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Asia-Pacific Journal of Operational Research Vol. 28, No. 6 (2011) 705–737 c World Scientific Publishing Co. & Operational Research Society of Singapore DOI: 10.1142/S021759591100351X
NOTES ON “LINEAR PROGRAMMING TECHNIQUE TO SOLVE TWO-PERSON MATRIX GAMES WITH INTERVAL PAY-OFFS”
DENG-FENG LI∗ School of Management, Fuzhou University No. 2, Xueyuan Road, Daxue New District Fuzhou District, Fuzhou, Fujian 350108, China
[email protected] [email protected]
The aim of this note is to point out and correct some vital mistakes in the paper by P K Nayak and M Pal, “Linear programming technique to solve two person matrix (games with interval pay-offs). Asia-Pacific Journal of Operational Research, 26(2), 285– 305”. Lots of serious mistakes on the definitions, conclusions, models, methods, proofs and computing results have been corrected and modified in this note. We also indicate inappropriate formulations regarding their proposed linear programming models for solving generic matrix games with interval pay-offs and suggest a pair of linear programming models with any minimal acceptance degree of the interval inequality constraints which may be allowed to violate. The lexicographic method is suggested so that a rational and credible solution of the generic matrix game with interval pay-offs can be achieved. Keywords: Mathematical programming; interval matrix game; saddle point; interval computation; fuzzy number.
1. Introduction Recently, Nayak and Pal (2009) investigated two-person matrix games with interval pay-offs, which are called “interval matrix games” for short. However, it was found that there were lots of vital mistakes on the definitions, conclusions, models, methods, proofs and computing results, which will be shown in this note. Revised definitions and corrected methods, models and conclusions are presented and justified. The rest of this note is organized as follows. Section 2 briefly reviews some necessary concepts and notations for sequent discussions. Section 3 shows that 2 × 2 interval matrix games are not always strictly determined. Modified definitions and corrected conclusions are presented for generic m × n interval matrix games. Section 4 shows that the method for 2 × 2 interval matrix games without saddle
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points (Nayak and Pal, 2009) are not always effective. A generic conclusion for any interval matrix game is presented. Corrected conclusion for Theorem 4.4 given by Nayak and Pal (2009) and proof are given in Sec. 5. Section 6 presents corrected linear programming models for generic interval matrix games and corrected computing results of the numerical example given by Nayak and Pal (2009). In Sec. 7, the acceptability index used by Nayak and Pal (2009) is modified. Hereby, for any minimal acceptance degree of the interval inequality constraints being allowed to violate, we construct a pair of bi-objective linear programming models which are solved through using the lexicographic method. This note concludes with some remarks in Sec. 8.
2. Some Concepts and Notations The notations, symbols, variables and expressions in Nayak and Pal (2009) are extremely confused, inconsistent and illogical. In this note, all these are unified to facilitate the sequent discussions and the reading for the reader. Therefore, this note will mainly focus on pointing out and correcting the mistakes on the definitions, conclusions, models, methods, proofs and computing results rather than the letters, symbols and notations. Let be the set of all real numbers. An interval may be expressed as a ¯ = [aL , aR ] = {a | aL ≤ a ≤ aR }, where aL ∈ and aR ∈ are called the lower and upper limits of the interval a ¯, respectively. If aL = aR then a ¯ = [aL , aR ] is ¯ reduced to a real number a, where a = aL = aR . Alternatively, an interval a can be expressed in mean-width or center-radius form as a ¯ = m(¯ a), w(¯ a), where a) = (aR − aL )/2 are the mid-point and half-width of m(¯ a) = (aL + aR )/2 and w(¯ the interval a ¯, respectively. The set of all intervals in is denoted by I(). Let a ¯ = m(¯ a), w(¯ a) and ¯b = m(¯b), w(¯b) be two intervals. For m(¯ a) ≤ m(¯b) ¯b is defined as follows and w(¯ a)+w(¯b) = 0, an acceptability index to the premise a ¯≤ ∼ (Nayak and Pal, 2009): ¯ a) ¯b) = m(b) − m(¯ , ψ(¯ a≤ ∼ ¯ w(b) + w(¯ a)
(2.1)
which is the value judgment or satisfaction degree of the decision maker (DM) or Player that the interval a ¯ is not superior to the interval ¯b (or ¯b is not inferior to a ¯) in terms of value. Here, “not inferior to” and “not superior to” are analogous to “not less than” and “not greater than” in the real number set , respectively. ” is defined. Thus, the orderings for two intervals a ¯ and ¯b Similarly, the symbol “≥ ∼ are defined as follows (Nayak and Pal, 2009): ¯ b ¯ a ¯ ∨ ¯b = a ¯b
¯b) > 0 if ψ(¯ a≤ ∼ ¯b) = 0 and w(¯ if ψ(¯ a≤ a) < w(¯b) and DM is pessimistic ∼ ¯b) = 0 and w(¯ if ψ(¯ a≤ a) < w(¯b) and DM is optimistic ∼
(2.2)
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¯ b if ψ(¯b ≤ a ¯) > 0 ∼ ¯ if ψ(¯b ≤ a ¯) = 0 and w(¯ a) > w(¯b) and DM is pessimistic a ¯ ∧ ¯b = a ∼ ¯b if ψ(¯b ≤ a ¯) = 0 and w(¯ a) > w(¯b) and DM is optimistic. ∼
707
(2.3)
In the sequent discussions, the max operator “∨” in Eq. (2.2) and the min operator “∧” in Eq. (2.3) are meant to be in the sense of Eq. (2.1) unless specially stated. 3. Modified Definitions and Corrected Conclusions Let Ai (i = 1, 2, . . . , m) and Bj (j = 1, 2, . . . , n) be pure strategies for Players (or DMs) A and B, respectively. If Player A adopts the pure strategy Ai (i.e., the row i) and Player B adopts pure strategy Bj (i.e., the column j), then the pay-off for Player A is expressed with a ¯ij = [aLij , aRij ], which is an interval on the real number set . The interval pay-off matrix of a matrix game is concisely expressed in the matrix form as follows:
¯ = (¯ D aij )m×n
A1 A2 = . .. Am
B1
B2
···
Bn
[aL11 , aR11 ] [aL21 , aR21 ] .. .
[aL12 , aR12 ] [aL22 , aR22 ] .. .
··· ···
[aL1n , aR1n ] [aL2n , aR2n ] .. .
··· [aLm1 , aRm1 ] [aLm2 , aRm2 ] · · ·
,
[aLmn , aRmn ]
which is referred to the interval matrix game. In the following, an interval matrix ¯ is called as the interval matrix game D ¯ for game with the interval pay-off matrix D short. In Nayak and Pal (2009), the position (k, r) of the interval pay-off matrix ¯ was defined as a saddle point of the interval matrix game D ¯ if D 1≤i≤m { 1≤j≤n {[aLij , aRij ]}} and 1≤j≤n { 1≤i≤m {[aLij , aRij ]}} exist and are equal, i.e.,
[aLkr , aRkr ] = {[aLij , aRij ]} = {[aLij , aRij ]} . 1≤i≤m
1≤j≤n
1≤j≤n
1≤i≤m
¯ is strictly determined if it has a saddle point. The claim An interval matrix game D ¯ is strictly determined (Nayak and Pal, 2009) is not that any interval matrix game D always correct. The reason is that some interval matrix games may have no saddle points in the sense of the pure strategies even if both 1≤i≤m { 1≤j≤n {[aLij , aRij ]}} and 1≤j≤n { 1≤i≤m {[aLij , aRij ]}} exist whereas they may not be always equal. Example 3.1. Let’s consider the 2 × 2 interval matrix game with the interval pay-off matrix as follows (Nayak and Pal, 2009):
¯ 1 = ([aLij , aRij ])2×2 = D
A1 A2
B1
B2
[−1, 1]
[1, 3]
[0, 2]
[−2, 0]
.
