CC BY
2Contributions to Game Theory and Management, XI, 73-102
Competitive and Cooperative Behavior in Distribution
Networks*
Yulia Lonyagina, Natalia Nikolchenko, Nikolay Zenkevich
St. Petersburg State University 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia E-mail: yuliya-lonyagina@yandex.ru, st057455@student.spbu.ru, zenkevich@gsom.pu.ru
Abstract This paper considers the problem of cooperation in supply networks. The model is based on distribution network, which includes several manufactures, single distributor and multi retailers, operated and competed in consumer markets that are functioning according to the Cournot model with the linear demand. All participants in a chain are trying to maximize their profit. A multi-stage hierarchic game was carried out. At the first step, we construct the competitive solution for such supply network as the perfect Nash equilibrium in the multi-step hierarchical game in the closed form. At the second step, we construct the cooperative solution for the network, where winnings of all participants in the found perfect Nash equilibrium are considered as the status quo point. Cooperative decision we calculate in the form of the weighted Nash bargaining solution, which comes down to the solution of a separable nonlinear programming problem. Numerical example for the network shows that cooperative decision is more profitable than competitive decision for all participants.
Keywords: distribution network, competitive and cooperative decisions, multi-stage hierarchical game, perfect Nash equilibrium, weighted Nash bargaining solution
1. Introduction
In the last several years, the evolution of supply chain management recognized that a business process consists of several decentralized firms and operational decisions of these different entities influence each other's profit, and thus the profit of the whole supply chain. With this understanding came a great deal of interest in modeling and understanding the impact of strategic operational decisions of the various players in supply chains. To effectively model and analyze decision making in such multiperson situation where the outcome depends on the choice made by every party, game theory is a natural choice. Researchers in supply chain management now use tools from game theory and economics to understand, predict, and help managers to make strategic operational decisions in complex multiagent supply chain systems (Nagarajan and Sosic, 2006).
This paper considers the problem of cooperation in distribution network, which includes several manufactures, single distributor and multi retailers, operated and competed in consumer markets that are functioning according to the Cournot model with the linear demand. Two types of behavior were considered in the research. As the competitive behavior in this paper we recognize the perfect Nash equilibrium
* This work is supported by the Russian Foundation for basic Research, projects No.
16-01-00805A and No. 17-07-00371
solution, and as the cooperative behavior - the weighted Nash bargaining solution. At the first step, we construct the competitive solution for such supply network as the perfect Nash equilibrium in the multi-step hierarchical game in closed form. At the second step, we construct the cooperative solution for the network, where winnings of all participants in the found perfect Nash equilibrium are considered as a point of the status quo. As cooperative decision we calculate the weighted Nash bargaining solution, which comes down to the solution of a separable nonlinear programming problem with concave payoff function, which has a unique solution.
This study can be divided into two main logical parts. The first one provides theoretical results; the second one presents the computational results of the case of Russian distribution network. The research organized as follows: theoretical background in cooperative games in supply chain management. The next two parts are devoted to description and formulation of the problem and finding equilibrium and cooperative solutions. Further, the case of Russian distribution network is considered. The next part provides comparative analysis of the results. Conclusion and limitations of the study and directions for future research are presented in the final part.
2. Theoretical background
Supply chains (SC) have been characterized as organizational networks that are linked through upstream and downstream processes and activities that produce value in the form of products and services delivered to the hands of the ultimate customer (Christopher, 1998). Chopra & Meindl defined the term of supply chain management as follow: supply chain management involves the management of flows between and among stages in a supply chain to maximize total profitability (Chopra and Meindl, 2001). Handfield and Nichols (1999) defined supply chain management as the integration of activities through improved supply chain relationships, to achieve a sustainable competitive advantage. As can be seen from the above definitions, all of them have more or less in common that supply chains are based on cooperation in order to generate a benefit. Some authors claim that, in the future, competition will take place between supply chains rather than between individual companies.
Supply chains have nowadays more and more complex structures, and may involve partners from different domains, size, countries, therefore of different cultures. In that context, the performance of the partnership can be assessed through technical criteria (Ounnar et al., 2007), but is also concerned with behavioral issues (Mollering, 2003). According to Sepulveda Rojas and Frein (2008), cooperation is the following level of the relationship: companies are more tightly tied together, sharing more information than they would even in an extended armlength relationship. In case of cooperation, there are fewer suppliers and longer-term supplier-customer relationships. Cooperation is therefore an upper level of relationship, determined by the deg ree of information sharing.
Many researchers have taken multiple perspectives and have developed many-theories to understand the activities involved in inter-organizational cooperation. Since the emergence of international cooperation and the development of vertical disintegration, managers have paid more attention to inter-firm spanning activities than to the optimization of interior processes (Buhman et al., 2005; Chen and Paulraj, 2004). The common objective of academics and practitioners is to deter-
mine how a firm can achieve a sustainable competitive advantage. As a concept for coordinating information and material between companies, supply chain management has a significant potential in creating competitive advantage for the companies involved. The great potential of supply chain management for competitiveness has often been mentioned in the literature (Chopra & Meindl, 2001). The main advantages that can be derived from choosing the right supply chain are an improvement in efficiency, e.g. due to high turns of inventory, or an increase in market responsiveness, e.g. by shorter lead time (Fisher, 1997). Another important benefit is to fight cooperatively against a phenomenon commonly referred to as the "bullwhip" effect which was first observed by logistic executives at P&G concerning disposable diapers (Lee et al., 1997; Forrester, 1958). By cooperation across the participants of supply chain, the bullwhip effect can be mitigated. In that sense, supply chain management is currently a major issue within the academic discussion.
In order to generate advantages, contracts for vertical cooperation are established within supply chains. Cooperative interactions in a supply chain have been comprehensively researched in the past. Cachón and Larivier (2005) investigated several types of supply chain contracts to promote cooperation between a manufacturer and a retailer. Li et al. (2000), Huang and Li (2001), and Zhang et al. (2012) discussed cooperative advertising models in a manufacturer-retailer supply-chain and investigated the effect of cooperation on investment effort levels. Leng and Parlar (2009) analyzed how the cooperative effect would influence cost savings from a supply chain with a manufacturer, a distributor and a retailer. The above studies aim at the issues of cooperation in forward supply chains. Even though it has been widely discussed in the academic literature, there is still a lack of applied rational methodologies analyzing supply chain management.
There is a shortage of research in cooperative models in supply chains. Operational research models are mathematical instruments to solve decision problems. Most of them deal with one decision maker situations. However, in real world, it is very common that the result of decisions depends also on other decision makers' choices, i.e. in the real world many decision situations are interactive. Operations management focused on single-firm analysis in the past. Its goal was to provide managers with suitable tools to improve the performance of their firms. Nowadays, business decisions are dominated by the globalization of markets and should consider the increasing competition among firms. Further, more and more products reach the customer through supply chains that are composed of independent firms. Following these trends, research in supply chain has shifted its focus from singlefirm analysis to multi-firm analysis, in particular to improving the efficiency and performance of supply chains under decentralized control. The main characteristics of such chains are that the firms in the chain are independent actors who try to optimize their individual objectives, and that the decisions taken by a firm do also affect the performance of the other parties in the supply chain. These interactions among firms' decisions ask for alignment and coordination of actions and, therefore, game theory is very well suited to deal with these interactions.
There is an increasing number of documents that apply tools, techniques, and models from game theory to supply chain problems. The authors discuss both non-cooperative and cooperative game theory in static and dynamic settings. Additionally, Cachón (1998) reviewed competitive supply chain inventory management, and Cachón (2003) reviewed and extends the supply chain literature on the manage-
merit of incentive conflicts with contracts. Papers using cooperative game theory to study supply chain management are scarce, but the use of cooperative games in this context is becoming more popular. Nagarajan and Sosic (2008) reviewed and extended the problem of bargaining and negotiations in supply chain relationships. A very recent survey on applications of cooperative game theory to supply chain management, the so-called supply chain collaboration, is Meca and Timmer (2008). Thus, one challenging field within operations research is that of game theoretical models in operations research.
