern j .
c
e
j
: Operational cost of pattern j
in overtime.
h : Holding cost of product per period
for the leader.
f
i
: Cost for unsatisfied-demand of
product i for the leader.
W : Warehouse capacity (in kg.)
for the leader.
t
j
: Operation time for cutting pattern j.
T
w
: Available hours in regular time.
T
e
w
: Available hours in overtime.
δ : Auxiliary parameter for better
control the available carcasses .
P : Purchase cost of carcasses.
d
it
: Demand of product i at each period t.
G : Warehouse capacity (in kg.)
for the follower.
Decision Variables:
v
it
: Quantity of product i to be sold in t.
x
it
: Total quantity of product i to have
in period t.
s
i
: Total quantity of excess product i at the
end of planning horizon.
H
t
: Number of carcasses to be processed
at each period t.
z
jt
: Number of times to perform the
cutting pattern j in period t in normal
work hours.
z
e
jt
: Number of times to perform the
cutting pattern j in period t, in
overtime.
I
it
: Quantity of product i to hold for the
leader in t.
u
it
: Unsatisfied-demand of product i in t.
I
r
it
: Quantity of product i to hold for the
follower in t.
max
∑
i∈P
T
∑
t=1
p
i
v
it
−
T
∑
t=1
PH
t
−
T
∑
t=1
∑
j∈J
∑
r∈R
(c
j
z
jrt
+ c
e
j
z
e
jrt
)
A Stackelberg Game Model between Manufacturer and Wholesaler in a Food Supply Chain
411

−
∑
i∈P
T
∑
t=1
hI
it
−
∑
i∈P
T
∑
t=1
f
i
u
it
(11)
s.t
α
r
H
t
−
∑
j∈J
k
(z
jrt
+ z
e
jrt
) = 0 ; t ∈ T, k ∈ K, r ∈ R (12)
x
it
−
∑
r∈R
∑
j∈J
k
ψ
i jr
(z
jrt
+ z
e
jrt
) = 0 ; i ∈ P, t ∈ T, k ∈ K
(13)
∑
r∈R
∑
j∈J
k
t
j
z
jrt
≤ T
w
; t ∈ T (14)
∑
r∈R
∑
j∈J
k
t
j
z
e
jrt
≤ T
e
w
; t ∈ T (15)
v
it
− x
it
− I
it
+ I
i,t+1
= 0 ; i ∈ P, t = 1, ..., T − 1 (16)
v
iT
− x
iT
+ s
i
− I
iT
= 0 ; i ∈ P (17)
T
∑
t=1
H
t
≤ H (18)
−
T
∑
t=1
H
t
≤ −δH (19)
∑
i∈P
I
it
≤ W ; t ∈ T (20)
v
it
≥ 0 , x
it
≥ 0 , s
i
≥ 0 , I
it
≥ 0 , H
t
integer,
z
jrt
integer , z
e
jrt
integer (21)
min
∑
i∈P
T
∑
t=1
u
it
(22)
s.t
v
it
+u
it
+I
r
it
− I
r
i,t+1
= d
it
; i ∈ P, t = 1, ..., T − 1 (23)
v
iT
+ u
iT
+ I
r
iT
= d
iT
; i ∈ P (24)
∑
i∈P
I
r
it
≤ G ; t ∈ T (25)
I
r
i,T
= 0 ; i ∈ P (26)
u
it
≥ 0 ; I
r
it
≥ 0 (27)
Where v
it
, s
i
, H
t
, z
jrt
, z
e
jrt
, I
it
are decision variables
of the leader and u
it
, I
r
it
are decision variables of the
follower. The manufacturer’s objective (16) is to max-
imize profits. Profits are understood as the difference
between total revenues from selling the products and
the following costs: inventory, production, purchases
of carcasses and unsatisfied-demand penalties. Con-
versely, the follower (22) is simply trying to maxi-
mize his service level.