(3.1)
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Obviously, 1≤i≤2 { 1≤j≤2 {[aLij , aRij ]}} = [aL11 , aR11 ] = [−1, 1] and 1≤j≤2 { 1≤i≤2 {[aLij , aRij ]}} = [aL21 , aR21 ] = [0, 2]. Hence, 1≤i≤2 { 1≤j≤2 {[aLij , ¯1 aRij ]}} = 1≤j≤2 { 1≤i≤2 {[aLij , aRij ]}}. Therefore, the interval matrix game D has no saddle points, which also was pointed out by Nayak and Pal themselves ¯ 1 is not strictly determined. (2009). Thus, the interval matrix game D
n Let m + and + be the m-dimensional and n-dimensional nonnegative Euclidean spaces, respectively. Denote x = (x1 , x2 , . . . , xm )T and y = (y1 , y2 , . . . , yn )T , respectively, where the symbol “T” denotes the transpose of a vector. Strategy spaces for Players A and B are denoted as SA = {x | x ∈ m + , xi ∈ m [0, 1] (i = 1, 2, . . . , m), i=1 xi = 1} and SB = {y | y ∈ n+ , yj ∈ [0, 1] (j = n 1, 2, . . . , n), j=1 yj = 1}, respectively. Vectors x ∈ SA and y ∈ SB are called mixed strategies of Players A and B, respectively. Definition 4.5 introduced by Nayak and Pal (2009:296) is incorrect in that there does not always exist a triad system (x∗ , y∗ , v¯) ∈ SA × SB × I() such that it ¯ ≥ v¯ and xT Dy ¯ ∗ ≤ v¯ for any strategies x ∈ SA and y ∈ SB , satisfies both x∗T Dy ∼ ∼ where v¯ = [vL , vR ] is an interval on the real number set . Thus, we introduce the ¯ as follows. concept of solutions of an interval matrix game D
Definition 3.1. Let υ¯ = [υL , υR ] and ω ¯ = [ωL , ωR ] be two intervals on the real number set . Assume that there exist strategies x∗ ∈ SA and y∗ ∈ SB . If for ¯ ≥ υ¯ and ¯ ) satisfies both x∗T Dy any strategies x ∈ SA and y ∈ SB , (x∗ , y∗ , υ¯, ω ∼ ∗ ∗ ¯ ∗ ≤ω ¯ , then (x , y , υ ¯ , ω ¯ ) is called a reasonable solution of the interval matrix xT Dy ∼ ¯ game D; υ¯ and ω ¯ are called reasonable values for Players A and B, respectively; x∗ ∗ and y are called reasonable strategies for Players A and B, respectively. Let U and W be the sets of reasonable values υ¯ and ω ¯ for Players A and B, respectively. It is worth noticing that Definition 3.1 only gives the notion of a reasonable solution rather than the notion of an optimal solution. In other words, the reasonable solution is not the solution of the interval matrix game, which is given as follows. ¯∗ ∈ Definition 3.2. Assume that there exist two reasonable values υ¯∗ ∈ U and ω ∗ υ = υ¯ ) and ω ¯ ∈ W (¯ ω = ω ¯ ∗) W . If there do not exist reasonable values υ¯ ∈ U (¯ ∗ ∗ ∗ ∗ ∗ ∗ such that they satisfy both υ¯ ≥ υ¯ and ω ¯ ≤ ω ¯ , then (x , y , υ¯ , ω ¯ ) is called a ∼ ∼ ∗ ¯ solution of the interval matrix game D; x is called an optimal (or a maximin) strategy for Player A and y∗ is called an optimal (or a minimax) strategy for ¯ ∗ are called Player A’s gain-floor and Player B’s loss-ceiling, Player B; υ¯∗ and ω respectively. Theorem 4.3 given by Nayak and Pal (2009:296) is wrong since x∈SA { y∈SB ¯ ¯ and y∈SB { x∈SA {xT Dy}} are not always equal although both of them {xT Dy}} exist.
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The corrected conclusion can be summarized as in Theorem 3.1. ¯ ¯ and y∈SB { x∈SA {xT Dy}} exist Theorem 3.1. Both x∈SA { y∈SB {xT Dy}} and the following inequality is valid
T¯ T¯ (3.2) {x Dy} ≤ {x Dy} . ∼ x∈SA
y∈SB
y∈SB
x∈SA
Proof. Obviously, for any strategies x ∈ SA and y ∈ SB , we have
¯ ¯ ≤ ¯ ≤ {xT Dy} xT Dy {xT Dy}. ∼ ∼ y∈SB
x∈SA
Hence, it follows that
x∈SA
¯ {xT Dy}
y∈SB
≤ ∼
¯ {xT Dy}.
x∈SA
Therefore, we obtain
x∈SA
y∈SB
T
¯ {x Dy}
≤ ∼
y∈SB
T
¯ . {x Dy}
x∈SA
Thus, the proof of Theorem 3.1 has been completed. It is easily seen from Definitions 3.1 and 3.2 that υ¯∗ = ¯ Then, we have and ω ¯ ∗ = y∈SB { x∈SA {xT Dy}}. ω ¯ ∗. υ¯∗ ≤ ∼
x∈SA {
T¯ y∈SB {x Dy}}
(3.3)
Thus, Theorem 3.1 means that Player A’s gain-floor “essentially cannot exceed” player B’s loss-ceiling. It is worthwhile to note that Eq. (3.3) is strictly valid for some inter¯ ∗ . For example, it is easily seen that val matrix games, i.e., υ¯∗ ω 1≤i≤2 { 1≤j≤2 {[aLij , aRij ]}} = [−1, 1] [0, 2] = 1≤j≤2 { 1≤i≤2 {[aLij , aRij ]}} ¯ 1 in Eq. (3.1). for the 2 × 2 interval matrix game with the interval pay-off matrix D (It is worthwhile noticing that pure strategies are special cases of mixed strategies and the pay-offs are intervals.) In other words, there may not always exist an ¯ ≥ v¯ and xT Dy ¯ ∗ ≤ v¯ for any interval v¯ ∈ I() such that it satisfies both x∗T Dy ∼ ∼ strategies x ∈ SA and y ∈ SB . That is to say, Definition 4.5 given by Nayak and Pal (2009) is not always rational. Hereby, Definitions 3.1 and 3.2 are suggested to replace Definition 4.5 as the concept of solutions of the interval matrix games. It is easily seen that SA and SB are finite, compact and convex sets. Hence, T¯ T¯ y∈SB {x Dy} and x∈SA {x Dy} will be attained at extreme points of the
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strategy spaces SA and SB , respectively. Thus, for any strategies x ∈ SA and y ∈ SB , we have m
¯ = {xT Dy} [aLij , aRij ]xi x∈SA
and y∈SB
y∈SB
x∈SA
¯ {xT Dy}
x∈SA
=
1≤j≤n
i=1
n y∈SB
1≤i≤m
[aLij , aRij ]yj
j=1
,
respectively. m ¯ = [ωL , ωR ] = Denote υ¯ = [υL , υR ] = 1≤j≤n { i=1 [aLij , aRij ]xi } and ω n { [a , a ]y }. Then, the maximin value for Player A and the miniLij Rij j 1≤i≤m j=1 max value for Player B can be obtained through solving the interval mathematical programming models as follows: max{¯ υ} m [aLij , aRij ]xi ≥ υ¯ (j = 1, 2, . . . , n) ∼ i=1 m s.t. xi = 1 i=1 xi ≥ 0 (i = 1, 2, . . . , m)
(3.4)
min{¯ ω} n [aLij , aRij ]yj ≤ ω ¯ (i = 1, 2, . . . , m) ∼ j=1 n yj = 1 j=1 yj ≥ 0 (j = 1, 2, . . . , n),
(3.5)
and
respectively. Nayak and Pal (2009) asserted that Eqs. (3.4) and (3.5) can be converted into a primal-dual pair of crisp equivalent linear programming models, which have optimal solutions x∗ ∈ SA , y∗ ∈ SB and their common value v¯∗ ∈ I(), respectively. Thus, Nayak and Pal (2009) rewrote Eqs. (3.4) and (3.5) as follows: m i=1
[aLij , aRij ]x∗i ≥ v¯∗ ∼
(j = 1, 2, . . . , n)
(3.6)
[aLij , aRij ]yj∗ ≤ v¯∗ ∼
(i = 1, 2, . . . , m),
(3.7)
and n j=1
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respectively. Namely, ¯ ≥ v¯∗ x∗T Dy ∼
for any strategy y ∈ SB
(3.8)
¯ ∗ ≤ v¯∗ xT Dy ∼
for any strategy x ∈ SA .