Game theoretic models of supply chains can be classified into non-cooperative (Cachon and Netessine, 2004) or cooperative (Slikker and Nouweland, 2001; Nagarajan and Sosic, 2008). The cooperative game studies intrachain relationships, which have three issues: what coalitions will form; how the outcome be divided; and whether the outcomes are stable and robust (Nagarajan and Sosic, 2008). Cooperative games may further be classified into coalitional form, alliance and negotiation game theoretic models. The coalitional form game assumes that there is a defined set of players, a combination of which form a coalition. The members in the coalition collectively generate a value that is independent from non-members or other coalitions. The feasible outcomes represent the total set of all possible outcomes that players may realize. Players may select their respective coalition from the set of feasible outcomes such that each player's respective payoff is maximized (Xue, 1998). However, there is a dynamic process of coalition formation. Once a player joins a coalition, he may join or form an alternative coalition with a higher payoff. This deviation process continues until a stable equilibrium is reached (Konishi and Ray, 2003). Alternatively, members may initially decide on and successfully form a coalition and subsequently negotiate allocation rules with chain members. These are alliance models. In one-echelon horizontal alliances, Gerchak and Gupta (1991) studied the cost allocation for centralized inventory between horizontal retailers, while Hartman and Dror (1996) studied centralized inventory between stores. Meca et al. (2004) studied a single inventory model with n retailers to develop a proportional rule to allocate joint-ordering costs. Plambeck and Taylor (2004) studied a two-echelon supply chain where a manufacturer negotiates and efficiently allocates its capacity among n buyers. Leng and Parlar (2009) analyzed the allocation of cost savings from sharing demand information in a three-echelon supply chain that includes a supplier, manufacture and retailer. Besides negotiating allocation rules, players in a successful coalition may negotiate the terms of trade, called negotiation models.
In this paper, we specifically investigate the problem of cooperation in distribution network. We constructed the perfect Nash equilibrium solution as a competitive behavior. Using the obtained solution as a point of the status quo, we constructed the weighted Nash bargaining solution as a cooperative solution in distribution network.
3. Description and formulation of the problem
Let us look at the tree-like graph G1 = (X1; F1) where X1 is a set of nodes and F1 is a function of alternatives (Petrosyan et al., 2014). The root node of this tree can be named as x*. Also let us look at the graph G2 = (X2, F2) such that:
1. There is unique node x* e X2 such that F2 (x*) = 0;
2. For all x € X2\x* : |F2 (x)| = 1, where |F2 (x)| means a cardinality of the set F2(x).
The example of such a graph with the root node x*= xn is depicted on the Fig.l.
Fig. 1. The graph with the root node x* = xn . Source: Authors' own
Consider the graph G = (X, F), where
x = Xi u X2;
F i Fi (x), x € Xi\x*; 1 F2 (x) , elsewhere.
G
graph is depicted below (Fig. 2).
In the set of nodes X let us define the set X of final nodes: X = {x € X | F (x) = 0}. Then in the set of nodes X\X we define the sets of X1;..., X; in the following way:
Xi = {x | $y € X : F (y) = x}; (1)
Xfc+i = (J (F (x) \X), if (J (F (x) \X) = 0, k = 1, 2,..., l - 1;
x, = X.
Definition 1. Subset of nodes Xj C X, i = 1,..., l will be named as the set of nodes of the level i.
We will denote the nodes x from the set X as xj, where the upper index is equal to the number of the level Xj this node is situated and the lower index to the order number of this node in the set Xj. Also by m» we will understand the number of the nodes of the level i, i.e. m» = |X»
Fig. 2. The example of an hourglasses supply chain.
Source: Authors' own
Definition 2. We will say that the decomposition X1;..., Xi of the set X, which was defined under the rule (1). is defining the supply chain with the hourglasses structure.
Definition 3. The sector of the node xj G X\Xi is the set of nodes F(xj).
The set Sj is the set of paired indexes of these nodes that are in the sector of the node xj: Sj = {(k, h) | xkh G F(xj)}.
Assume that every node xj, i = 1,... j = 1,..., mi of a supply chain consists of a finite set of elements {xjkfor which the set of numbers is defined {vijk}fc=i> vijk > 0, where nij is a number of elements. This set of elements is a group of competitive firms that are producing and consuming the homogeneous product as well as having the different production costs (the production power is meant to be unrestricted). For each firm xjk M us define the variable qijk > 0 that is characterizing the production quantity of this firm as well as the integrated quantity of the homogeneous product that was produced by all firms {xjk in the node xj let us call Qij = Y1qijfc-
Then for the sector of each node in a supply chain, the following condition is considered to be fulfilled:
nij nr h
Qij = ^2 qijk = Qrh = X^qrht. (2)
k=1 (r,h)eSj (r,h)eSj t=1
That means that there is no deficit or surplus of production in the supply chain. For each node xj G X let us work in the variable pij that is equivalent the prices according to that firm are selling the unit of the produced good. It is considered
that for the every of the final nodes xj € X; there is the following linear demand function prescribed
pij = aij - bjQij, (3)
where ay > 0, by > 0 are known parameters.
Definition 4. The set ({qjjk }j j k, {pjj} j) is defining the commodity flow d in the supply chain.
d
Pij > 0, Qij > 0, j = 1,..., m;.
Let the set D is the set of all feasible flows in a supply chain. For each firm let us define the profit function as the following:
{qijk (pij - vijk), if i = 1; qijk (aij - bijQij - Prh - vijk), if i = l; qjjk (pjj - Prh - Vjjk), elsewhere;
where prh : xj €
X
then the nodes of the second level, then the third level, fourth level and up to the final inclusively, i.e. we will receive the arranged system {xi, x^ , ..., x2, x2, ..., xlmi} This arranged set of all nodes (let us denote it with N) of supply chain we will consider as the set of players. The set Ujj = {ujj} the strategy of the player xj will be considered as the set of all the possible vectors ujj € D, where:
= (qjjl,..., qjjnij ,pjj) € D, xj € N i = 1,..., l - 1 j = 1,..., mjj , . u* j j — / \ j V /
L \qiji,...,qijnl^ € D, xj € N, j = 1,...,mjj.
We assume that each of the supply chains participants is acting independently from each other and exclusively in favor of his own interests. Such model and corresponding solution will be named decentralized.
Definition 6. The feasible flow d* will be called optimal if it is fulfilled:
njjk (d*) > njjk (djj) for all i, j, k,
djj ujj
xjj
In the terms of game theory, the optimal solution is equal to Nash equilibrium in a multi-step hierarchical game with the complete information
r = ^N, {Ujj}j j, {njjk}j j k^ on the graph G.
4. Behavior models in distribution network
In this section, theoretical statements of problems for competitive and cooperative solutions are formulated. As a model of competitive behavior we consider the perfect Nash equilibrium solution, as a model of cooperative behavior we consider the weighted Nash bargaining solution.
4.1. Nash equilibrium in a multilevel decentralized model
The search for an optimal solution will be carried out with consideration of the final nodes. Let us analyze the revenue function of the firm k from the node xj :
nijk = qijk (pij - Pit - vijk), Pit : (l,j) G S¡.
(5)
Let us substitute in the revenue formula (5) the formula for the variable pij, using the equation (3):
nijk = qijk (aij — bijQij — pit — vijk ).
(6)
Having done (5)-(6) for all k = !,..., nij and having applied the necessary maximum condition:
dnij
ijk
dqijk
= 0, k = 1,
nij,
(7)
we will come to the following system:
/211 1 2 1
( bj (aij — Pit — vijl ) \ bj (aij — Pit — vij2 )
\ 111 • • • 2 J \qijnh ) yb- (aij — Pit — vijnl3)
\ ( qij1 \
* qij =
/ \qijnij
The system (8) is solvable due to it has the non-singular matrix /211 1\ 12 1 ••• 1
V 1 1 1 ••• 2 J , v 1
V / [nij xnijJ
The unique solution is:
qij1 qij
\lijn- )
-1
n-__
nij + 1 nij + 1
nij +1
-1 -1 nij \ nij + 1 nij + 1 nij + 1 /
( bj (aij — Pit — vij1) \
bj (aij — Pit — vij2 ) \ b; (aij — Pit — vijnij )
after the multiplication:
( qij1 \ qij
\lijnij )
1
bij (nij + 1)
/
bij (nij + 1)
/
aij — [Pit + nijvij1 — J2 vijh
h=2
aij
\
Pit + nij vij2 — 2 vijh h = 1
V h = 2 /
^ j (nj + 1) laij — [Pit + nij vijnij — h¿ vijh
\
/
Competitive and Cooperative Behavior in Distribution Networks 81 For the node xj the following equation holds true as well:
Q ^ m,(aij - p,,) - sn=i Vijk „„,
Q" = £ j =-bjTcniT+T)-• <10)
Let us fulfill the same analogical operations (5)-(10) for all the final nodes
. a j
xj G Xi.