A feasible solution of the model satisfies a differ-
ent set of leader’s constraints. Constraint (12) ensures
a balance between cutting patterns and the number of
carcasses to be processed at each time period. Equal-
ity is forced because it is not possible to leave unpro-
cessed raw material. Constraint (13) calculates the to-
tal kilograms of each product retrieved by all the cut-
ting patterns applied at each time period. Constraint
(14) ensures that the labor time does not exceed the vi-
able working hours of regular time. Constraints (15)
ensure that the labor time does not exceed the viable
working hours during overtime. Constraints (16) and
(17) determine the quantity of product to be processed
and held considering the excess product. This is the
amount that the manufacturer sells when the whole-
saler does not purchase all products at the end of the
planning horizon (without revenues because it is a
sunk cost). Constraints (18) and (19) impose a lower
and upper limit according to the animal availability
from suppliers and a given percentage δ to allow an
extra flexibility in the total number of carcasses to be
processed. Constraint (20) ensures that the holding
capacity for products is never exceeded. Constraint
(21) defines the domain of decision variables.
On the other hand, the set of follower’s constraints
are the following. Constraints (23) and (24) ensure
that the requested level of each product is addressed,
allowing the existence of unsatisfied-demand if the
manufacturer does not provide enough products to
satisfy the demand. Constraint (25) ensures that the
capacity for holding products is never exceeded. Con-
straint (26) satisfies the condition to not holding prod-
ucts at the end of the planning horizon. Constraint
(27) defines the domain of decision variables.
In this paper, we solve a linear relaxation of model
(11)-(27) using the equivalent reformulation describes
in (5)-(10), obtaining a nonlinear optimization prob-
lem. To solve this last model, (Fortuny-Amat and Mc-
Carl, 1981) propose an equivalent mixed-integer lin-
ear program. This formulation adds (IT + T + I) bi-
nary variables and 2(IT + T + I) new constraints that
replace nonlinear constraints of type (8). The result-
ing model can be solved using a mixed-integer solver
only for small size instances. To solve medium and
larger instances can be solved by the algorithm pro-
posed by (Bard and Moore, 1990) .
3 RESULTS
In this section, a case study is presented to illustrate
the suitability and advantages of the proposed bilevel
optimization model. Basic parameters (such as prices,
costs and warehouse capacities) were created using
market information gathered from different pork pro-
ducers. Different countries use different cutting pat-
terns for producing meat products according to their
ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems
412

history and gastronomic culture. The case study con-
siders cutting patterns used by a Mexican pork firm
that must plan its production over a time horizon.
First, pork carcasses were split up into 5 sections, and
for each section a set of cutting patterns was assigned.
In total, the company operates with 17 cutting pat-
terns, and manages 40 pork products.
The case study represents a batch of fattened pigs
arriving every day to the manufacturer to be slaugh-
tered and later processed as carcasses. It is assumed
the available amount of carcasses during the whole
horizon is fixed and known. The total amount of car-
casses available over a time horizon was set at 5000.
The yield matrix for a carcass per product, section and
cutting pattern was obtained from production lines
(in kg). Considering the large amount of products,
we selected 15 products with the highest prices, rep-
resenting 67% of total demand. Labour capacity is
considered as 8 hours per day in normal time, and 3
hours per day of overtime. To perform each cutting
pattern, a specific amount of labour time is required.
The solved instance considered 10 planning periods.
In addition, the shelf life was 10 days, not enough to
overcome during the planning horizon.
Results in a bilevel linear programming model,
for which reformulations are presented, becomes a
mixed-integer linear problem with 700 and 300 deci-
sion variables for the leader and follower, 2825 con-
straints and 475 dual variables. The first instance rep-
resents the coordination and integration of the two
supply chain agents. Moreover, it is also assumed
that demand is given, and the available amount of car-
casses is fixed and known. Table 1 summarizes the
achieved results:
Table 1: Results.
Profit [US$] 3.557.190
Service level 73,7%
Unsatisfied demand [u] 44.522
Carcasses acquired [u] 4.424
Carcasses acquired rate 88%
Excess Product [u] 38.523
Results from the case study showed a net profit
of $3.557.190 dollars for the leader a service level
of 73,7% for the follower. Under this solution 88%
of whole carcasses available in the planning horizon
(4.424 carcasses) were used. This demonstrated that
the demand was not completely satisfied. It also notes
that the manufacturer did not reach its maximum ca-
pacity, and the occupancy rate carcasses during the
planning horizon was less than 100%, meaning that
there was still raw material to be processed.. Table 2
shows the acquisition of carcasses during the planning
horizon (10 days):
Table 2: Value of H in each period of time.
Horizon H
t
1 550
2 516
3 548
4 415
5 555
6 447
7 427
8 506
9 336
10 124
The quantity of carcasses acquired at each period
did not show major changes during the planning hori-
zon, except for the last day where the carcasses ac-
quired were to process and sell just for that period,
without holding products. The amount of unsatis-
fied demand had a total of 44.522 products during the
planning horizon, and the excess product at the end of
the planning horizon is 38.523. The following table
shows the detail for each of the products.