(3.9)
and
However, as stated earlier, Nayak and Pal’s assertion and the models (i.e., Eqs. (3.6)–(3.9)) are incorrect according to Theorem 3.1. 4. Corrected Methods for 2 × 2 Interval Matrix Games Without Saddle Points Assume that the interval pay-off matrix of the 2 × 2 matrix game without saddle points is given as follows: ¯ = ([aLij , aRij ])2×2 = A1 D A2
B1 B2 [aL11 , aR11 ] [aL12 , aR12 ] . [aL21 , aR21 ] [aR22 , aR22 ]
(4.1)
Let λij = aRij − aLij
(i, j = 1, 2),
(4.2)
where all values λij are required to be positive, i.e., λij > 0. Thus, the interval ¯ in Eq. (4.1) is normalized as follows: pay-off matrix D B1 L11 aR11 A1 , λ11 λ11 ¯ D = aL21 aR21 , A2 λ21 λ21 a
B2 aL12 aR12 , λ12 λ12 , aR22 aR22 , λ22 λ22
(4.3)
¯ = ([bLij , bRij ])2×2 , where which is denoted by D aLij aRij [bLij , bRij ] = , (i, j = 1, 2). λij λij
(4.4)
Using the similar method for solving the classical 2 × 2 matrix games without saddle points, optimal strategies for Players A and B and the value of the interval ¯ in Eq. (4.3) were obtained by Nayak and Pal (2009) as follows: matrix game D T bL22 − bL21 bL11 − bL12 , , (4.5) x∗ = (x1 , x2 )T = bL11 + bL22 − bL12 − bL21 bL11 + bL22 − bL12 − bL21 y∗ = (y1 , y2 )T = and
bL22 − bL12 bL11 − bL21 , bL11 + bL22 − bL12 − bL21 bL11 + bL22 − bL12 − bL21
T
bL11 bL22 − b12 b21 bR11 bR22 − bR12 bR21 , , v¯ = [vL , vR ] = bL11 + bL22 − bL12 − bL21 bR11 + bR22 − bR12 − bR21 ∗
(4.6)
(4.7)
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respectively. However, it is found that Eqs. (4.5) and (4.6) could provide optimal ¯ = ([aLij , aRij ])2×2 strategies for Players A and B in the 2×2 interval matrix game D in Eq. (4.1) only if its intervals [aLij , aRij ] satisfy the conditions: aRij = aLij + µ (i, j = 1, 2), i.e., all the intervals are of the same length µ. In other words, the solution provided by Eqs. (4.5)–(4.7) is a solution to the 2 × 2 interval matrix game ¯ in Eq. (4.3) whereas it is not always a solution to the 2 × 2 interval matrix game D ¯ D in Eq. (4.1). Therefore, Eqs. (4.5)–(4.7) are not generic formulae for solving any ¯ = ([aLij , aRij ])2×2 . 2 × 2 interval matrix game D Example 4.1. Let’s consider the 2 × 2 interval matrix game with the interval pay-off matrix as follows:
¯ 2 = ([aLij , aRij ])2×2 = D
A1
A2
B1
B2
[2, 3] [6, 16]
.
[3, 6] [4, 14]
(4.8)
It is easily verified that 1≤i≤2 { 1≤j≤2 {[aLij , aRij ]}} = 1≤j≤2 { 1≤i≤2 {[aLij , ¯ 2 has a saddle aRij ]}} = [aL21 , aR21 ] = [3, 6]. Therefore, the interval matrix game D point (2, 1), i.e., the optimal strategies for Players A and B are the pure strategies ¯ 2 is v¯∗ = [3, 6]. A2 and B1 , respectively. The value of the interval matrix game D ¯ 2 is normalized as follows: Using Eq. (4.4), the interval pay-off matrix D
¯ D 2
= ([bLij , bRij ])2×2 =
A1
A2
B1
B2
[2, 3] [0.6, 1.6]
.
[1, 2] [0.4, 1.4]
(4.9)
¯ 2 has a saddle point (1, 2) since It is easily seen that the interval matrix game D 1≤i≤2 { 1≤j≤2 {[bLij , bRij ]}} = 1≤j≤2 { 1≤i≤2 {[bLij , bRij ]}} = [bL12 , bR12 ] = [0.6, 1.6]. Namely, the optimal strategies for Players A and B are the pure strategies A1 and B2 , respectively, which are remarkably different from those for Players A ¯ is ¯ 2 . The value of the interval matrix game D and B in the interval matrix game D 2 ∗ v¯ = [0.6, 1.6] which corresponds to the interval [aL12 , aR12 ] = [6, 16] in the original ¯ 2 . As a result, the interval matrix game D ¯ 2 has two different interval matrix game D values. This shows that [aL12 , aR12 ] = [6, 16] obtained through using Eq. (4.4) is not ¯ 2 . In other words, the transform method the value of the interval matrix game D proposed by Nayak and Pal (2009) is not always effective for solving interval matrix games. Example 4.2. Let’s consider the 2 × 2 interval matrix game with the interval pay-off matrix as follows:
¯ 3 = ([aLij , aRij ])2×2 = D
A1 A2
B1
B2
[2, 3]
[6, 16]
[4, 14]
[3, 9]
.
(4.10)
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It is easily verified that 1≤i≤2 { 1≤j≤2 {[aLij , aRij ]}} = [aL22 , aR22 ] = [3, 9] = 1≤j≤2 { 1≤i≤2 {[aLij , aRij ]}} = [aL21 , aR21 ] = [4, 14]. Therefore, the interval ¯ 3 has no saddle points in the sense of the pure strategies. matrix game D ¯ 3 is normalized as follows: Using Eq. (4.4), the interval pay-off matrix D
¯ 3 = ([bLij , bRij ])2×2 = D
A1 A2
B1 [2, 3]
B2 [0.6, 1.6]
[0.4, 1.4] [0.5, 1.5]
.
(4.11)
¯ has a saddle point (1, 2) since It is easily seen that the interval matrix game D 3 1≤i≤2 { 1≤j≤2 {[bLij , bRij ]}} = 1≤j≤2 { 1≤i≤2 {[bLij , bRij ]}} = [bL12 , bR12 ] = [0.6, 1.6]. Namely, the optimal strategies for Players A and B are the pure strategies A1 and B2 , respectively. As stated earlier, the method for solving the 2 × 2 interval matrix game with the ¯ = ([aLij , aRij ])2×2 can be effective and meaningful only interval pay-off matrix D if all the following conditions are satisfied: µ > 0 and aRij = aLij + µ (i, j = 1, 2). In other words, the method proposed by Nayak and Pal (2009) is not effective and meaningful if either µ = 0 or aRij = aLij + µ (i, j = 1, 2). That is to say, the method is not effective if one of the values λij = aRij −aLij is not equal to the same constant, i.e., the intervals are not of the same length. In this case, Eq. (4.4) cannot ensure ¯ = ([bLij , bRij ])2×2 ¯ = ([aLij , aRij ])2×2 and D that the 2 × 2 interval matrix games D are equivalent. However, we can draw a generic conclusion for any interval matrix game. Theorem 4.1. Assume that there are two interval matrix games, whose interval ¯ = ([a , a ])m×n , ¯ = ([aLij , aRij ])m×n and D pay-off matrices are given as D Lij Rij respectively, where [aLij , aRij ] = [cL , cR ][aLij , aRij ] = [cL aLij , cR aRij ] and [cL , cR ] ¯ and D ¯ have the is a positive interval, i.e., cL > 0. The interval matrix games D υ∗, ω ¯ ∗ = [cL , cR ]¯ ω∗, same optimal strategies for Players A and B, and υ¯∗ = [cL , cR ]¯ ∗ ∗ ¯ where υ¯ and υ¯ are Player A’s gain-floors in the interval matrix games D and ¯ , ω ¯ and D ¯ , respectively. D ¯ ∗ and ω ¯ ∗ are Player B’s loss-ceilings in D Proof. Since [cL , cR ] is a positive interval, it is easily derived from Eqs. (3.4) and (3.5) that max{[cL , cR ]¯ υ} m [cL , cR ][aLij , aRij ]xi ≥ [cL , cR ]¯ υ ∼ i=1 m s.t. xi = 1 i=1 xi ≥ 0 (i = 1, 2, . . . , m)
(j = 1, 2, . . . , n)
