Game theory approach for supply chains optimization Текст научной статьи по специальности «Математика»

Mansur G. Gasratov1 and Victor V. Zakharov2

1 St.Petersburg State University,

Faculty of Applied Mathematics and Control Processes, Universitetskaya pl. 35, St.Petersburg, Petergoph, 198504, Russia gasratovmans@mail.ru

2 St.Petersburg State University,

Faculty of Applied Mathematics and Control Processes, Universitetskaya pl. 35, St.Petersburg, Petergoph, 198504, Russia mcvictor@mail.ru

Abstract. In this paper game theoretic mathematical models of inventory systems are treated. We consider a market, where several distributors are acting. Each distributor has warehouse for storage goods before supply to customers. Assume that demand for their goods has deterministic nature and depends on total supply or on prices of distributors. So we will consider quantitative and price competition among distributors. Distributors are considered as players in non-cooperative game. First we treat quantitative competition in context of model of Cournot. Then to consider price competition we use modified model of Bertrand. For modeling of control of inventory system we use the relaxation method of inventory regulation with admission of deficiency.

Keywords: Nash equilibrium, optimal, internal strategy, external strategy, demand, distributor.

1. Basic Model for Inventory Policy with Backordering

Let’s consider a single-product inventory control system of stocks with a deficiency assumption. We will use relaxation method of regulation of stocks to minimize long-run inventory costs (Grigoriev M.N., Dolgov A.P., Uvarov S.A., 2006), (Hedly J., Uaitin T., 1969). Assume the demand for items of product is known and uniform during a period of planning. The dynamic of inventory system is illustrated in Figure 1.

The following parameters are used to establish a mathematical model for this problem

K - fixed cost per order.

c - unit cost of procurement an item of product. h - cost per holding item in inventory during the period of planning. g - cost of being short one item during the entire period of planning. a - demand per inventory circle. y - order quantity.

S - maximal inventory level.

D - demand for the period of planning.

Variables y and S are controlled variables in the problem and may be assumed as strategies.

The usual objective of an inventory policy is to minimize cost of inventory system. To meet a given constant demand for the period of planning distributor has to

Fig. 1.

make m = D/y orders. The total cost of inventory system for the period of planning Tplan is the following

D ( S2 (y — S)2 \

L(y, S) = m.Lcycle{y, S) = — Ik + cy + h— + g------------——J (1)

2. Description of Non-cooperetive Game

Following (Vorobiev N.N., 1985) a system

r = {N, [Xi]ieN, [ni]iEN) , (2)

is called a non-cooperative game, where N = {1, 2, - set of players,

Xi - set of strategies of player i,

ni - payoff function of player iwhich provides a mapping from the set of strategies of players X = nn=i Xi to R1.

Players make an interactive decisions simultaneously choosing their strategies x,, from strategy sets X,. Vector x = (x1,x2,..., xn)is called situation in the game. As a result players are paid payoff ni = n,(x). We call situation x* = (xi,x2, ...,xn) a Nash equilibrium if for all admissible strategies x, G X,, i = 1, ...,n the following inequalities hold

ni(x*) > ni(x1 ,x2,...,x*-i,xi,x*+i,....,x*n). (3)

Theorem 1 (Kukushkin N.S., Morozov V.V., 1984). In game (4) there exists Nash equilibrium in pure strategies if for each i G N strategy set X, is compact

and convex, and payoff function ni(x) is concave with respect to x, and continuous

on X.

Assume for any i G N the function n, (x) is twice continuously differentiable with respect to x,. From (Jean Tirole, 2000) we can see that first-order necessary condition for Nash equilibrium is the following

2f^2=°,.CiV. (4)

Suppose the payoff function n,(x), i = 1, ...,n is concave for all x, G X,, that is

d2 n(x *)

dx2

< 0,,i G N. (5)

In this case solution of system (4) appears to be a Nash equilibrium in pure strategies in non-cooperative game r = {N, {X,}ieN, {n}ieN).