Table 3: Unsatisfied demand and excess products.
i Unsatisfied demand [u] Excess products [u]
1 0 0
2 0 0
3 0 4.672
4 4.321 0
5 0 0
6 24.105 1.226
7 6.910 0
8 5.012 0
9 1.297 0
10 0 23.946
11 436 0
12 0 728
13 0 0
14 0 7.950
15 2.380 0
The pattern cuts used resulted in different prod-
ucts. Products in inventory at the end of the planning
horizon are excess products, and the manufacturer
sells them without revenues because it’s a salvage
value. This implies that the leader produces to achieve
a high follower service level as long as revenues are
higher than the production cost of each of product.
When the cost exceeds income from sales, the leader
will prefer not to produce, and pay a penalty. In this
game, the agents related to two levels of the supply
chain, and it was observed that decisions have an in-
terdependent relationship. The amount of unsatisfied
demand selected by the wholesaler has an effect on
A Stackelberg Game Model between Manufacturer and Wholesaler in a Food Supply Chain
413

the profit function of the leader Deciding how much
to produce has an effect on the quantity of unsatisfied
demand by the follower.
In order to see the advantages of the bilevel pro-
gramming model, these results are compared with the
decision making of the leader in Table 4.
Table 4: Comparison of two models.
Models Bilevel Leader
Profit [US$] 3.557.190 1.805.780
Service level 73,7% 72,5%
Carcasses acquisition rate 88% 86%
Excess Product [u] 38.523 38.886
A new instance arises when the leader and fol-
lower are not coordinated nor integrated. In this case,
the leader’s model optimizes his operations according
to (11) - (21), adding the demand requirement from
the wholesaler. Table 4 shows that the leader does
not use whole carcasses available. In addition, ser-
vice level is 72,5% and the production capacity of
the plant are not at the maximum. Comparing both
models, when the two players are coordinating and
integrating information and activities, the profits and
service level increases. The carcasses acquisition rate
rises because the leader is producing more products
so that the unsatisfied demand decreases. In relation
to excess product; in the bilevel programming model,
the amount at the end of the planning horizon is less
than the final quantity in the one-level decision model.
In this case, we show the importance of coordi-
nating and integrating different agents in the supply
chain. The leader maximizes profits, and even if the
demand is given, profits and service level are lower
than when leader and follower are coordinated and in-
tegrated. The difference lies mainly in inventory cost;
when two players work together, products are held in
two warehouses. In this way, costs are shared, unlike
when the leader works alone. Coordination and inte-
gration is not only for better profits, but also increases
service level and productivity.
All instances described in the computational re-
sults were implemented and solved using CPLEX
12.6.2.0 as a solver in a Macbook Pro Retine Display,
i5-5257U Broadwell and 8Gb RAM with the software
optimization package IBM ILOG CPLEX.
4 CONCLUSIONS
This work presents a novel Stackelberg production
game between two players in the Pork Industry. The
players are manufacturer and wholesaler, representing
leader and follower, respectively. This game a coor-
dinates and integrate this link of the supply chain. To
represent the game, we propose a linear bilevel pro-
graming model, where the leader maximizes profits,
and the follower maximizes their service level.
Our model was applied to a case study in the pork
industry, concluding that coordinating and integrat-
ing both players in the supply chain is a better strat-
egy than previously proposed solutions. In effect, this
coordination obtains higher profits and better service
level. Furthermore, the reformulations used to solve
this bilevel model are useful in this context. They
ensure global optimum with computational time re-
quired to solve the mixed linear programming prob-
lem instances less than 1 minute, proving to be effi-
cient for dimensions resolved in this work.
Game theory in supply chain management proves
to be a powerful method that allows modeling games
between different players within the food industry.
Future research in this area focus on resolution tech-
niques from a bilevel linear mixed-integer program-
ming model as well as the development of a three-
level model: supplier, producer and distributor.
ACKNOWLEDGEMENTS
This research was partially supported by DGIIP
(Grant USM 28.15.20) and Departamento de Indus-
trias, Universidad Técnica Federico Santa María. Au-
thors also wish to acknowledge to Ibero-American
Program for Science and Technology for Develop-
ment (CYTED 516RT0513).
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