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and min{[cL , cR ]¯ ω} n [cL , cR ][aLij , aRij ]yj ≤ [cL , cR ]¯ ω ∼ j=1 n s.t. yj = 1 j=1 yj ≥ 0 (j = 1, 2, . . . , n),
(i = 1, 2, . . . , m)
respectively. Namely, υ} max{[cL , cR ]¯ m [cL aLij , cR aRij ]xi ≥ [cL , cR ]¯ υ ∼ i=1 m s.t. xi = 1 i=1 xi ≥ 0 (i = 1, 2, . . . , m)
(j = 1, 2, . . . , n)
and min{[cL , cR ]¯ ω} n [cL aLij , cR aRij ]yj ≤ [cL , cR ]¯ ω ∼ j=1 n s.t. yj = 1 j=1 yj ≥ 0 (j = 1, 2, . . . , n),
(i = 1, 2, . . . , m)
which are just about a pair of interval mathematical programming models for the ¯ . Thus, the proof of Theorem 4.1 has been completed. interval matrix game D Obviously, the assertion given by Nayak and Pal (Nayak and Pal, 2009:300) is only a special case of Theorem 4.1, i.e., [cL , cR ] = 1/µ, where µ is a positive number. ¯ = ([aLij , aRij ])2×2 , if its Corollary 4.1. For the 2 × 2 interval matrix game D intervals [aLij , aRij ] satisfy the conditions: aRij = aLij + µ (i, j = 1, 2), where µ > 0, then the optimal strategies for Players A and B and the value of the interval matrix game are obtained as follows: x∗ = (x∗1 , x∗2 )T =
aL22 − aL21 aL11 − aL12 , aL11 + aL22 − aL12 − aL21 aL11 + aL22 − aL12 − aL21
T
,
(4.12)
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y∗ = (y1∗ , y2∗ )T =
aL22 − aL12 aL11 − aL21 , aL11 + aL22 − aL12 − aL21 aL11 + aL22 − aL12 − aL21
T (4.13)
and ∗
v¯ =
∗ ∗ [vL , vR ]
aL11 aL22 − aL12 aL21 aR11 aR22 − aR12 aR21 = , , aL11 + aL22 − aL12 − aL21 aR11 + aR22 − aR12 − aR21
(4.14)
respectively. Proof. Obviously, λij = aRij − aLij = µ > 0 since aRij = aLij + µ (i, j = 1, 2). Using Eq. (4.4), it directly follows that bRij =
aLij aRij aLij + µ = + 1 = bLij + 1 (i, j = 1, 2). = λij µ µ
Hence, it is easily verified that the following two equations are valid: bL22 − bL21 bR22 − bR21 = bL11 + bL22 − bL12 − bL21 bR11 + bR22 − bR12 − bR21 and bL22 − bL12 bR22 − bR12 = . bL11 + bL22 − bL12 − bL21 bR11 + bR22 − bR12 − bR21 Therefore, the optimal strategies for Players A and B and the value of the interval ¯ = ([bLij , bRij ])2×2 are obtained as follows (Nayak and Pal, 2009): matrix game D ∗ T x∗ = (x∗ 1 , x2 )
=
bL22 − bL21 bL11 − bL12 , bL11 + bL22 − bL12 − bL21 bL11 + bL22 − bL12 − bL21
T
,
(4.15)
y∗ = (y1∗ , y2∗ )T =
bL22 − bL12 bL11 − bL21 , bL11 + bL22 − bL12 − bL21 bL11 + bL22 − bL12 − bL21
T (4.16)
and ∗
v¯ =
∗ ∗ [vL , vR ]
respectively.
bL11 bL22 − bL12 bL21 bR11 bR22 − bR12 bR21 = , , bL11 + bL22 − bL12 − bL21 bR11 + bR22 − bR12 − bR21
(4.17)
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According to Theorem 4.1, the optimal strategies x∗ and y∗ for Players A and ¯ are the same as those for Players A and B in the B in the interval matrix game D ¯ interval matrix game D = ([aLij , aRij ])2×2 , i.e., x∗ = x∗ =
bL22 − bL21 bL11 − bL12 , bL11 + bL22 − bL12 − bL21 bL11 + bL22 − bL12 − bL21
T (4.18)
and ∗
∗
y =y =
bL22 − bL12 bL11 − bL21 , bL11 + bL22 − bL12 − bL21 bL11 + bL22 − bL12 − bL21
T
¯ is obtained as follows: Also the value v¯∗ of the interval matrix game D bL11 bL22 − bL12 bL21 bR11 bR22 − bR12 bR21 ∗ ∗ v¯ = µ¯ v = µ ,µ . bL11 + bL22 − bL12 − bL21 bR11 + bR22 − bR12 − bR21
.
(4.19)
(4.20)
Using Eq. (4.4), i.e., bRij = aRij /µ and bLij = aLij /µ (i, j = 1, 2), Eqs. (4.8)– (4.20) can be rewritten as Eqs. (4.12)–(4.14). Thus, we have finished the proof of Corollary 4.1. Remark 4.1. If aRij = aLij +µ (i, j = 1, 2), then Eqs. (4.12)–(4.14) can be directly obtained through solving the systems of equations as follows: aL11 x1 + aL21 (1 − x1 ) = aL12 x1 + aL22 (1 − x1 ) aR11 x1 + aR21 (1 − x1 ) = aR12 x1 + aR22 (1 − x1 ) and
aL11 y1 + aL12 (1 − y1 ) = aL21 y1 + aL22 (1 − y1 ) aR11 y1 + aR12 (1 − y1 ) = aR21 y1 + aR22 (1 − y1 )
,
respectively. 5. Corrected Conclusion for Theorem 4.4 and Proof ¯ = ¯ = ([aLij , aRij ])m×n and D Assume that there are two interval matrix games D ([a Lij , a Rij ])m×n , where [cL , cR ] is an interval and [a Lij , a Rij ] = [cL , cR ] + [aLij , aRij ] = [cL + aLij , cR + aRij ]. ¯ Nayak and Pal considered the relation between the interval matrix games D ¯ and D and gave Theorem 4.4 (Nayak and Pal, 2009: 300), which asserted that the optimal strategies for Players A and B remained the same and v¯ ∗ = [cL , cR ] + v¯∗ , ∗ ¯ and D ¯ , respectively. where v¯ and v¯∗ are the values of the interval matrix games D It is found that Theorem 4.4 and its proof are incorrect. The proof of Theorem 4.4 is incorrect in that Nayak and Pal (2009) used the wrong formulae (i.e., Eqs. (4.5)– (4.7)), which can be correct and meaningful only if aRij = aLij + µ (i, j = 1, 2).
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Furthermore, it is easily seen that the solutions of generic m × n interval matrix games cannot always be obtained through solving the systems of linear equalities. ¯ We reach a generic conclusion on the relation between the interval matrix games D ¯ and D , which is summarized and proven as follows. Theorem 5.1. Assume that [a Lij , a Rij ] = [cL + aLij , cR + aRij ]. Interval matrix ¯ = ([a Lij , a Rij ])m×n have the same optimal ¯ = ([aLij , aRij ])m×n and D games D ∗ ∗ ¯ = [cL , cR ] + ω ¯ ∗ , where strategies for Players A and B, and υ¯ = [cL , cR ] + υ¯∗ , ω ∗ ∗ ¯ ¯ , ω ¯∗ υ¯ and υ¯ are Player A’s gain-floors in the interval matrix games D and D ∗ ¯ and D ¯ , respectively. and ω ¯ are Player B’s loss-ceilings in D Proof. Since [cL , cR ] is a positive interval, it is easily derived from Eqs. (3.4) and (3.5) that max{[cL , cR ] + υ¯} m ([cL , cR ] + [aLij , aRij ])xi ≥ [cL , cR ] + υ¯ ∼ i=1 m s.t. xi = 1 i=1 xi ≥ 0 (i = 1, 2, . . . , m)
(j = 1, 2, . . . , n)
and min{[cL , cR ] + ω ¯} n ([cL , cR ] + [aLij , aRij ])yj ≤ [cL , cR ] + ω ¯ ∼ j=1 n s.t. yj = 1 j=1 yj ≥ 0 (j = 1, 2, . . . , n),
(i = 1, 2, . . . , m)
respectively. Namely, max{[cL , cR ] + υ¯} m ([cL + aLij , cR + aRij ])xi ≥ [cL , cR ] + υ¯ ∼ i=1 m s.t. xi = 1 i=1 xi ≥ 0 (i = 1, 2, . . . , m)
(j = 1, 2, . . . , n)
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and min{[cL , cR ] + ω ¯} n ([cL + aLij , cR + aRij ])yj ≤ [cL , cR ] + ω ¯ ∼ j=1 n s.t. yj = 1 j=1 yj ≥ 0 (j = 1, 2, . . . , n),
(i = 1, 2, . . . , m)
which are just about a pair of interval mathematical programming models ¯ . Thus, the proof of Theorem 5.1 has been for the interval matrix game D completed.
