Page 187 - DCOM303_DMGT504_OPERATION_RESEARCH

Operations Research

Notes Linear Programming Method

When 2 person game has more than two options we can make use of linear programming
method to establish the opposing player’s strategies. Let A and B be two players with m and n
options respectively then the payoff matrix is given by,
Player B
b 1 b 2……….. b n
A 1 a 11 a 12…………..a ln
a 2 a 21 a 22 ………….a 2n
Player A . .
. .
. .
. .

a m a ml a m2………….a mn
The problem is to find the fixed optimal strategies. Let x , x … X and y , y ……..Yn denote the
1 2 m 1 2
probability that the player A chooses the strategies a , a ……….a and B chooses the strategies
1 2 m
b , b ………..b such that     and if the value of the game is V, then it is
1 2 n
 
possible to formulate LPP as below.
Maximise V = Expected payoff
Subject to a X + a X + ……….. a X  V
11 1 21 2 m1 m
a X + a X + ……….. am X  V
12 1 22 2 2 m
.
.
. (1)
.
.

a X + a X + ………….. + a X  V
ln 1 2n 2 mn m
X X X ……………….. X  0
1 2 3 m
Objective function can be expressed as,

Max. V = Min. (V > 0)

= Min.

1
1
1
= (x + x …………… + x ) where x = V = 1 to m.
1
1 2 m 1

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