Now let us analyze the firm k from xj i. Its revenue function has the following form:
n(j-i)jk = q(j-i)jk (p(j-i)j - pj, - V(j-i)jk) , k = 1,n(i_i)j, (11) where p,, : (l - 1, j) € S,j.
xj-1
(2) let us have the formula:
n(i-i)j
q(l_i)jk = Q(j_i)j = Qlh =
/ \ nih (ajh -p(i-i)j) - S Vjhr
= _r = i
= ^ , 1 bjh (njh + 1) :
h:(l,h)esj-1
from that it is possible to express the variable p(l-i)j in explicit form:
p(i-1)j - /(i-1)j (q(i-1)j1; • • • ; q(i-1)jn(i-i)^ -
nih
n(i-i)j "ihaih-I] "!hr
- q(i-1)jk + ^ , , bih("i'h=11) k=1 h:(i,h)es'-1
E
(12)
bih("ih + 1)
Let us substitute (12) in the revenue formulas (11)
n(i-1)jfc = q(i-1)jfc (/(i-1)j - Pit - v(i-1)jfc) ; k — 1; n(i-1)j; (13) and let us apply the maximum condition of necessity to the formulas (13):
dn(i-1)jfc /, \ dq-= (/(i-1)j - Pit - v(i-1)j^ +
-1
q(i-1)jfc-^- nih = 0 k = 1;n(i-1)j ;
h:(i,h)eS'-1
k=1 h:(i,h)eS1-1
nih
or in the matrix form:
/211 1 2 1
V 111 /
1
2
( q(i—i)ji q(i_1)j2
\q(i-1)jn(i_1)j/
h:(i,h)eS'"
^ bih(nih + 1) ( - «-íhPií - níhv(í-1)j1 - E Vhr
r = 1 "ih
f nih \ S bih(nih + 1) níh«íh - níhPjí - nlhv(l_1)j2 - E Vhr >.)psi-1 V r=1 /
^ bih("Íh + 1) ( - nihPii - n¡hv(í-1)j„(i_i)3 - E vhr
'i-1 \ r = 1 /
. (14)
V r = 1 / y
This system is solvable as its matrix is non-singular. As a result, (14) could be solved in a one-valued way in relation to the variables q(l—1)jk, k = 1,n(l_1)¿:
q(í-1)jfc
n(i_1)j + 1
1 I "ih
/, TI-TTT nihah - níhPií - / J Vhr -
^ -i bih(nh + 1M
h:(i.h)eSi
\
"(i-i)j
-n(i_1)j n;h v(í_1)jfc + nih V(l_1)je
e = 1
e = k /J
k = 1, n
(i_1)j •
There are could be further calculated the value of Q(l-i)j:
"(i-i)j
Qc_1)j = E qc_1)jk =
1
k=1
+ 1
E
1
i-1 bh (n;h + 1)
h:(i.h)£Si
/ / "ih \ "(i-1)j * n(i_1)j n(hah - níhPií - Vhr J - níh V(i_1)jfc
fc=1
• (15)
Let us repeat the process (11)-(15) for all the remained nodes xj i from the level l-1
Then in the similar way we will analyze the nodes x, from sets X^ i = (l - 2),
(l - 3), • • •, t - 1, where t is a number of level the node x* is situated.
x*
xi. The revenue function of a firm t from this node as follows:
nm = qt1fc | Pt1 - E P*h - vt1k
k =1, • • •, nt1 •
*
From the previous step we have explicit form for the variable pti (denote it as fti(qtii,..., qtint1,... ), it is easy to see that ft1 — is a linear function of its variables) and we can substitute it to the formula above:
nm = qtik I fti (qtii,..., qtinti,...) - 53 Pih - Vnk I ' k = 1, ...,nti. V (M)esi J
As before we apply the necessary maximum condition and solve the obtained system in relation to variables qtik, k = 1,nti. Then we also calculate the integrated quantity Qti. Using the condition (2) we obtain the following system of equations:
Qti = Qih, i,h :(t, 1) e Sh, pih
quantity and from each other.
Next, we proceed to the analysis of the nodes from Xt-i. Let us consider the following systems:
I n(t-i)jk = q(t-i)jk p(t-i)j - E pih - v(t-i)jk , ■ = t—--
S _ V (t-i,j)esj J j = 1,—(t-i).
[k = 1,n(t-i)j
Substituting the corresponding variable p(t-i)j by its explicit form, we apply the necessary maximum condition for each system and solve the obtained systems in relation to variables q(t-i)jk, k = 1,n(t-i)j, j = 1,—j. After that we calculate Q(t-i)j, substitute these expressions to the expressions for p(t-i)j and solve the system, getting p(t-i)j depending only on prices of suppliers' nodes.
Acting the same way, we move up level by level towards to the root nodes. For these nodes the revenue functions are:
i nijk = qijk (pij - vijk), k = 1, ..., nij ; \j = 1,—i.
pi j
maximum condition and solve the system, getting solutions, which depend only on known parameters of the chain. Moving back from the root nodes to the final ones we will found values for all variables of quantity and prices.
4.2. The weighted Nash bargaining solution
Suppose we have a multi-level distribution supply chain G = (X, F) with a centralized model of behavior of participants, i.e. all participants in this chain join the coalition and act centrally to achieve a common goal. We will consider the weighted Nash function as the objective function.
Let there is a game in the normal form, i.e. a set r = (N, [Ui]leN, {Hi}ieN), where N = {1, 2, ..., n} - a non-empty set of players, Ul - the set of strategies of the player I, Hl - payoff function of player I, defined on the Cartesian product of sets {Ul}leN players' strategies Y = ni£N Yu Hl : Y ^ R (Grossman, Hart, 1983).
Definition 7. A weighted Nash bargaining solution for the game with weights «1, a2, ..., an : a > 0 Vi = 1,n, En=i a = 1 we a vector such
y' = (yi, y2, • • •, ^n) e Y, which maximize function:
n
arg max II (Hi (yi, •••, yn) - #i)ai = y'. (16)
yi, yn i=i
The point d = (01;..., 0n) , where i = 1, n are known parameters, is also called "status quo" point for the problem (16).
As a set of players, we take an ordered set of graph nodes, as a set of strategies - a set Ujj, and as the functions of winning - profit function. The status quo point will be the value of the profit function on the decentralized solution of the same supply chain (denote it by n* ). Then the weighted Nash bargaining solution of this cooperative game will be the solution of the following optimization problem:
(i m* n»j \
nnn(nijfc(qiji,viji-njfc)ai3fc ) (i7)
i=ij=ik=i J
pth _1_(_i,j ) e ; _
nijfc > nj, i =1,1, j = 1, mi, k = 1, nij;
Pij = aij - bij qijk, j = 1, mi ; (18)
k=i
nth nij
= 53 j' t,h: eXi; (is)
r=i i,j:(i,j)eSh k=i
j > 0, i = 1,1, j = 1, mi, k =1,nij;
Pij > 0, j = 1, mi. where aijk - given weights, such that:
«¿jfc > 0, i = 1,l, j = l,mj, k = l,nj,
l mi
13 13 13^ = 1-
i=1 j=1 k=1
The existence and uniqueness of the solution are proved by the fact that the Nash multiplication is a continuous convex function, and the constraints set a compact, hence, by the Weierstrass theorem, the maximum of the function exists and is unique.
5. Competitive and cooperation behavior in GTM distribution network
This section considers the case of the Russian distribution network. Solutions for competitive and cooperative behavior models are explored.
5.1. GTM network description
We will use "GTM" as a name of distributor and distribution network. The data for the research was provided by the GTM distribution company operating in Russia. The company is presented in more than 150 cities in different regions of Russia from the North-West to the Far East. As a major player in the market, the company has its own intra-organizational supply chain network including 8 distribution centers. There are more than 60 sales departments with full category B warehouses. The number of employees is nearly six thousand. The number of suppliers having a valid contract is more than 600 by the end of 2016. Among suppliers, there are more than 400 manufacturers. The main suppliers of the company are manufacturers representing electrical industry divided into six parts, namely: Cable production; Industrial electrical equipment; Lighting products; Installation electrical equipment; Safety systems and Fasteners and Plumbing.