6. Corrected Linear Programming Models for Interval Matrix Games and Computation Results of the Numerical Example Based on the Moore’s concept of set inclusions, two satisfactory crisp equivalent forms of interval inequalities were given as follows: aR z ≤ b R ¯ (6.1) b ⇒ m(¯ a ¯z ≤ az) − m(¯b) ∼ ≤α ¯ w(¯ az) + w(b) and aL z ≥ b L ¯b ⇒ a ¯z ≥ m(¯b) − m(¯ az) ∼ ¯ ≤ α, w(b) + w(¯ az)
(6.2)
where α ∈ [0, 1] represents the minimal acceptance degree of the inequality constraints which may be allowed to violate. Let’s consider two interval linear programming models given as follows: min
K
[cLk , cRk ]zk
k=1
K [a s.t.
k=1
Ltk , aRtk ]zk
z ≥ 0 k
≥ [bLt , bRt ] (t = 1, 2, . . . , T0 ) ∼
(k =, 1, 2, . . . , K)
(6.3)
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and max
K
[cLk , cRk ]zk
k=1
K s.t.
[aLtk , aRtk ]zk ≤ [bLt , bRt ] (t = 1, 2, . . . , T0 ). ∼
k=1 zk ≥ 0
(6.4)
(k =, 1, 2, . . . , K)
To employ standard linear programming technique to solve the above interval optimization problems, Nayak and Pal (2009) considered only the means of the interval objective functions in Eqs. (6.3) and (6.4). Thus, using Eqs. (6.1) and (6.2), Eqs. (6.3) and (6.4) were converted into the linear programming models as follows (Nayak and Pal, 2009): min
K
1 (cLk + cRk )zk 2
k=1
K aLtk zk ≥ bLt (t = 1, 2, . . . , T0 ) k=1 K [(1 − α)aLtk + (1 + α)aRtk ]zk s.t. k=1 ≥ (1 + α)bLt + (1 − α)bRt (t = 1, 2, . . . , T0 ) zk ≥ 0 (k =, 1, 2, . . . , K)
(6.5)
and max
K
1 (cLk + cRk )zk 2
k=1
K aRtk zk ≤bRt (t = 1, 2, . . . , T0 ) k=1 K [(1 + α)aLtk + (1 − α)aRtk ]zk s.t. k=1 ≤ (1 − α)bLt + (1 + α)bRt (t = 1, 2, . . . , T0 ) zk ≥ 0 (k =, 1, 2, . . . , K), respectively.
(6.6)
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Using Eqs. (6.5) and (6.6), Eqs. (3.4) and (3.5) for the interval matrix game ¯ D = ([aLij , aRij ])m×n should be transformed into the linear programming models as follows: max
s.t.
1 (υL + υR ) 2
m aLij xi ≥ υL (j = 1, 2, . . . , n) i=1 m [(1 − α)aLij + (1 + α)aRij ]xi i=1 ≥ (1 + α)υL + (1 − α)υR m xi = 1 i=1 xi ≥ 0 (i = 1, 2, . . . , m)
(6.7)
(j = 1, 2, . . . , n)
and 1 (ωL + ωR ) 2 n aRij yj ≤ ωR (i = 1, 2, . . . , m) j=1 n [(1 + α)aLij + (1 − α)aRij ]yj j=1
min
s.t.
≤ (1 − α)ωL + (1 + α)ωR n yj = 1 j=1 yj ≥ 0 (j = 1, 2, . . . , n),
(6.8)
(i = 1, 2, . . . , m)
respectively. Obviously, Eqs. (6.7) and (6.8) are not a primal-dual pair of the linear programming problems for any given parameter α ∈ [0, 1]. However, Nayak and Pal (2009) wrongly employed Eq. (6.1) and used υL and ωL to replace the objective functions (υL + υR )/2 and (ωL + ωR )/2 in Eqs. (6.7) and (6.8), respectively. As a result, for a fixed parameter α = 1,
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Eqs. (3.4) and (3.8) were wrongly converted into the linear programming models as follows: max{υL } m aLij xi ≥ υL (j = 1, 2, . . . , n) i=1 m m 1 1 (aLij + aRij )xi + (aRij − aLij )xi ≥ υL 2 i=1 2 i=1 s.t. m xi = 1 i=1 xi ≥ 0 (i = 1, 2, . . . , m)
(j = 1, 2, . . . , n)
(6.9) and min{ωL } n aLij yj ≤ ωL (i = 1, 2, . . . , m) j=1 n n 1 1 (aLij + aRij )yj + (aRij − aLij )yj ≤ ωL 2 j=1 s.t. 2 j=1 n yj = 1 j=1 yj ≥ 0 (j = 1, 2, . . . , n),
(i = 1, 2, . . . , m)
(6.10) respectively. Remark 6.1. Obviously, Eqs. (6.9) and (6.10) can be rewritten as follows: max{υL } m aLij xi ≥ υL (j = 1, 2, . . . , n) i=1 m aRij xi ≥ υL (j = 1, 2, . . . , n) s.t. i=1 m xi = 1 i=1 xi ≥ 0 (i = 1, 2, . . . , m)
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and min{ωL } n aLij yj ≤ ωL (i = 1, 2, . . . , m) j=1 n aRij yj ≤ ωL (i = 1, 2, . . . , m) s.t. j=1 n yj = 1 j=1 yj ≥ 0 (j = 1, 2, . . . , n), respectively, which are equivalent to the simpler linear programming models as follows: max{υL } m aLij xi ≥ υL (j = 1, 2, . . . , n) i=1 m s.t. xi = 1 i=1 xi ≥ 0 (i = 1, 2, . . . , m)
(6.11)
min{ωL } n aRij yj ≤ ωL (i = 1, 2, . . . , m) j=1 n s.t. yj = 1 j=1 yj ≥ 0 (j = 1, 2, . . . , n),
(6.12)
and
respectively. Let pi =
xi υL
(i = 1, 2, . . . , m),
qj =
yj ωL
(j = 1, 2, . . . , n).
(6.13)
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Then, Eqs. (6.9) and (6.10) were transformed into the linear programming models as follows (Nayak and Pal, 2009): m min pi i=1
m aLij pi ≥ 1 (j = 1, 2, . . . , n) i=1 m m s.t. 1 1 (aLij + aRij )pi + (aRij − aLij )pi ≥ 1 2 i=1 2 i=1 pi ≥ 0 (i = 1, 2, . . . , m)
(j = 1, 2, . . . , n)
(6.14) and max
n
j=1
qj
n aLij qj ≤ 1 (i = 1, 2, . . . , m) j=1 n n s.t. 1 1 (a + a )q + (aRij − aLij )qj ≤ 1 (i = 1, 2, . . . , m) Lij Rij j 2 j=1 2 j=1 qj ≥ 0 (j = 1, 2, . . . , n), (6.15) respectively, i.e., min
m
pi
i=1
m aLij pi ≥ 1 (j = 1, 2, . . . , n) i=1 m s.t. aRij pi ≥ 1 (j = 1, 2, . . . , n) i=1 pi ≥ 0 (i = 1, 2, . . . , m)
(6.16)
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and
n
qj j=1 n aLij qj ≤ 1 (i = 1, 2, . . . , m) j=1 n s.t. aRij qj ≤ 1 (i = 1, 2, . . . , m) j=1 qj ≥ 0 (j = 1, 2, . . . , n).
max
(6.17)
Remark 6.2. Noticing the fact that aLij ≤ aRij , pi ≥ 0 and qj ≥ 0 (i = 1, 2, . . . , m; j = 1, 2, . . . , n), it easily follows that Eqs. (6.16) and (6.17) are equivalent to the linear programming models as follows: m pi min i=1
m aLij pi ≥ 1 (j = 1, 2, . . . , n)
s.t.