Let us consider two type of oligopoly: quantitave competition ( model of Kournot) and price competition (model of Bertrand) (Jean Tirole, 2000).

In the model of Kournot n players (distributors) make simultaneously interactive decisions about Q, G Q,, quantities of product to be supplied (produced for) to the market. Their cost functions Li(Qi), i = 1, ...,n are the same like in (3). Suppose Q-i = (Qi, Q2,..., Qi-1, Qi+1,..., Qn) is a vector of expected value of quantities of product of other players. Demand is defined by decreasing inverse demand function p(Qi,Q-i) = P (2 n=i Qk). Than payoff function of player i can be expressed as follows

n(Qi, Q-i) = p (Qi, Q-i) - Li(Qi). (6)

Thus we have non-cooperative game

rK = {N, {Qi}n=i, {n}n=i). (7)

In modified model of Bertrand we use demand function Q, = Di(pi,p-i)of price p, G Q assigned by player i and prices of competitors p-i = (p1,p2, ...,pi-1,pi+1, ...,pn). In this case payoff function of player i will be of the form

n,(pi,p-i) = p,Di (pi,p-i) - Li(Di(pi,p-i)). (8)

We denote the game (4) with payoff functions (8):

rB = {N, {Qi}n=i, {n}n=i). (9)

3. Game Theory Model for Inventory Decision. Quantity Competition

3.1. Statement of the Problem

Let us consider Kournot model (7). Suppose that n distributors supply uniformly product to market making decisions about quantitatis Q, and variables (yi,Si) to maximize profit (payoff functions)

n,(Qi, Q-i) = p (Qi, Q-i) Qi - Li(Qi), i = 1,..., n, (10)

where total cost Li(Qi) is the following

Li(Qi) = Li(Qi, yj, Si) = — f Ki + Cjj/j + —h 9i ^ 0- ^ • (11)

yi \ 2ai 2 ai J

Notice that demand per inventory circle a,i(-)may be constant or considered as a function of price per item of product. Let ai(p) = ai(p(Qi,Q-i)) = bi(Qi,Q-i), i = 1, ...,n. In this case cost function (11) will be in the form

T ... G \ _ ( JZ , , I, Si , „ (Vi-Si)2 \

Li(Qi, Q~i, 2/», Si) — I Ki + Cit/i + a* . . + . . I . (12)

2bi(Qi,Q-i) 2bi(Qi, Q-i))

Substituting (12) to expression (11) we get

n(Qi, Q-i) = Ui(Qi, Q-i, yi, Si) =

= p(Qu Q-i)Qi - Ki + ayi + hi 2bi(yQlQ_i) + 9i2bi(QuQ_i)]- (13)

Function (13) for each i, i = 1, ...,n is continuously differentiable on yi and S\ and concave with respect to vector of variables (yi,Si) on the set [0, to) x [0, to) when admissible vector (Qi,Q-i)is fixed. Notice also that according to (13) profit of a distributor i is influenced by supply strategy of all competitors and variables

(yi, Si) as well.

Definition 1. We will call pair (yi, Si) internal strategy and quantity Qi external strategy of player i.

Denote strategy sets of player i, i = 1,..., n by : &(1) = {Qi | Qi G [ai1^, bi1^] C

[0,to)}; ft(2) = {yi | yi G [af),bf)] C (0,to)}; fti3) = {Si 1 Si G [af),bf)] C [0, to)}. Let aij) ^ bij) for all i = 1,..., N j = 1, 2, 3.