For the research one district of distribution company was selected. We limited the network to four suppliers, which are manufactures of industrial electrical equipment, distributor's center and retailers, operated in Central Region of Russia. Each manufacture (supplier) supply only one product which is used to form a portfolio. Costs of each manufacture and the share of its product in portfolio are presented in the table 1.
Table 1. Input data for computational results. Suppliers.
Suppliers Supplier's costs (cost price), Rub. Share of the product in portfolio
Supplier 1.1 5 742 0.222
Supplier 1.2 2 441 0.148
Supplier 1.3 11 399 0.444
Supplier 1.4 14 010 0.185
The table 2 shows the costs that the distributor incurs for the purchase of products from suppliers and the costs that the distributor incurs for the organization of logistics.
Table 2. Input data for computational results. Distributor.
Suppliers Distributor's costs (cost price), Rub. Distributor's logistic costs per unit of portfolio
Distributor 2.1 48 778 1 340.411
In each region, there are a certain number of retailers. Logistics costs per unit of production vary for each retailer. For ease of computation, we have accepted that demand is a linear function. Demand function was constructed for each region based on data from previous periods. The retailers compete according to the Cournot model. The data presents in the table below.
Based on the data presented in the tables, we construct the structure of the network. The model includes sixteen nodes: four manufactures, one distribution center and eleven regions with compete retailers. The structure of GTM's supply-network is presented on the Fig. 3.
Table 3. Input data for computational results. Retailers.
Region Retailers Logistic costs for 1 portfolio, Rub. Demand function
3.1 Belgorod region BelRet .3.1.1 923 P = 59 104-0.17Q
BelRet 3.1.2 818
BelRet 3.1.3 263
3.2 Vladimir region VladRet 3.2.1 947 P = 60399 0.02Q
VladRet 3.2.2 554
VladRet 3.2.3 1 060
VladRet 3.2.4 1 046
VladRet 3.2.5 439
VladRet 3.2.6 583
VladRet 3.2.7 647
VladRet 3.2.8 735
3.3 Voronezh region Vor Ret 3.3.1 513 P = 59866 - 0.08Q
VorRet 3.3.2 820
Vor Ret 3.3.3 1 125
VorRet 3.3.4 671
VorRet 3.3.5 800
3.4_Kaluga region KalRet 3.4.1 847 P = 60488 2.06Q
KalRet 3.4.2 794
3.5 Kursk region KurRet 3.5.1 463 P = 64469 - 0.72Q
KurRet 3.5.2 278
KurRet 3.5.3 253
3.6_Lipetsk region LipRet_3.6.1 1 265 P = 61802 0.38Q
LipRet_3.6.2 805
LipRet_3.6.3 739
LipRet_3.6.4 1 237
LipRet_3.6.5 521
LipRet_3.6.6 919
3.7 Orel region OreRet 3.7.1 543 P = 61120 1.44Q
OreRet 3.7.2 934
3.8 _ Ryazan region RyazRet_3.8.1 338 P = 61364 - 0.23Q
RyazRet_3.8.2 229
RyazRet_3.8.3 470
RyazRet_3.8.4 183
RyazRet_3.8.5 620
3.9 Tambov region TamRet 3.9.1 201 P = 62133 1.90Q
TamRet 3.9.2 515
3.10_Tula region TulRet 3.10.1 609 P = 58236 0.07Q
TulRet 3.10.2 652
TulRet 3.10.3 736
3.11 Yaroslavl region YarRet 3.11.1 341 P = 61773 - 1.66Q
YarRet 3.11.2 607
YarRet 3.11.3 608
Suppliers
Distributor
Retailers (regions)
Fig. 3. The GTM network structure Source: Authors' own
Distributor takes the goods from the supplier's warehouse, so on the supplier-distributor branch logistics costs are incurred by distributor. Retailers incur the logistics costs on distributor retailer branches. The distributor determines the volume of product order from the manufacturer. The distributor forms a portfolio of suppliers ' products and further supplies retailers only in the amount of at least one portfolio. Partial delivery of the portfolio is not allowed. This scheme of supply network is designed in order to provide a presence of products of all four suppliers in the markets.
5.2. Perfect Nash equilibrium
We present the results of competitive solution (the perfect Nash equilibrium) only for one node of retailers of GTM network. The whole computations are presented in the Appendix. The first region was selected for presenting results. We construct profit functions for end vertices, i.e. for retailers.
^311 = 3311 (P31 - P21 - 923);
^312 = 3312 (P31 - P21 - 818);
^313 = 3313 (p31 - P21 - 263);
In these functions, we substitute expressions for market prices, using the demand functions, and apply the necessary maximum condition:
58 181 - 0,17 (3311 + 3312 + 3313) - P21 - 0,17 3311 = 0; 58 286 - 0,17 (3311 + 3312 + 3313) - P21 - 0,17 3313 = 0; 58 841 - 0,17 (3311 + 3312 + 3313) - P21 - 0,17 3313 = 0.
We solve systems with respect to quantity variables:
3311 = 84 435, 2941 - 1,4706 P21;
qsi2 = 85 052, 9412 - 1, 4706 p2i; qsis = 88 317, 6471 - 1, 4706 p2i.
Thereby, the quantity for distributor is:
q2ii = Q2i = 4 660 598,4499 - 78, 4969 p2i.
So, we get that:
p2i = 59 373,0052 - 0, 0127 q2ii.
(20)
Distributor's profit function:
n2ii = q2ii (P2i - pii - Pi2 - Pi3 - Pi4 - 1340). Let's put an expression in it (20):
n2ii « q2ii (59 373 - 0.0127 q^ii -pii - Pi2 - pis - Pi4 - 1340) and apply the necessary condition of the maximum:
58 033, 0052 - pii - pi2 - pis - pi4 - 0, 0255 q2ii = 0. from which we get:
q2ii = 2 277 706, 3057 - 39, 2485 (pn + pi2 + pis + pi4). Using the condition of absence of shortage and surplus, we have a ratio:
qiii = 0, 222 * q2ii « 505 650, 7999 - 8.7132 (pn + pi2 + pis + Pi4) ;
qii2 =0,148 * q2ii « 337 100, 5333 - 5, 8088 (pn + pi2 + pis + Pi4) ; qiis = 0, 444 * q2ii « 1 011 301, 5997 - 17,4263 (pn + pi2 + pis + pi4) ; qii4 = 0, 185 * q2ii « 421 375, 6666 - 7, 261 (pn + pi2 + pis + pi4) .
pii, pi2, pis pi4
pii = 58033, 0052 - pi2 - pis - pi4 - 0,1148qm; pi2 = 58033, 0052 - pn - pis - pi4 - 0,1148qi2i; pis = 58033,0052 - pn - pi2 - pi4 - 0,1148qm; pi4 = 58033, 0052 - pn - pi2 - pis - 0,1148qi4i.
(21)
Construct the profit functions for suppliers:
niii = qiii (pii - 5742) ; ni2i = qi2i (p2i - 2 441) ; nisi = qisi (psi - 11 399) ; ni4i = qi4i (pi4 - 14 010)
and substitute them in (21):
niii = qiii (58 033, 0052 - pi2 - pis - pi4 - 0,1148 qm - 5742);
ni2i = qi2i (58 033, 0052 - pn - pis - pi4 - 0,1148 q2n - 2441); nisi = qisi (58 033, 0052 - pn - pu - pi4 - 0,1148 q2U - 11399); ni4i = qi4i (58 033,0052 - pn - pi2 - pis - 0,1148 q2ii - 14010). Apply the necessary condition of the maximum:
52 291,0052 - pi2 - pis - pi4 - 0, 2295 qiii = 0;
55 592,0052 - pii - pis - pi4 - 0, 3443 qi2i = 0; 46 634,0052 - pii - pi2 - pi4 - 0,1148 qisi = 0; 44 023,0052 - pii - pi2 - pis - 0, 2755 qi4i = 0. Then, we solve equations with respect to quantity:
qiii = 277 809, 9205 - 4, 3566 (pi2 + pis + pi4);
qi2i = 161 460, 6596 - 2,9044 (pii + pis + pM); qisi = 406 329, 5007 - 8,7132 (pii + pi2 + pw); qi4i = 159 824, 7678 - 3, 6305 (pii + pi2 + pis). Substitute the values obtained in equality (21):
pii = 31 887, 5026 - 0, 5 (pi2 + pis + pi4);
pi2 = 30 237,0026 - 0, 5 (pn + pis + pM); pis = 34 716,0026 - 0, 5 (pii + pi2 + pM); pi4 = 36 021, 5026 - 0, 5 (pn + pi2 + pis). Solving the system, we get:
pii « 10 630, 2010; pi2 « 7 329; pis « 16 287; pi4 « 18 898.