and
s.t.
i=1
pi ≥ 0
n
(i = 1, 2, . . . , m)
qj j=1 n aRij qj ≤ 1
max
(6.18)
(6.19)
(i = 1, 2, . . . , m)
j=1
qj ≥ 0 (j = 1, 2, . . . , n),
respectively. Nayak and Pal (2009) asserted that Eqs. (6.16) and (6.17) constitute a primal-dual pair of linear programming problems, which are wrong in that there are the following two major mistakes. Minimizing the objective function ωL in Eq. (6.10) cannot always ensure minimizing the interval objective function [ωL , ωR ] in Eq. (3.5). Usually min{[ωL , ωR ]} should be written as min{ωR }. On the other hand, according to Eq. (6.1), the constraints n aLij yj ≤ ωL (i = 1, 2, . . . , m) j=1 n n 1 1 (a + a )y + (aRij − aLij )yj ≤ ωL Lij Rij j 2 2 j=1 j=1
(i = 1, 2, . . . , m)
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in Eq. (6.10) should be written as follows: n aRij yj ≤ ωR (i = 1, 2, . . . , m) j=1 n n 1 1 (a + a )y − (aRij − aLij )yj ≤ ωR Lij Rij j 2 2 j=1 j=1
(i = 1, 2, . . . , m).
Thus, Eq. (6.10) should be corrected as follows: min{ωR } n aRij yj ≤ ωR (i = 1, 2, . . . , m) j=1 n n 1 1 (aLij + aRij )yj − (aRij − aLij )yj ≤ ωR 2 j=1 2 j=1 s.t. n yj = 1 j=1 yj ≥ 0 (j = 1, 2, . . . , n).
(i = 1, 2, . . . , m)
(6.20) Namely, min{ωR } n aRij yj ≤ ωR (i = 1, 2, . . . , m) j=1 n aLij yj ≤ ωR (i = 1, 2, . . . , m) s.t. j=1 n yj = 1 j=1 yj ≥ 0 (j = 1, 2, . . . , n), which is equivalent to the linear programming model as follows: min{ωR } n aRij yj ≤ ωR (i = 1, 2, . . . , m) j=1 n s.t. yj = 1 j=1 yj ≥ 0 (j = 1, 2, . . . , n).
(6.21)
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Remark 6.3. Equations (6.11) and (6.21) show that the optimal strategy and the gain-floor for Player A only depend on the left limits/bounds of the intervals [aLij , aRij ] whereas the optimal strategy and the loss-ceiling for Player B only depend on the right limits/bounds of the intervals [aLij , aRij ]. However, υR does not appear in Eq. (6.9) (or (6.11)). As a result, Eq. (6.9) or Eq. (6.11) cannot determine υR . Similarly, Eq. (6.20) or Eq. (6.21) cannot determine ωL . Therefore, the gain-floor and the loss-ceiling for Players A and B cannot be bounded intervals, which show that the linear programming models proposed by Nayak and Pal (2009) cannot be employed to solve generic m × n interval matrix games. Let qj = yj /ωR (j = 1, 2, . . . , n), then Eq. (6.21) is converted into the linear programming model as follows: n qj max j=1 n (6.22) aRij qj ≤ 1 (i = 1, 2, . . . , m) s.t. j=1 qj ≥ 0 (j = 1, 2, . . . , n). Obviously, Eqs. (6.18) and (6.19) (or (6.22)) are not a primal-dual pair of linear programming problems. m n Nayak and Pal (2009) asserted that v¯∗ = i=1 j=1 [aLij , aRij ]x∗i yj∗ is a common value for Players A and B, which was defined as the value of the interval ¯ = ([aLij , aRij ])m×n . However, according to Eq. (3.3), we only matrix game D have the inequality relation: υ¯∗ ≤ v¯∗ ≤ ω ¯ ∗ , which means that in some situations ∼ ∼ ∗ ∗ ∗ ¯ is not valid. In other words, there may appear one of the possible υ¯ = v¯ = ω ¯ ∗ , υ¯∗ = v¯∗ ω ¯ ∗ , υ¯∗ v¯∗ ω ¯ ∗ . Thus, v¯∗ is only an equilibcases: υ¯∗ v¯∗ = ω ∗ ∗ ¯ rather than the common value rium (or a compromise) value between υ¯ and ω for Players A and B. Example 6.1. Let’s consider the following 2 × 2 interval matrix game given by Nayak and Pal (Section 6, Nayak and Pal, 2009:300):
¯ 4 = ([aLij , aRij ])2×2 = D
A1
B1
B2
[−3, −1]
[4, 6]
[6, 8]
[−7, −5
A2
.
¯ 4, a Adding an interval [8, 10] to each element of the interval pay-off matrix D positive interval pay-off matrix was obtained as follows:
¯ D 4
= ([bLij , bRij ])2×2 =
A1 A2
B1
B2
[5, 9]
[12, 16]
[14, 18]
[1, 5]
.
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Using Eqs. (6.16) and (6.17), Nayak and Pal (2009) obtained the linear programming models as follows: 1 = p1 + p2 min υL 5p1 + 14p2 ≥ 1 12p1 + p2 ≥ 1 (6.23) s.t. 9p1 + 18p2 ≥ 1 16p1 + 5p2 ≥ 1 p1 ≥ 0, p2 ≥ 0 and
1 = q1 + q2 ωL
max
5q1 + 12q2 ≤ 1 14q + q2 ≤ 1 1 s.t. 9q1 + 16q2 ≤ 1 18q1 + 5q2 ≤ 1 q1 ≥ 0, q2 ≥ 0.
(6.24)
Applying the Simplex method for linear programming, Nayak and Pal (2009) ∗ ∗ ) and (q1∗ , q2∗ , ωL ) of Eqs. (6.23) and (6.24), respecobtained the solutions (p∗1 , p∗2 , υL tively, where p∗1 =
13 , 199
p∗2 =
7 , 199
∗ υL =
199 20
(6.25)
q1∗ =
11 , 199
q2∗ =
9 , 199
∗ ωL =
199 , 20
(6.26)
and
which are wrong. In fact, it is easily verified that 5p∗1 + 14p∗2 =
5 × 13 14 × 7 163 + = < 1, 199 199 199
12p∗1 + p∗2 =
7 163 12 × 13 + = < 1, 199 199 199
9q1∗ + 16q2∗ =
243 9 × 11 16 × 9 + = >1 199 199 199
18q1∗ + 5q2∗ =
243 18 × 11 5 × 9 + = > 1. 199 199 199
and
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Thus, the obtained results (p∗1 , p∗2 )T and (q1∗ , q2∗ )T given in Eqs. (6.25) and (6.26) are not feasible solutions to Eqs. (6.23) and (6.24), respectively. Certainly, they are not optimal solutions of Eqs. (6.23) and (6.24). Remark 6.4. In the numerical example of Sec. 6, Nayak and Pal (2009) denoted ∗ ∗ and ωL by p∗ and q ∗ , respectively. υL Using Eq. (6.13), Nayak and Pal (2009) obtained 13 , 20 11 ∗ y1∗ = q1∗ ωL , = 20
∗ = x∗1 = p∗1 υL
7 , 20 9 ∗ q2∗ = q2∗ ωL = 20
∗ x∗2 = p∗2 υL =
(6.27) (6.28)
and v¯∗ =
2 2
p∗i [aLij , aRij ]qj∗ =
i=1 j=1
3 43 , . 20 20
(6.29)
Indeed, (x∗1 , x∗2 )T and (y1∗ , y2∗ )T are the optimal strategies for Players A and B, v¯∗ is ¯ 4 defined by Nayak and Pal (2009). However, the value of the interval matrix game D as stated earlier, the obtained results (p∗1 , p∗2 )T and (q1∗ , q2∗ )T have been proven to be wrong. Therefore, it is just a coincidence that Eqs. (6.27)–(6.29) provided the ¯ 4 . Again, it is easily seen that Eqs. (6.23) solution to the interval matrix game D ∗ ∗ ∗ ∗ and (6.24) cannot determine υR and ωR . In other words, the intervals [υL , υR ] ∗ ∗ and [ωL , ωR ] cannot be obtained through solving Eqs. (6.23) and (6.24). Therefore, Eqs. (6.23) and (6.24) (hereby Eqs. (6.16) and (6.17)) are not feasible. That is to say, the linear programming method of interval matrix games proposed by Nayak and Pal (2009) is not feasible. Remark 6.5. It is easily seen that Eqs. (6.23) and (6.24) are equivalent to the linear programming models as follows: 1 = p1 + p2 min υL (6.30) 5p1 + 14p2 ≥ 1 s.t. 12p1 + p2 ≥ 1 p1 ≥ 0, p2 ≥ 0 and
max
1 = q1 + q2 ωL
9q1 + 16q2 ≤ 1 s.t. 18q1 + 5q2 ≤ 1 q1 ≥ 0, q2 ≥ 0,
(6.31)
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∗ ∗ ∗ ∗ ∗ respectively. We easily obtain the solutions (p∗ 1 , p2 , υL ) and (q1 , q2 , ωL ) of Eqs. (6.30) and (6.31), where
p∗ 1 =
13 , 163
p∗ 2 =
7 , 163
∗ υL =
163 20
(6.32)
q1∗ =
11 , 243
q2∗ =
9 , 243
∗ ωL =
243 . 20
(6.33)
and
∗ T Using Eq. (6.13), we obtain the optimal strategies (x∗ and (y1∗ , y2∗ )T for 1 , x2 ) ¯ 4 , which are the Players A and B and the value v¯∗ of the interval matrix game D same as those obtained by Nayak and Pal (2009). This fact further shows that Nayak and Pal (2009) essentially solved Eqs. (6.30) and (6.31) rather than Eqs. (6.23) and (6.24).