3.2. Internal optimization problem

Considering vector of external strategies (Qi,Q-i) G &(1) = ftl1 X &21 X ... X C R+ as given player i chooses internal strategy (yi,Si) G &(2) X &(3) to miximize payoff function

IIi(Qi,Q-i,yi,Si) ^(yitSi) max, yi G ft(2\ Si G ft(3K (14)

Solving problem (14) for each i = 1, ...,n we get

y* =y*(Qi,Q-i) = \2Ki(^ + hi)bi(Qi,Q-i), (15)

V higi

St = S* (Qi, Q-i) = y hifai+ihi-)bi(Qi’Q-i)- (16)

3.3. Existence of Nash equilibrium in pure strategies

Substituting optimal internal strategy of player i (y* ,S*), i =1,..., n defined by (15) and (16) we get the following expression for payoff functions

n(Qi, Q-i, y*(Qi, Q-i), S*(Qi, Q-i)) = $i(Qi, Q-i) =

Qip(Qi, Q-i) - \ ~r~ ' /, ~ Qi°i- (17)

V hi+9i y/bi(Qi,Q-i)

These functions are only influenced by strategies (Qi, Q-i). Thus we have now noncooperative n-person game of type (7) (model of Kournot)

rK = N, [Qi}n=i, [$i}n=i) . (18)

The following theorem takes place

Theorem 2. Suppose the following conditions hold for i = 1, ..., n:

1) reverse demand function p(Qi,Q-i) is twice differentiable, decreasing and concave with respect to Qi G &( 1) for any fixed Q-i G &(1)/&(

2) function —^=2i=== is continuous on convex with respect to Qi G

for any fixed Q-i G &(1)/&( ^ or the following inequality holds (if bi(Qi,Q-i) -twice differentiable)

3Q, ^-----J > 46,(<3„ Q_,)---------^----------+ 2QMQ„ Q-,)------------------------

3) there exists Q G (ai1\bi1'))such that

r\ ^ ^ j^Kigihi bi(Qi,Q-i) l/2Qi ^ „ /1A\

P\Qi-,Q—i) ^ \ I 7 * s/2/ \ ci (19)

v gi + hi bj (Qi, Q-i)

for Qi > Q.

i

Then in game (18) there exists Nash equilibrium inpure strategies (Q1, Q2, ..., Qn),

and Q* G [ai1^, Q), i = 1, ..., n.

i

The proof of this theorem is based on the first-order (4) and second-order (5) conditions. According to the proof of the theorem Nash equilibrium is a solution of the following system

dni(Qi,Q-i)

= 0, i = 1, ...,n

dQi

or the same

9p(Qi,Q-i) n (n n s jzKigihi HQi, Q-i) l/2Qi ^ „

-Qi +P(Qi, Q-i) ~ -i I----;------------—-------------------Q = 0,

dQi

gi + hi

b3/2(Qi,Q-i)

(20)

i = 1,..., n.

Substituting solution of this system to (15) and (16) to calculate optimal internal strategies (y*,S*) where y* = y*(Q1,Q2,...,Q*n), S* = S*(Q1 ,Q2,...,Q„), we will finally get optimal distributor’s strategies U* = (Q*, y*, S1*), i = 1,...., n, which maximize profit and support Nash equilibrium in the game.

4. Game Theory Model for Inventory Decision. Price Competition

4.1. Statement of the problem

To formalize game theory model of price competition we consider model of Bertrand described in chapter 2. According to (8) profit of distributor i = 1, ...,nis expressed as follows

Hi(pi,p-i) = Di(pi,p-i)pi - Li(Di(pi,p-i)), (21)

where Li(Di(pi,p-i)) is the cost function of inventory system. From (3) we get the following expression for the cost function

Li(Di(pi,p-i)) = Li(pi,p-i,yi,Si) =

Di(pi,p-i) ( Sf (yi-Si)2\

-K-i + Ciyi + hi---\-fji--------- . (22)

yi i i i i 2ai i 2ai In a similar manner as in chapter 3 we consider demand function per inventory circle ai(-) of the form ai = bi(pi,p-i), i = 1, ...,n. Thus function (22) can be rewritten in the form

t /rw„ „ ^ _ Di(j>i,p-i) ( ^ L Sf | ^ (yi-Si)2 ^ /00^

Li(Di(pi,p—i)) I Ki ~\~ Ciyi ~\~ hi . \ ^ oi / w ' ( )

yi V 2bi(pi,p-i) 2bi(pi,p-i) J

Substituting (23) in (21) we get profit function as follows

IIi(pi,p-i) = IIi(pi,p-i,yi,Si) =

_ rw„ „ Di{Pi,P-i) f Ts , , u S? , (yi-Si)2\

— Di(pi,p-i)pi ^ {K* + c^ + h*2h{pijP_i)+^2HpijP_i))- (24)