Next, find the values of all other variables. The obtained values of all other variables are presented in the Table 4.
The results, presented in the table 4, show the solution of a non-cooperative game involving network members in which each member of the network is assumed to know the equilibrium strategies of the other members, and no member has anything to gain by changing only their own strategy. Obtained results reflect the performance of network participants in condition of competitive behavior. These results are used as a status quo point for cooperative game.
Table 4. The perfect Nash equilibrium solution
Node Equilibrium solution
Quantity Price Profit
X11 9111 s 42 592 P11 s 10 630 n111 s 208 196 659
X12 9112 s 28 394 P12 s 7 329 ^121 s 138 797 773
X13 9113 s 85 183 P13 s 16 287 n131 s 416 393 318
X14 9114 s 35 493 P14 s 18 898 ^141 s 173 497 216
X21 9211 ~ 191 854 P21 s 56 929 ^211 s 468 911 394
X31 9311 s 716 ^311 s 87 229
9312 s 1 334 P31 s 57 974 ^312 s 302 508
9313 s 4 599 ^313 s ¿ 3 595 120
9321 s 7 545 ^321 s ¿ 1 138 533
9322 s i 7 195 ^322 s ¿ 1 035 353
9323 s 1 895 ^323 s 71 819
X32 9324 s 9325 s 2 595 32 945 P32 s 58 027 ^324 ^325 s s 134 678 21 707 426
9326 s 25 745 ^326 s 13 256 074
9327 s 22 545 ^327 s 10 165 517
9328 s 18 145 ^328 s ¿ 6 584 802
9331 s 7 892 ^331 s ¿4 982 523
9332 s i 4 054 ^332 s ¿ 1 315 030
X33 9333 s 242 P33 s 58 073 n333 s 4 680
9334 s i 5 917 n334 s ¿ 2 800 744
9335 s 4 304 n335 s ¿ 1 482 205
X34 9341 9342 s 430 s 456 P34 s 58 662 ^341 ^342 s 381 380 s 428 353
9351 s 2320 ^351 s ¿3 875 903
X35 9352 s 2577 P35 s 59 062 ^352 s ¿4 781 901
9353 s 2612 ^353 s ¿ 4 911 625
9361 s 565 ^361 s 121 499
9362 s 1 776 ^362 s ¿ 1 198 554
X36 9363 s 9364 1 950 s 639 P36 s 58 408 ^363 s ^364 ¿ 1 444 446 s 155 227
9365 s 2 523 ^365 s ¿2 419 561
9366 s 1 476 ^366 s 827 832
X37 9371 9372 s 935 s 663 P37 s 58 818 ^371 s ^372 ¿ 1 258 819 s 633 835
9381 s 3 078 ^381 s ¿ 2 178 481
9382 s 3 552 ^382 s ¿ 2 901 055
X38 9383 s 2 504 P38 s 57 974 ^383 s ¿ 1 441 749
9384 s 3 752 ^384 s ¿3 236 995
9385 s 1 852 ^385 s 788 468
X39 9391 9392 s 933 s 768 P39 s 58 902 ^391 s ^392 s ¿ 1 653 304 ¿ 1 119 384
93101 s 3 100 ^3101 s 672 848
X310 93102 s 2 486 P310 s 57 754 ^302 s 432 633
93103 s 1 286 ^3103 s 115 776
93111 s 758 ^3111 s 954 904
X311 93112 s 598 P311 s 58 528 ^3112 s 594 034
93113 s 598 ^3113 s 592 838
5.3. Weighted Nash bargaining solution
To find a cooperative solution, we used the MATLAB application package. We used the extremum search function of the constraint function based on the method of sequential quadratic programming, which is an iterative method for constrained nonlinear optimization. It is one of the most effective methods for nonlinearly constrained optimization problems. The method generates steps by solving quadratic subproblems; it can be used both in line search and trust-region frameworks. Sequential quadratic programming is appropriate for small and large problems and it is well suited to solving problems with significant nonlinearities.
To find a cooperative solution, weights were assigned to each member of the network. According to the total number of vertices equal to 16, the first node level (suppliers) was assigned the weight of 8/16, the second one (distributor) - 4/16, the third one (retailers) - 4/16. The weight of each supplier is the weight of the first level nodes (8/16) multiplied by the share of its product in the portfolio. The weight of each retailer is the weight of the nodes of the third level (4/16) divided by 8 (the number of nodes at the level) and the number of retailers at the node. Retailers have the least weight per retailer. This is because before retailers cooperate with a distributor, they need to come to an agreement within the region. Thus, the weight of each individual retailer in cooperation with the distributor is not as important as the weight value of the region.
Let's formulate and solve the following optimization problem:
max [(9311 (psi - P21 - 923) - 87 361)0'019* (9312 (P31 - P21 - 818) - 302 576)0'019* qijk ,
*(q313 (P31 - P21 - 263) - 3 594 527)0'019 * (9321 (ps2 - P21 - 947) - 1 137 958)0'0007* *(9322 (P32 - P21 - 554) - 3 915 571)0'0007 * (9323 (p32 - P21 - 1 060) - 72 430)0'0007* *(9324 (P32 - P21 - 1046) - 134 873)0'0007 * (9325 (P32 - P21 - 439) - 21 711 969)0'0007* *(9326 (P32 - P21 - 583) - 13 244 049)0'0007 * (9327 (P32 - P21 - 647) - 10 166 800)0'0007* *(9328 (P32 - P21 - 735) - 6 592 280)0'0007 * (9331 (P33 - P21 - 513) - 4 985 510)0'0011 *
*(9332 (P33 - P21 - 820) - 1 316 821)0'0011 * (9333 (P33 - P21 - 1 125) - 4 603)0'0011 * *(9334 (P33 - P21 - 671) - 2 801 497)0'0011 * (9335 (P33 - P21 - 800) - 1 480 311)0'0011 *
*(9341 (P34 - P21 - 847) - 381 524)0'0028 * (9342 (P34 - P21 - 794) - 428 195)0'0028* *(9351 (P35 - P21 - 463) - 3 875 142)0'0019 * (935 (P35 - P21 - 278) - 4 780 888)0'0019* *(9353 (P35 - P21 - 253) - 4 910 880)0'0019 * (9361 (P36 - P21 - 1 265) - 121 471)0'0009* *(9362 (P36 - P21 - 805) - 1 198 779)0'0009 * (9363 (P36 - P21 - 739) - 1 444 701)0'0009* *(9364 (P36 - P21 - 1 237) - 155 266)0'0009 * (9365 (P36 - P21 - 521) - 2 419 493)0'0009* *(9366 (P36 - P21 - 919) - 828 314)0'0009 * (9371 (P37 - P21 - 543) - 1 259 126)0'0028* *(9372 (P37 - P21 - 934) - 633 835)0'0028 * (9381 (P38 - P21 - 338) - 2 177 757)0'0011 * *(9382 (P38 - P21 - 229) - 2 899 771)0'0011 * (9383 (P38 - P21 - 470) - 1 442 984)0'0011 * *(9384 (P38 - P21 - 183) - 3 236 514)0'0011 * (9385 (P38 - P21 - 620) - 789 291)0'0011 * *(9391 (P39 - P21 - 201) - 1 653 011)0'0028 * (9392 (P39 - P21 - 515) - 1 119 008)0'0028* *(93101 (P310 - P21 - 609) - 673 995)0'0019 * (93102 (P310 - P21 - 652) - 433 140)0'0019* *(93103 (P310 - P21 - 736) - 115 445)0'0019 * (93111 (P311 - P21 - 341) - 954 613)0'0019*
*(93ii2 (psii - P2i - 607) - 594 287)0-0019 * (93113 (p3ii - P21 - 608) - 592 804)0'0019* *(?211 (p2i - pii - pi2 - pi3 - pi4 - 1 340) - 4 688 322 609)0'25* *(qiii (pii - 5 742) - 208 187 438)0'1526 * (gm (pi2 - 2 441) - 138 792 459)0'1017* *(qi3i (pi3 - 11 399) - 416 350 845)0'3063 * (qi4i (pi4 - 14 010) - 173 503 931)0'1272]; psi = 59 104 - 0,17Qsi; ps5 = 64 469 - 0, 72Qss; ps9 = 62 133 - 1,9Qsg;
ps2 = 60 399 - 0,02Qs2; pse = 61 802 - 0,38Qse; psio = 58 236 - 0,07Qsio;
pss = 59 866 - 0,08Qss; psr = 61 120 - 1,44Qsr; psii = 61 773 - 1,66Qsii; ps4 = 60 488 - 2,06Qs4; pss = 61 364 - 0, 23Qss;
Qsi + • • • + Qsii = Q2i; Qii = 0, 222Q2i; Qis = 0,444Q2i Qi2 =0,148Q2i; Qi4 = 0,185Q2i; qijfc > 0, pij > 0. Values of variables and profits of all firms are given in the Table 5.
