A Graphical Method for Solving Interval Matrix Games

and Applied Analysis 3 Let a and b be two cost intervals, and the minimum cost interval is to be chosen. i If the decision maker DM is optimistic, then he/she will prefer the interval with maximum width along with the risk of more uncertainty giving less importance. ii If DM is pessimistic, then he/she will pay more attention on more uncertainty. That is, on the right hand points of the intervals, and he/she will choose the interval with minimum width. The case will be reverse when a and b represent profit intervals. Therefore, we can define the ranking order of a and b as a ∨ b ⎧ ⎨ ⎩ a, if the DM is optimistic, b, if the DM is pessimistic. 1.5 Here, the notation “ ∨ ” denotes the maximum among the interval numbers a and b. Similarly, we can write a ∧ b ⎧ ⎨ ⎩ b, if the DM is optimistic, a, if the DM is pessimistic. 1.6 Likewise, the notation “ ∧ ” denotes the minimum among the interval numbers a and b. 1.2.3. Partially Overlapping Intervals The above-mentioned order relations cannot explain ranking between two overlapping closed intervals. Here, we use the acceptability index idea suggested by 8, 10 . This comparison method is mainly based on the mid-points of the intervals. Let IR be the set of all interval numbers. The function A : IR × IR −→ R, A a ≺ b m b −m a r a r b , 1.7 r a r b / 0 is called acceptability function. For m b ≥ m a , the number A a ≺ b is called the grade of acceptability of the interval number a to be inferior to the second interval number b. By the definition of A, for any interval numbers a and b, we have i A a ≺ b ≥ 1 for m b > m a and a ≤ b−, ii A a ≺ b ∈ 0, 1 for m b > m a and b− ≤ a , iii A a ≺ b 0 for m b m a . In this case, if r a r b , then a is identical with b. If r a / r b , then the intervals a and b are noninferior to each other. In this case, the acceptability index becomes insignificant, so DM has to negotiate with the widths of a and b. 4 Abstract and Applied Analysis Table 1 Optimistic outlook Pessimistic outlook Profit intervals DM will prefer the interval a instead of b, because DM will pay more attention on the highest possible profit of 18 unit ignoring the risk of minimum profit of 2 unit. DM will prefer the interval b, because his/her attention will be drawn to the fact that the minimum profit of 5 unit will never be decreased. Cost intervals DM will pay more attention on the minimum cost of 2 unit, that is, the left hand points both of the cost intervals a and b, and select a instead of b. DM will again prefer the interval b, because DM will pay more attention on the maximum cost of 15 unit will never be increased. Let a and b be two cost intervals, and minimum cost interval is to be chosen. If the DM is optimistic, then he/she will prefer the interval with maximum width along with the risk of more uncertainty giving less importance. Similarly, if the DM is pessimistic, then he/she will pay more attention on more uncertainty, that is, on the right end points of the intervals, and will choose the interval with minimum width. The case will be reverse when a and b represent profit intervals. Example 1.1. Let a −5,−3 and b 2, 6 be two intervals. Then, A a ≺ b 4 − −4 1 2 8 3 > 1. 1.8 Thus, the DM accept the decision that a is less than b with full satisfaction. Example 1.2. Let a 2, 8 and b 1, 15 be two intervals. Then, A a ≺ b 8 − 5 3 7 0.3 ∈ 0, 1 . 1.9 Hence, the DM accept the decision that a is less than bwith grade of satisfaction 0.3. Example 1.3. Let a 2, 18 and b 5, 15 be two intervals. Then, A a ≺ b 0. Here, both of the intervals are noninferior to each other. In this case, the DM has to negotiate with the widths of a and b as listed inTable 1 . These can be written explicitly as a ∨ b ⎧ ⎪⎨ ⎪⎩ b, if A a ≺ b > 0, a, if A a ≺ b 0, r a < r b , DM is pessimistic, b, if A a ≺ b 0, r a < r b , DM is optimistic, 1.10 Abstract and Applied Analysis 5 where, a ∨ b denotes the maximum of the interval numbers a and b. Similarly,and Applied Analysis 5 where, a ∨ b denotes the maximum of the interval numbers a and b. Similarly, a ∧ b ⎧ ⎪⎨ ⎪⎩ b, if A b ≺ a > 0, a, if A b ≺ a 0, r a > r b , DM is optimistic, b, if A b ≺ a 0, r a > r b , DM is pessimistic. 1.11 Here, the notation a ∧ b represents the minimum of the interval numbers a and b. Proposition 1.4. The interval ordering by the acceptability index defines a partial order relation on IR. Proof. i If A a ≺ b 0 and r a r b , then a ≡ b, and we say that a and b are noninferior to each other. Hence, it is reflexive. ii For any interval number a and b, A a ≺ b > 0, and A b ≺ a > 0 implies a b. Therefore, A is antisymmetric. iii For any interval numbers a,b, c ∈ IR if A a ≺ b m b −m a r b r a ≥ 0, A b ≺ c m c −m b r c r b ≥ 0, 1.12 then m c −m a m c −m b m b −m a ≥ 0. Hence, A a ≺ c m c −m a r c r a ≥ 0. 1.13 Thus, A is transitive. On the other hand, acceptability index must not interpreted as difference operator of real analysis. Indeed, whileA a ≺ b ≥ 0 andA b ≺ c ≥ 0, the inequality A a ≺ c ≥ max{A a ≺ b ,A b ≺ c } 1.14 may not hold. Actually, let a −2, 2 , b −1, 21/20 and c −1/2, 3/4 , then A a ≺ b 1 121 , A b ≺ c 2 33 , 1.15 but A a ≺ c 1 21 < 2 33 . 1.16 For any two interval numbers a and b from IR, either A a ≺ b > 0, or A a ≺ b A b ≺ a 0, or A b ≺ a > 0. Also, A a ≺ b < 0 can be interpreted as the interval number b is inferior to the interval number a, since A b ≺ a > 0. 6 Abstract and Applied Analysis Additionally, for any interval numbers a, b, c, and d, the following properties of acceptability index is obvious. If A a ≺ b ≥ 0, A c ≺ d ≥ 0 ⇒ A a c ≺ b d ≥ 0, A a ≺ b ≥ 0 ⇒ A a c ≺ b c ≥ 0, A a ≺ b ≥ 0 ⇒ A −1 · b ≺ −1 · a ≥ 0. 1.17 1.3. Matrix Games Game theory is a mathematical discipline which studies situations of competition and cooperation between several involved parties, and it has many applications in broad areas such as strategic warfares, economic or social problems, animal behaviours, political voting systems. It is accepted that game theory starts with the von Neumann’s study on zero-sum games see 11 , in which he proved the famous minimax theorem for zero-sum games. It was also basis for 12 . The simplest game is finite, two-person, zero-sum game. There are only two players, player I and player II, and it can be denoted by a matrix. Thus, such a game is called matrix game. More formally, a matrix game is an m × n matrix G of real numbers. A mixed strategy of player I is a probability distribution x over the rows of G, that is, an element of the set Xm { x x1, . . . , xm ∈ R : xi ≥ 0, m ∑