Analogously to chapter 3 we call (yi, Si) internal strategy, pi - external strategy of distributor i, i = 1, ...,n.

Strategy sets of player i, i = 1, ...,n are as follows

&i1) = {pi I pi G K(1),b(1)] C [0, to)}; &(2) = {yi | yi G K(2),bi(2)] C (0, to)}; &(3) = {Si | Si G [ai3-1, bi3-1] C [0, to)}. Thus we have two level optimization problem for each distributor. It is also can be presented as combination of internal and external problems.

4.2. Internal optimization problem

In internal optimization problem player i chooses internal strategy (yi, Si) G &(2) X &(3) with fixed given prices of all players (pi,p-i) G &( 1) = &11) X &21) X ... X &i1^ to maximize profit function

Hi(pi,p-i,yi,Si) ^(yi,Si) max, yi G &f\ Si G &(3). (25)

Note that profit function (24) for each i is continuously differentiable with respect to yi and Si taken separately and also is convex with respect to (yi,Si) on

2) 3)

the set [0, to) x [0, to) (in particular on &( ) X &( )) when vector of prices (pi,p-i) is fixed.

Solving problem (25) we get the following internal strategies of players

Vi = ViipuP-i) = \ll^ + h^h(Pi,P-i), (26)

i i higi

^=^^^)=Vrao6<(p<’p-<)’ (27)

i =1, ..., n.

4.3. Nahs equilibrium in external problem

Substituting internal strategies (26), (27) in (24) we can find payoff functions of players in external game

ni (pi,p-i,y* (pi,p-i),S* (pi,p-i)) = $i(pi,p-i) =

= PiDi(pi,p-i) - Ci-^d=^= - Di{pi,p-i)ci, (28)

Vbi(pi,p-i)

where

^ 2Kigihi ■

Si 4 "T | ; * 1; •• •; n-

V hi + gi

Notice that functions (28) depend only on external strategies of players (pi,p-i) G

&(1) = &11) X &21) X ... X &N C R+. Thus we are getting modified Bertrand model as non-cooperative game

rs = (N, {$i }n=1 ,[&\. (29)

To calculate Nash equilibrium in this game we can be guided by the following theorem

Theorem 3. Suppose the following conditions hold for i = 1, ..., n:

1) demand function Di(pi,p-i) is differentiable, decreasing, convex with respect to pi on the set & 1) for any fixed admissible p-i and continuous on the set & 1);

2) function Di(pi,p-i)/y/bi(pi,p-i) is convex with respect to pi on the set for any fixed admissible p-i and continuous on the set & 1);

3) function piDi(pi,p-i) is cocave with respect to pi on the set &( 1) for any fixed admissible p-i.

4) if for each i = 1,..., n there exists pi G (a'i1^,^^1^ such that the following inequality holds

---w,—lbi(Pi,P-i)-Di(Pi,P-i) dp% , dDiipup-i)

Di(pi,p-i) < &--------11----------irr-------------------11-----1------z------Ci

2b3 (pi,p-i) dpi

i (30)

Then in game (29) there exists Nash equilibrium in pure strategies (p1,p2, . .. ,pN) and p* G [ai1^, p), i = 1,..., n..

As in Theorem 2 the proof of Theorem 3 is based on the first-order (4) and second-order (5) conditions.