Table 5. The weighted Nash bargaining solution
Node Equilibrium solution
Quantity Price Profit
Xii qiii « 47 794 pii « 10 158 niii « 211 069 495
Xi2 qii2 « 31 863 pi2 « 6 853 ni2i « 140 580 514
xis qiis « 95 588 pis « 16 268 nisi « 465 388 411
Xi4 qii4 « 39 828 pi4 « 18 654 ni4i « 184 974 523
X2i q2ii « 215 288 p2i « 56 487 n2ii « 468 322 609
xsi qsii « 672 nsii « 287 401
qsi2? « 1 412 psi « 57 837 nsi2 « 751 483
qsis? 5 369 nsis « a 5 836 947
qs2i? « 9 125 ns2i « a 1 275 419
qs22? « 8 717 ns22 « a 4 647 429
qs2s? 2 674 ns2s « 72 681
xs2 qs24 ? qs25 « 3 443 38 646 ps2 « 57 574 ns24 ns25 « « 140 907 25 042 313
qs2e « 30 460 ns26 « 15 333 081
qs27 « 26 690 ns27 « 11 741 319
qs2s « 21 520 ns2S « a 7 580 734
qssi? 9 450 nssi « a 6 699 543
qss2? 5 874 nss2 « a 1 958 995
xss qsss « 390 pss « 57 708 nsss « 37 520
qss4? 7 089 nss4 « a 3 903 870
qss5 « « 5 170 nss5 « a 2 177 095
X34 9341 ~ 316 9342 ~ 358 p34 « 59 099 n341 « 558 523 n342 « 650 770
X35 9351 ~ 1033 9352 « 1321 9353 ~ 1360 P35 « 61 795 n351 « 5 003 330 n352 « 6 645 457 n353 « 6 876 004
X36 9361 « 108 9362 ~ 1 386 9363 ~ 1 581 9364 ~ 115 9365 ~ 2 228 9366 ~ 1 048 P36 « 59 345 n361 « 172 321 n362 « 2 845 741 n363 « 3 351 998 n364 « 185 716 n365 « 5 207 520 n366 « 2 032 443
X37 9371 « 810 9372 ~ 588 p37 « 59 107 n371 « 1 681 812 n372 « 992 292
X38 9381 « 3 153 9382 ~ 3 709 9383 « 2 482 9384 ~ 3 944 9385 « 1 723 P38 « 57 911 n381 « 3 425 495 n382 « 4 433 475 n383 « 2 370 941 n384 « 4 897 086 n385 « 1 387 695
X39 9391 « 1 115 9392 ~ 949 P39 « 58 212 n391 « 1 698 889 n392 « 1 147 671
X310 93101 ~ 3 971 93102 ~ 3 225 93103 ~ 1 771 P310 « 57 608 n3101 « 2 037 296 ^3102 « 1 515 255 ^3103 « 682 368
X311 93111 « 480 93112 ~ 407 93113 ~ 407 p311 « 59 625 n3111 « 1 343 314 n3112 « 1 030 373 n3113 « 1 029 138
The results of the weighted Nash bargaining solution presented in the table 5 reflect the possible performance of all participants in the cooperative behavior. For each of the network participants the value of profit in terms of cooperation is calculated. The price is set for the region in which the cooperation of retailers is carried out. For each of the retailers, the sales volume is calculated in terms of cooperation behavior maximizing their profits. To find a cooperative solution that will increase the profit of each participant relative to the equilibrium, the Nash equilibrium solution is taken as the status quo point. Thus, the network participants get profit better or at least not worse than in the equilibrium solution. This statement is well illustrated by the presented results. Table 5 shows that each of the network participants, including the manufacturer and distributor, and not just retailers, gained a profit value higher or at least not worse than in equilibrium.
6. Comparative analysis of competitive and cooperative behavior
In this study, we considered the perfect Nash equilibrium solution as competitive behavior and the weighted Nash bargaining solution as a cooperative solution. To find a cooperative solution, the Nash equilibrium solution was taken as the status quo point.
The Nash equilibrium solution reflects the results of the decision of a non-cooperative game involving in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy According to this the algorithm for foundation the optimal solution, which in terms of game theory is Nash equilibrium in a multistep hierarchical game with complete information on a graph G with an hourglasses structure was provided. At the first step of the research, the perfect Nash equilibrium solution was found as a competitive behavior of participants within the network. In order to improve the results of the Nash equilibrium solution, i.e. to increase the profit of each participant in the chain, cooperation is necessary. In general, cooperation allows achieving better results from interaction, than in the case where companies operate independently. To find a cooperative solution that will increase the profit of each participant relative to the equilibrium we included all participant in coalition and the Nash equilibrium solution was taken as the status quo point. Thus, the network participants get profit better or at least not worse than in the equilibrium solution. Since retailers are working in the region markets, the solution was found for each region separately. The comparative analysis of retailers' profits for each region is presented in the table 6.
Table 6. The comparative analysis of retailers' profits for competitive and cooperative behavior
Regions (nodes) The profit of retailers obtained in case of competitive behavior The profit of retailers obtained in case of cooperative behavior The deviation of cooperative solution from competitive solution
3.1 Belgorod region 3 984 464 6 875 831 72.6%
3.2 Vladimir region 56 975 930 65 833 884 15.5%
3.3 Voronezh region 10 589 442 14 777 023 39.5%
3.4_Kaluga region 809 719 1 209 308 49.3%
3.5 Kursk region 13 566 910 18 524 790 36.5%
3.6_Lipetsk region 6 168 024 13 795 648 123.7%
3.7 Orel region 1 892 961 2 674 104 41.3%
3.8_Ryazan region 10 546 316 16 514 692 56.6%
3.9 Tambov region 2 772 019 2 846 560 2.7%
3.10_Tula region 1 222 580 4 234 919 246.4%
3.11 Yaroslavl region 2 141 704 3 402 825 58.9%
Total 110 670 069 150 689 584 36.2%
The obtained results show that each of the network retailers gained a profit value higher or at least not worse than in equilibrium. At the same time, we see that the obtained results do not deviate from the equilibrium solution so much as to talk about their unattainability. For instance, the profit of all retailers of Belgorod region increased by 72.6%. For Ryazan region the retailers' profit in case of cooperative solution increased by 56.6%. At the same time the results of cooperative solution in Tambov region is higher than the results of competitive solution only by 2.7%. It is the worse result however it is still better than the results obtained in condition of competitive behavior. The only minus of such results is that the motivation for
cooperation of Tambov retailers will be low for effective and stable relationships. The total profit of all retailers in the network is higher for cooperative solution by 36.2% that can be considered as a motivation factor for all retailers to cooperate and maintain stable relationships within the network.
The sale price of the portfolio in the market is also set for the region. As retailers compete according to Cournot model, the optimal quantity for each retailer in the region was found. The comparative analysis of obtained price and quantities for competitive and cooperative behaviors is presented in table 7 and table 8.