Remark 6.6. It is easily seen that aRij = aLij + 2 (i, j = 1, 2) for all the inter¯ 4 . Thus, using Eqs. (4.12)–(4.14), vals [aLij , aRij ] in the interval pay-off matrix D ∗ T and we directly obtain the correct results, i.e., the optimal strategies (x∗ 1 , x2 ) ∗ ∗ T ∗ ¯ 4, (y1 , y2 ) for Players A and B and the value v¯ of the interval matrix game D which are given by Eqs. (6.27)–(6.29), respectively. 7. Suggested Auxiliary Linear Programming Models and Lexicographic Method for Generic Interval Matrix Games ¯b lacks of mathematiThe concept of an acceptability index to the premise a ¯≤ ∼ ¯ cal rigor and is not rational since ψ(¯ a≤ b) may be any nonnegative number, i.e., ∼ ¯b) ∈ [0, +∞). In fact, the statement “¯ ¯b) may be a is not superior to ¯b” (i.e., a ¯≤ ψ(¯ a≤ ∼ ∼ regarded as a fuzzy relation between the intervals a ¯ and ¯b in terms of the fuzzy set. ¯b” is regarded as a fuzzy set, whose membership function Namely, the premise “¯ a≤ ∼ is defined as follows (Collins and Hu, 2008): 1 if aR ≤ bL − if aL ≤ bL ≤ aR ≤ bR and w(¯ a) > 0 1 ¯ b) = ϕ(¯ a≤ (7.1) b R − aR ∼ a) if bL ≤ aL ≤ aR ≤ bR and w(¯b) > w(¯ 2(w(¯b) − w(¯ a)) 0.5 if aL = bL and w(¯ a) = w(¯b), where “1− ” is a fuzzy number being less than 1, which indicates the fact that the interval a ¯ is weakly not superior to the interval ¯b. a ¯) as follows: We can define ϕ(¯b ≥ ∼ ¯b). ϕ(¯b ≥ a ¯) = 1 − ϕ(¯ a≤ ∼ ∼
(7.2)
It is easily proven that the ordering relations defined by Eqs. (7.1) and (7.2) are continuous except a special case, i.e., aL = bL and w(¯ a) = w(¯b). Moreover, it is
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easily proven that the following properties are valid (Collins and Hu, 2008): (a) (b) (c) (d)
¯b) ≤ 1; 0 ≤ ϕ(¯ a≤ ∼ a ¯) = 0.5; ϕ(¯ a≤ ∼ ¯b) + ϕ(¯b ≥ a ¯) = 1; ϕ(¯ a≤ ∼ ∼ ¯b) ≥ 0.5 and ϕ(¯b ≤ c¯) ≥ 0.5 then ϕ(¯ a≤ c¯) ≥ For any intervals a ¯, ¯b and c¯, if ϕ(¯ a≤ ∼ ∼ ∼ ¯ ¯ b) ≤ 0.5 and ϕ( b ≤ c ¯ ) ≤ 0.5 then ϕ(¯ a ≤ c ¯ ) ≤ 0.5. 0.5; or if ϕ(¯ a≤ ∼ ∼ ∼
” and “≤ ” establish fuzzy partial orders for intervals (Collins and Hu, Thus, “≥ ∼ ∼ 2008). Equations (7.1) and (7.2) may provide quantitative methods to determine the exact degrees of membership for ordering two intervals. Using Eqs. (7.1) and (7.2), according to Eqs. (6.1) and (6.2), Eqs. (3.4) and (3.5) are transformed into the bi-objective mathematical programming models as follows: υL + υR max υL , 2 m aijL xi ≥ υL (j = 1, 2, . . . , n) i=1 m υR − i=1 aijR xi m m ≤ α (j = 1, 2, . . . , n) (7.3) (υR − υL ) − ( i=1 aijR xi − i=1 aijL xi ) s.t. υL ≤ υR m xi = 1 i=1 x ≥ 0 (i = 1, 2, . . . , m) i and
ωL + ωR min ωR , 2 n aijR yj ≤ ωR (i = 1, 2, . . . , m) j=1 n j=1 aijL yj − ωL n ≤ α (i = 1, 2, . . . , m) n (ωR − ωL ) − ( j=1 aijR yj − j=1 aijL yj ) s.t. ωL ≤ ωR n yj = 1 j=1 yj ≥ 0 (j = 1, 2, . . . , n),
respectively.
(7.4)
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Remark 7.1. Equations (7.3) and (7.4) adopt the crisp satisfactory forms as follows: max{¯ a} s.t. a ¯ ∈ Ω1
⇒
a)} max{aL , m(¯ s.t. a ¯ ∈ Ω1
and min{¯ a} s.t. a ¯ ∈ Ω2
⇒
a)} min{aR , m(¯ s.t. a ¯ ∈ Ω2 ,
¯ respectively, where Ω1 and Ω2 are sets of constraints that the interval variable a should satisfy according to requirements in the real situations. Equations (7.3) and (7.4) are further simplified as follows: υL + υR max υL , 2 m aijL xi ≥ υL (j = 1, 2, . . . , n) i=1 m m (1 − α)a x + αaijL xi ≥ (1 − α)υR + αυL (j = 1, 2, . . . , n) (7.5) ijR i i=1 i=1 s.t. υL ≤ υR m xi = 1 i=1 xi ≥ 0 (i = 1, 2, . . . , m) and
ωL + ωR min ωR , 2 n aijR yj ≤ ωR (i = 1, 2, . . . , m) j=1 n n αa y + (1 − α)aijL yj ≤ αωR + (1 − α)ωL ijR j j=1 j=1 s.t. ωL ≤ ωR n yj = 1 j=1 y ≥ 0 (j = 1, 2, . . . , n), j respectively.
(i = 1, 2, . . . , m)
(7.6)
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Equations (7.5) and (7.6) are bi-objective linear programming models, which may be solved using the existing methods for multi-objective programming. In this note, we suggest a lexicographic method for solving Eqs. (7.5) and (7.6), which is summarized as follows (Li, 2008): Step 1: Solve the linear programming models as follows: max{υL } m aijL xi ≥ υL (j = 1, 2, . . . , n) i=1 m m (1 − α)a x + αaijL xi ≥ (1 − α)υR + αυL ijR i i=1 i=1 s.t. υL ≤ υR m xi = 1 i=1 xi ≥ 0 (i = 1, 2, . . . , m)
(j = 1, 2, . . . , n)
(7.7) and min{ωR } n aijR yj ≤ ωR (i = 1, 2, . . . , m) j=1 n n αa y + (1 − α)aijL yj ≤ αωR + (1 − α)ωL ijR j j=1 j=1 s.t. ωL ≤ ωR n yj = 1 j=1 yj ≥ 0 (j = 1, 2, . . . , n),
(i = 1, 2, . . . , m)
(7.8) respectively, where the parameter α ∈ [0, 1] is given by Player (or DM). 0 0 0 0 , υR ) and (y0 , ωL , ωR ), Denote optimal solutions to Eqs. (7.7) and (7.8) by (x0 , υL respectively.