Nash equilibrium (p1,p2,... ,p*N) can be found as a solution of the following system

dDiipi^p-i)

-----a--------Oi(Pi,P-i) + Di[pi,p-i) =

dpi

r2aDi%£,~i)bi(pi,P-i)-Di(pi,p-i)abi{f£-i) dDi(pi,p-i) .

= £i---------------------oTn------------------------------1------R---------Ci, 1=1 ,...,71.

2b3/2(pi,p-i) dpi

i (31)

In the case when period of planning is constant, that is Tj = ^ =const ,

we have

Di(j>i,p-i) Di{jpup-i) /7^7V-------------------------

/, , , = /n/ wifT = y/TiDiipup-i),

Vbi(pi,p-i) v Di(pi,p-i)/Ti

and profit function (28) will be in the form

@i(Pi,P-i) =PiDi(pi,p-i) - pi^/Di{pi,p-i) - Di(pi7p-i)ci7 (32)

where

2TiKigihi .

Pi = \ ~r~,---------> t = l,...,n.

V hi + gi

Then Nash equilibrium (p\,p2,...,p*n) could be defined as a solution of the system

dDi(pi,p-i\ , , , n , s dDi(pi,p-i) 1 dDi(pi,p-i)

—5— WP..P-. )+D,(p,,„-,) = ft—^^-7== + —^--------------------------------------------------

(33)

i =1, ..., n

if there exists rpi G ^ai1^, b, i = 1, ...,n such that

n / N ^ dDi(pi,p-i) 1 dDi(pi,p-i)

D,(P„P-,) < + ——c,

for pi > pi. Moreover p* G [ai1'^,pi), V i.

Internal strategies of players as optimal reaction to equilibrium external strategies (p\,p2,...,p*n) by the formulas

* ( 2Ki(gi + hi)bi(p*,p*_^

Vi =yi(Pi,P-i) = \ --------------v-----------------------------------, (34)

V higi

c* £,*{ * * ^ l2Ki9ibi(Pi,P*-i) ( ^

s,=s,(P„P_i) = J hi(si + hi) ,

i = 1,..., n.

5. Example of inventory system in case of price competition

Suppose that demand function Di(pi,p-i) and demand per inventory circle function bi(pi,p-i) for each distributor i has the same properties of depending on external strategies. Analogously to (Jean Tirole, 2000) we introduce these functions as follows

Pi1 Pi2 Pi,i-1 Pi,i+1 8in

rw ^ ,Pl P2 ■■■Pi-1 Pi+1 ■■■Pn Di(pi,p-i) = di--------------------j—---------------, (36)

pi '

„Pi1 rpi'i-1 rPi’i+1 nPn

b-(v. P .) - ePl P2 Pi+1 ■■■Pn (37)

ui\Pii P—i) — ^ 1+a* 5 v°V

p^ 1

where di and ei - a positive constants , ai > (3ij > 0 V j = i, i = 1,..., n.

Elasticity of distributor’s demand function Di(pi,p-i)with respect to his price is negative eii = —1 — ai < 0, i = 1,...,n, and with respect to competitors prices are positive and the following inequalities hold eij = (3ij > 0.V j = i, i = 1, ...,n So demand is uniform during the period of planning and a circle as well we notice that period of planning is equal to Di(pi,p-i)/bi(pi,p-i) = di/ei.

Let us consider internal and external problems in this case.

Internal problem. We can rewrite formulas for yi and Si in the form

Vi (pi,p-i) =

2Ki(gi + hi)eiplpi1 pf22.. .pf-i 1 pf+i+1.. .pn

higip

i+a-

Si (pi,p-i) =

2 Kigieip^p$“ .. .pfrr^f1 ...pt K{gi + hi)p]+ai

Denote Yi(p-i) = pPi1 pPi2 .. .pf-l 1 pf+i+1 .. .pnn. Than we get for i = 1,...,

*/„ „ \ 2Ki(gi + hi)ei^i(p-i)