Table 7. The comparative analysis of retailers' volumes for competitive and cooperative behavior
Regions (nodes) The quantity of retailers obtained in case of competitive behavior The quantity of retailers obtained in case of cooperative behavior The deviation of cooperative solution from competitive solution
3.1 Belgorod region 6 649 7 454 12.1%
3.2 Vladimir region 118 610 141 274 19.1%
3.3 Voronezh region 22 409 26 973 20.4%
3.4_Kaluga region 886 674 -23.9%
3.5 Kursk region 7 509 3 714 -50.5%
3.6_Lipetsk region 8 930 6 465 -27.6%
3.7 Orel region 1 598 1 398 -12.5%
3.8_Ryazan region 14 736 15 011 1.9%
3.9 Tambov region 1 700 2 064 21.4%
3.10_Tula region 6 872 8 967 30.5%
3.11 Yaroslavl region 1 954 1 294 -33.8%
Total 191 854 215 288 12.2%
Table 8. The comparative analysis of retailers' prices for competitive and cooperative behavior
Regions (nodes) The price of retailers obtained in case of competitive behavior The price of retailers obtained in case of cooperative behavior The deviation of cooperative solution from competitive solution
3.1 Belgorod region 57 974 57 837 -0.2%
3.2 Vladimir region 58 027 57 574 -0.8%
3.3 Voronezh region 58 073 57 708 -0.6%
3.4_Kaluga region 58 662 59 099 0.7%
3.5 Kursk region 59 062 61 795 4.6%
3.6_Lipetsk region 58 409 59 345 1.6%
3.7 Orel region 58 818 59 107 0.5%
3.8_Ryazan region 57 975 57 911 -0.1%
3.9 Tambov region 58 902 58 212 -1.2%
3.10_Tula region 57 755 57 608 -0.3%
3.11 Yaroslavl region 58 529 59 625 1.9%
As can be seen from table 8, the price remained unchanged in all regions. Deviations for all regions are minor. The increase in efficiency was mainly due to the redistribution of volume between the network participants in such a way that the total costs were minimal and with a slight increase in volume (by 12%), the growth of total profit was significant.
The optimal price and quantity were found for distributor and then - for each supplier. Thus, we understand that the profit gained from the equilibrium solution is better result or at least not worse for each of the participants of the presented chain than in the conditions of competition.
The obtained results for suppliers and distributor are presented in the table 9 and table 10 respectively. The result reflects only the deviation in profits for competitive and cooperative solutions. This is due to the fact that demand is formed in the final nodes, i.e. retailers. The distributor forming a portfolio provides only distribution of volumes between retailers, without affecting the total demand.
Table 9. The comparative analysis of competitive and cooperative behavior of suppliers
Suppliers The supplier's profit obtained in case of competitive behavior The supplier's profit obtained in case of cooperative behavior The deviation of cooperative solution from competitive solution
Supplier _ 1.1 208 187 437 211 069 495 1.4%
Supplier _ 1.2 138 792 459 140 580 514 1.3%
Supplier_1.3 416 350 844 465 388 410 11.8%
Supplier_1.4 173 503 931 184 974 523 6.6%
Table 10. The comparative analysis of competitive and cooperative behavior of distributor
Distributor The distributor's profit obtained in case of competitive behavior The distributor's profit obtained in case of cooperative behavior The deviation of cooperative solution from competitive solution
Distributor_2.1 184 974 523 468 832 610 147.5%
The results presented in table 9 show that a cooperative solution is better than a competitive solution for only two suppliers. For the other two suppliers, the cooperative solution, we can say, is no worse than the Nash equilibrium. In general, all suppliers benefit from cooperative behavior, however, the motivation for cooperation of the latter two suppliers will be higher. This means that in the process of coalition formation, these suppliers are more likely to take a positive decision to join the coalition while the other two suppliers will take a neutral position. Table 10 clearly shows that the distributor benefits significantly from cooperation. Its profit in terms of cooperative behavior is growing by 47.5%. With a slight change in price, the total number of purchased and sold products in the cooperative solution increased by 12%, but the greatest effect was given by changes in the distribution of products between retailers.
7. Conclusion and limitations of the research
The research has two main results. First, we construct the perfect Nash equilibrium for such supply network as the competitive solution in the multi-step hierarchical game in closed form. Second, we construct a cooperative behavior for the network and found the unique weighted Nash bargaining solution with the perfect Nash equilibrium as a point of the status quo. The weighted Nash bargaining solution comes down to the solution of a separable nonlinear programming problem with concave payoff function.
The obtained results show that cooperative behavior is more profitable than competitive one for all participants. The supply chain profit in conditions of cooperative behavior is higher than the profit in conditions of competitive behavior by 31.57%. The distributor has the greatest increase in profit in terms of cooperative behavior. It means that distributor is motivated for organization of cooperation most. In the network considered in this study, the distributor has one of the most important roles. The distributor has relationships with suppliers, retailers, and acts as a focal company of the network. The total profit of all retailers in the network is higher for cooperative solution by 36.2%. It means that most of retailers are motivated to cooperate with distributor. At the same time, the total of suppliers is higher for cooperative behavior only by 6.96%. On the one hand, it can be considered that suppliers are not motivated enough for cooperation, but on the other hand if suppliers refuse to join a coalition, the distributor may revise the portfolio. In this case, suppliers who have not joined the coalition may not only make a profit worse than in case of equilibrium, but even lower.
Therefore, we considered the network in two behavior models: competitive behavior and cooperative behavior. The results show that all members of supply network are motivated for coalition formation. This result is important for several reasons. First, it confirms that cooperation in supply networks gets better results for each member of the network and for the whole network. Second, it shows quite clearly that in distribution networks the organization of cooperation in the network is the responsibility of the distributor as a focal company, because the distributor has a greater motivation and greater weight than, for instance, retailers with similar motivation. Moreover, the obtained results are essential for future research in supply networks and research in coalition formation problem.
This study has some limitations, which are imposed on the one hand by the research methods used, on the other hand - the study model. The first one is that the price of products within the region is the same for all retailers. This situation is often typical for chain stores of one company. However, in conditions of high competition, such situation can be observed. Another limitation of the study is the assumption that the demand function is linear. This assumption is made for ease of calculation and display of the results of the study. To calculate demand functions in each region were used the data about sales of previous periods. The next limitation of the study-is that one product is selected. In our case, this product is the portfolio that is formed by distributor. This limitation is related to the chosen method of finding the solution. On the one hand, this limitation makes it possible to draw conclusions only within one product network, on the other hand, we understand that the distributor, having a significant weight in the network, can form a certain assortment matrix (portfolio) for interaction with retailers. In this case, consideration of multi-product networks becomes possible.
As a direction for future research, the coalition formation problem can be considered. To find the cooperative solution we included all participant of supply network in coalition. The results of a cooperative solution are better than the results of a decentralized solution, in that case, the question with which of the participants of the coalition will be the most profitable arises.
Appendix
The computations of the perfect Nash equilibrium for GTM network
We construct profit functions for end vertices, i.e. for retailers.
fl"311 = q311 (P31 - P21 - 923) n36i = q36i (p36 - P21 - 1 265) ;
^312 = q312 (P31 - P21 - 818) n362 = q362 (p36 - P21 - 805) ;
^313 = q313 (P31 - P21 - 263) n363 = q36 (p36 - P21 - 739) ;
n321 = q321 (p32 - P21 - 947) n364 = ©64 (P36 - P21 - 1 237) ;
^322 = q322 (P32 - P21 - 554) n365 = ©65 (P36 - P21 - 521);
^323 = q323 (P32 - P21 - 1 060); n366 = q366 (p36 - P21 - 919);
^324 = q324 (P32 - P21 - 1 046);
^325 = q325 (P32 - P21 - 439) n371 = ©71 (P37 - P21 - 543) ;
^326 = q326 (P32 - P21 - 583) n372 = ©72 (p37 - P21 - 934) ;
^327 = q327 (P32 - P21 - 647)
^328 = q328 (P32 - P21 - 735) ^381 = ^382 = ©81 (p38 ©82 (p38 - P21 - - P21 - 338) ; 229) ;
^331 = q331 (P33 - P21 - 513) ^383 = ©83 (p38 - P21 - 470) ;
^332 = q332 (P33 - P21 - 820) ^384 = ©84 (p38 - P21 - 183);
^333 = q333 (P33 - P21 - 1 125); ^385 = ©85 (P38 - P21 - 620) ;
^334 = q334 (P33 - P21 - 671)
^335 = q335 (P33 - P21 - 800) ^391 = ^392 = ©91 (p39 ©92 (p39 - P21 - - P21 - 201); 515);
^341 = q341 (P34 - P21 - 847)
^342 = q342 (p34 - P21 - 794) ^3101 = ^3102 = = ©101 (p3 = ©102 (P3 10 - P21 10 - P21 - 609) ; - 652) ;
^351 = q351 (P35 - P21 - 463) ^3103 = = q3i03 (p3 10 - P21 - 736) ;
^352 = q35 (P35 - "P21 " 278) ;
^353 = q353 (P35 - P21 - 253) n3iii = ^3112 = ^3113 = = ©111 (p3 = ©112 (p3 = ©113 (p3 11 - P21 11 - P21 11 - P21 - 341); - 607) ; - 608).