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Step 2: Solve the linear programming models as follows: υL + υR max 2 m aijL xi ≥ υL (j = 1, 2, . . . , n) i=1 m m (1 − α)a x + αaijL xi ≥ (1 − α)υR + αυL ijR i i=1 i=1 s.t. υL ≤ υR 0 υL ≥ υL m xi = 1 i=1 x ≥ 0 (i = 1, 2, . . . , m)
733
(j = 1, 2, . . . , n)
i
(7.9) and min
ωL + ωR 2
n aijR yj ≤ ωR (i = 1, 2, . . . , m) j=1 n n αaijR yj + (1 − α)aijL yj ≤ αωR + (1 − α)ωL j=1 j=1 s.t. ωL ≤ ωR 0 ωR ≤ ωR n yj = 1 j=1 yj ≥ 0 (j = 1, 2, . . . , n),
(i = 1, 2, . . . , m)
(7.10) ∗ ∗ respectively. Denote optimal solutions to Eqs. (7.9) and (7.10) by (x∗ , υL , υR ) and ∗ ∗ ∗ (y , ωL , ωR ), respectively. ∗ ∗ ∗ ∗ , υR ) and (y∗ , ωL , ωR ) are Pareto optimal It is not difficult to prove that (x∗ , υL ∗ ∗ ∗ , υR ] are the solutions to Eqs. (7.5) and (7.6), respectively. Thus, x and υ¯∗ = [υL ∗ ∗ ∗ ∗ ¯ = [ωL , ωR ] are the maximin strategy and the gain-floor for Player A, y and ω minimax strategy and the loss-ceiling for Player B, respectively.
¯ 4 given in Example 6.1 Example 7.1. Let’s consider the interval matrix game D using the above lexicographic method. Taking α = 0, which indicates that the
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inequality constraints are not allowed to violate, according to Eqs. (7.7) and (7.8), we obtain the linear programming models for as follows: max{υL } −3x1 + 6x2 ≥ υL 4x1 − 7x2 ≥ υL −x1 + 8x2 ≥ υR s.t. 6x1 − 5x2 ≥ υR υL ≤ υR x1 + x2 = 1 x1 ≥ 0, x2 ≥ 0
(7.11)
min{ωR } −3y1 + 4y2 ≤ ωL 6y1 − 7y2 ≤ ωL −y1 + 6y2 ≤ ωR s.t. 8y1 − 5y2 ≤ ωR ω L ≤ ω R y + y2 = 1 1 y1 ≥ 0, y2 ≥ 0,
(7.12)
and
respectively. Applying the Simplex method for linear programming, we obtain the solutions 0 0 0 0 , υR ) and (y10 , y20 , ωL , ωR ) of Eqs. (7.11) and (7.12), respectively, where (x01 , x02 , υL x01 =
13 , 20
x02 =
7 , 20
y10 =
11 , 20
y20 =
9 , 20
0 υL =
3 , 20
0 υR =
43 20
(7.13)
0 ωL =
3 , 20
0 ωR =
43 . 20
(7.14)
and
Remark 7.2. In the numerical example of Sec. 6 (Nayak and Pal, 2009), Nayak and Pal took α = 1, which indicates that the acceptance/satisfactory degree of the inequality constraints being violated is equal to 1. According to Eqs. (6.1) and (6.2), this seems to be irrational and unrealistic. Thus, we show that Eq. (7.1) is more reasonable than Eq. (2.1).
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According to Eqs. (7.9) and (7.10), combining with Eqs. (7.11) and (7.12), we obtain the linear programming models as follows: υL + υR max 2 −3x1 + 6x2 ≥ υL 4x1 − 7x2 ≥ υL −x1 + 8x2 ≥ υR (7.15) 6x1 − 5x2 ≥ υR s.t. υL ≤ υR υL ≥ 3/20 x1 + x2 = 1 x1 ≥ 0, x2 ≥ 0 and
ωL + ωR 2 −3y1 + 4y2 ≤ ωL 6y1 − 7y2 ≤ ωL −y1 + 6y2 ≤ ωR 8y1 − 5y2 ≤ ωR s.t. ω L ≤ ω R ωR ≤ 43/20 y1 + y2 = 1 y1 ≥ 0, y2 ≥ 0, min
(7.16)
respectively. Using the Simplex method for linear programming, we obtain the solutions ∗ ∗ ∗ ∗ , υR ) and (y1∗ , y2∗ , ωL , ωR ) of Eqs. (7.15) and (7.16), respectively, where (x∗1 , x∗2 , υL x∗1 =
13 , 20
x∗2 =
11 , 20
y2∗ =
7 , 20
∗ υL =
3 , 20
∗ υR =
43 20
(7.17)
and 3 43 ∗ , ωR . = 20 20 ¯ 4 is obtained as follows: Hence, the value of the interval matrix game D 2 2 3 43 ∗ ∗ ∗ , xi [aLij , aRij ]yj = v¯ = . 20 20 i=1 j=1 y1∗ =
9 , 20
∗ ωL =
(7.18)
(7.19)
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Thus, we obtain the optimal strategy (x∗1 , x∗2 )T and the gain-floor υ¯∗ for Player A, ¯ ∗ for Player B as well as the the optimal strategy (y1∗ , y2∗ )T and the loss-ceiling ω ∗ ∗ ¯ ¯ ∗ = v¯∗ = [3/20, 43/20]. value v¯ of the interval matrix game D4 , where υ¯ = ω ∗ ∗ Remark 7.3. The proposed lexicographic method can determine both υR and ωL and is different from the Nayak and Pal’s method (Nayak and Pal, 2009), i.e., Eqs. (6.9) and (6.10).
Remark 7.4. It is worthwhile to notice that Eq. (7.11) has unique optimal solution. Therefore, Eqs. (7.15) and (7.11) have the same optimal solution. Similarly, Eqs. (7.12) and (7.16) have the same optimal solution.
8. Concluding Remarks Nayak and Pal (2009) discussed how to solve interval matrix games through using linear programming technique. However, it is found that there are lots of serious mistakes on the definitions, conclusions, models, methods, proofs and computing results, which can only be meaningful and effective if they are revised. The corrected generic conclusions and models are presented and justified by strictly mathematical logic proofs and specific examples. Modified definitions and methods are suggested so that a rational and credible solution of the interval matrix game can be achieved. In addition, the reader can correctly understand and use the research results (Nayak and Pal, 2009) only if the mistakes are corrected.
Acknowledgments The author would like to thank the valuable reviews and also appreciate the constructive suggestions from the anonymous referees and the Area Editor. This research was sponsored by the Natural Science Foundation of China (No. 70871117, No. 70902041 and No. 71171055) and Program for New Century Excellent Talents in University (the Ministry of Education of China, NCET-10-0020).
References Collins, WD and CY Hu (2008). Studying interval valued matrix games with fuzzy logic. Soft Computing, 12, 147–155. Li, DF (2008). Lexicographic method for matrix games with payoffs of triangular fuzzy numbers. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 16(3), 371–389. Nayak, PK and M Pal (2009). Linear programming technique to solve two person matrix games with interval pay-offs. Asia-Pacific Journal of Operational Research, 26(2), 285–305.
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Deng-Feng Li received the BSc and MSc degrees in applied mathematics from the National University of Defense Technology, Changsha, China, in 1987 and 1990, respectively, and the PhD degree in system science and optimization from the Dalian University of Technology, Dalian, China, in 1995. From 2003 to 2004, he was a Visiting Scholar with the School of Management, University of Manchester Institute of Science and Technology, Manchester, UK. He is currently a Professor with the School of Management, Fuzhou University, Fuzhou, China. He has authored or co-authored more than 180 journal papers and three monographs. He has coedited one proceeding of the international conference. His current research interests include fuzzy decision analysis, group decision making, fuzzy game theory, supply chain, fuzzy sets and system analysis, fuzzy optimization, and differential game.