Vi(Pi,P-i) = \l---------- 1+a<--------, (38)

h i gi p i

Qi / \ I 2Ktyiei !i(p—i) /on\

Si(Pi,P-i) = \l—--------, , , i+a, • (39)

2Kigiei'yi(p-i) hi(gi + hi)pi+

External problem. We substitute internal strategies (38), (39) in (32). Due to (36) and (37) we get

. , \ diYi(p-i) di 2KigihiYi(p-i) diCiYi(p-i)

--tjtz------------------------------(40)

Thus we state model of Bertrand for our case as non-cooperative game

rK = (n, {41}>n=i, mn=i), (41)

where i) - the set of external strategies of player i, i) = {pi\ai1'> < pi < b^}, i = 1,..., n.

One can notice that for our case the following corollary takes place

n

Corollary 1. For any ai > 0, i = 1, ...,n in game rK (41) there exists unique Nash equilibrium in pure strategies. If 0 < ai < 1, i = 1, ...,n, then payoff functions on equilibrium strategies has positive values.

Solving of the problem . It is easy to see that second-order condition for function (40) is fulfilled 9 < 0 and from the first order condition (4) we

get the following system

d<I>i(pi,p-i) _ ajdijijp-i) dj{l + oti) ZKigihijijp-i) 1 dpi p\+ai 2 y ei(gi + hi) p(3+«i)/2

+ (l + „,)^,(i,-,)c,=0i , = 1| (42)

pi 1

It can also be written in the form

i 1 ^2Kigihi (i+c(i)/2 , , n n ■ 1 /'/iQ'i

-o.iPi -A-------—* —--------- -----rp\ " + (1 + on)Ci = 0, t = l(43)

2 V ei(gi + hi)Yi(p-i)

Denoting

1 + ai / 2Kigihi

(44)

2 Y ei(gi + hi)

we get system

iiP^~ = 'Jpi^P^2 • • -Pi-i^Pi+i’1 • • -Pnn (ompi - (1 + a.i)ci), i = 1,

This system can be only solved by numerical methods.

Let us consider game of two players i.e. n = 2. For this case we will have the following system to find Nash equilibrium

“1+1 r~z

ZlPl 2 - \JP212 (alPl - («1 + l)ci) = 0,

<*2 + 1 r~R V )

&P2 2 ~ yPl ia2P2 ~ («2 + 1)C2) = 0.

Let parameters of the model are as follows

Ki = 400 USD, ci = 10 USD, hi = 10 USD/h, di = 100000, ei = 10000, gi = 5 USD/h, ai = 1/2, /3i2 = 1/4, K2 = 400 USD, c2 =8 USD, h2 = 8 USD/h, d2 = 100000, e2 = 10000, g2 =6 USD/h, a2 = 1/2, 32i = 1/4.

System (45) will be of the form

0, 3872983344pf - p\ - 15) = 0,

,2'

0, 3927922024p| - pf (±p2 - 12) = 0.

With the solution pi = 37, 68643585 USD and p2 = 30, 47300925 USD. These prices are values of external strategies in Nash equilibrium. Profit of distributors for this situation are <Fi (p|,p2) = 26744, 43112 and &2 (p|,p2) = 22422, 48488.

Using (38) and (39) we also calculate internal strategies y|, Si ? y|, S|: y| «

156, Si « 52 and y*2 « 185, S2* « 79.

6. Conclusion

In this paper we have discussed two models of control of inventory systems in case of quantitative and price competition.

For each model necessary and sufficient conditions for existence of Nash equilibrium in pure strategies are proposed. Methods for finding Nash equilibrium in pure strategies are discussed.

Using these methods it is possible to get analytical expressions for internal and external strategies in deterministic models. When it is not the case we can find solutions numerically.

References

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Vorobiev, N. N. (1985). The Game Theory for The economist-kibernetik. M.: Nauka, 272 p.

Kukushkin, N. S., Morozov, V. V. (1984). The nonantagonistic Game Theory. M.: Moscow State University, 103 p.

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