In these functions, we substitute expressions for market prices, using the demand functions, and apply the necessary maximum condition:
r 58 181 - 0, 17 (qsii + q312 + qsis) - P21 - 0, 17q3l1 = 0; I 58 286 - 0,17 (qsii + q3i2 + q3i3) - P21 - 0,17 ®13 = 0; [ 58 841 - 0,17 (q3ii + q3i2 + q3i3) - P21 - 0,17 ®i3 = 0;
59 452 - 0, 02 (4321 + • • + 4328) - P21 - 0, 02 4321 =0
59 445 - 0, 02 (4321 + • • + 4328) - P21 - 0, 02 4322 =0
59 339 - 0, 02 (4321 + • • + 4328) - P21 - 0, 02 4323 =0
59 353 - 0, 02 (4321 + • • + 4328) - P21 - 0, 02 4324 =0
59 960 - 0, 02 (4321 + • • + 4328) - P21 - 0, 02 4325 =0
59 816 - 0, 02 (4321 + • • + 4328) - P21 - 0, 02 4326 =0
59 752 - 0, 02 (4321 + • • + 4328) - P21 - 0, 02 4327 =0
59 664 - 0, 02 (4321 + • • + 4328) - P21 - 0, 02 4328 =0
59 353 - 0, 08 (4331 + 4332 + 4333 + 4334 + 4335 ) - P21 - 0,08 4331 = 0;
59 046 - 0, 08 (4331 + 4332 + 4333 + 4334 + 4335 ) - P21 - 0, 08 4332 = 0;
58 741 - 0, 08 (4331 + 4332 + 4333 + 4334 + 4335 ) - P21 - 0, 08 4333 = 0;
59 195 - 0, 08 (4331 + 4332 + 4333 + 4334 + 4335 ) - P21 - 0, 08 4334 = 0;
59 066 - 0, 08 (4331 + 4332 + 4333 + 4334 + 4335 ) - P21 - 0, 08 4335 = 0;
f 59 641 - 2, 06 (4341 + 4342) - P21 - 2,06 4341 = 0; \ 59 694 - 2, 06 (4341 + 4342) - P21 - 2,06 4342 = 0;
64 006 - 0, 72 (4351 + 4352 + 4353) - P21 - 0, 72 4351 = 0 64 191 - 0, 72 (4351 + 4352 + 4353) - P21 - 0, 72 4352 = 0 64 216 - 0, 72 (4351 + 4352 + 4353) - P21 - 0, 72 4353 = 0
60 537 - 0, 38 (4361 + 4362 + 4363 + 4364 + 4365 + 4366) - P21 - 0, 38 4361 = 0
60 997 - 0, 38 (4361 + 4362 + 4363 + 4364 + 4365 + 4366) - P21 - 0, 38 4362 = 0
61 063 - 0, 38 (4361 + 4362 + 4363 + 4364 + 4365 + 4366) - P21 - 0, 38 4363 = 0
60 565 - 0, 38 (4361 + 4362 + 4363 + 4364 + 4365 + 4366) - P21 - 0, 38 4364 = 0
61 281 - 0, 38 (4361 + 4362 + 4363 + 4364 + 4365 + 4366) - P21 - 0, 38 4365 = 0 60 883 - 0, 38 (4361 + 4362 + 4363 + 4364 + 4365 + 4366) - P21 - 0, 38 4366 = 0
60 577 - 1, 44 (4371 + 4372) - P21 - 1,44 4371 = 0; 60 186 - 1, 44 (4371 + 4372) - P21 - 1,44 4372 = 0;
' 61 026 - 0, 23 (4381 + 4382 + 4383 + 4384 + 4385) - P21 - 0, 23 4381 = 0
61 135 - 0, 23 ( 4381 + 4382 + 4383 + 4384 + 4385) - P21 - 0, 23 4382 = 0
< 60 894 - 0, 23 (4381 + 4382 + 4383 + 4384 + 4385) - P21 - 0, 23 4383 = 0
61 181 - 0, 23 ( 4381 + 4382 + 4383 + 4384 + 4385) - P21 - 0, 23 4384 = 0
^ 60 744 - 0, 23 (4381 + 4382 + 4383 + 4384 + 4385) - P21 - 0, 23 4385 = 0
f 61 932 - 1, 9 (4391 + 4392) - P21 - 1, 9 4391 = 0; \ 61 618 - 1, 9 (4391 + 4392) - P21 - 1, 9 4392 = 0;
57 627 - 0, 07 (43101 + 43102 + 43103) - P21 - 0, 07 43101 = 0; 57 584 - 0, 07 (43101 + 43102 + 43103) - P21 - 0, 07 43102 = 0; 57 500 - 0, 07 (43101 + 43102 + 43103) - P21 - 0, 07 43103 = 0;
61 432 - 1, 66 (43111 + 43112 + 43113) - P21 - 1, 66 43111 = 0; 61 166 - 1, 66 (43111 + 43112 + 43113) - P21 - 1, 66 43112 = 0; 61 165 - 1, 66 (43111 + 43112 + 43113) - P21 - 1, 66 43113 = 0.
We solve systems with respect to quantity variables: 93H = 83 435, 2941 - 1, 4706 p21;
9312 = 85 052, 9412 - 1, 4706 p21; q313 = 88 317, 6471 - 1, 4706 p21;
q321 = 323 816, 3334 - 5, 5556 p21; q322 = 323 466, 6667 - 5, 5556 p21; q323 = 318 1 66, 6667 - 5, 5556 p21; q324 = 318 866, 667 - 5, 5556 p21; q325 = 349 216, 6667 - 5, 5556 p21; q326 = 342 016, 6667 - 5, 5556 p21; q327 = 338 816, 6667 - 5, 5556 p21; q328 = 334 416, 6667 - 5, 5556 p21;
q331 = 126 493, 75 - 2, 0833 p21; q332 = 122 656, 25 - 2, 0833 p21; q333 = 118 843, 75 - 2, 0833 p21; q334 = 124 518, 75 - 2, 0833 p21; q335 = 122 9 06, 25 - 2, 08 33 p21;
q341 = 9 642,0712 - 0,1618 p21; q342 = 9 667, 7994 - 0,1618 p21;
q351 = 22 087,1528 - 0, 3472 p21; q352 = 22 344,0972 - 0, 3472 p21; q353 = 22 378, 8194 - 0, 3472 p21;
q361 = 21 967, 2932 - 0, 3759 p21; q362 = 23 1 77, 8 1 96 - 0, 37 59 p21; q363 = 23 351, 5038 - 0, 3759 p21; q364 = 22 040, 9774 - 0, 3759 p21; q365 = 23 925,1880 - 0, 3759 p21; q366 = 22 877, 8196 - 0, 3759 p21;
q371 = 14 112, 963 - 0, 2315 p21;
9372 = 13 841,4352 - 0, 2315 p21;
q381 = 44 330,4348 - 0, 7246 p21; q382 = 44 804, 3478 - 0, 7246 p21; q383 = 43 756, 5217 - 0, 7246 p21; q384 = 45 004, 3478 - 0, 7246 p21; q385 = 43 104, 3478 - 0, 7246 p21;
q391 = 10 920, 3509 - 0,1754 p21; q392 = 10 755,0877 - 0,1754 p21;
q3101 = 206 417, 8571 - 3, 5714 p21;
q3102 = 205 803, 5714 - 3, 5714 p21;
93103 = 204 603, 5714 - 3, 5714 P21;
q3111 = 9 332,0783 - 0,1506 p21; 93112 = 9 171, 8374 - 0,1506 P21; q3111 = 9 171, 2349 - 0,1506 p21.
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