Advanced
Matrix Game with Z-numbers
Matrix Game with Z-numbers
International Journal of Fuzzy Logic and Intelligent Systems. 2015. Mar, 15(1): 60-71
Copyright © 2015, Korean Institute of Intelligent Systems
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • Received : February 28, 2015
  • Accepted : March 18, 2015
  • Published : March 25, 2015
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
Sibasis Bandyopadhyay
Department of Mathematics, Visva Bharati, Santiniketan-731235, India
Swapan Raha
Department of Mathematics, Visva Bharati, Santiniketan-731235, India
Prasun Kumar Nayak
Bankura Christian College, Bankura-722101, India

Abstract
In this paper, a matrix game is considered in which the elements are represented as Z-numbers. The objective is to formalize the human capability for solving decision-making problems in uncertain situations. A ranking method of Z-numbers is proposed and used to define pure and mixed strategies. These strategies are then applied to find the optimal solution to the game problem with an induced pay off matrix using a min max, max min algorithm and the multi-section technique. Numerical examples are given in support of the proposed method.
Keywords
1. Introduction
In the modern world, one is faced with innumerable problems arising from existing socioeconomic conditions that involve varying degrees of imprecision and uncertainty. We often need to take decisions in conflicting situations based on uncertain, ambiguous, or incomplete information. Human beings have a tremendous capability to form rational decisions based on such imprecise information. It is hard to formalize these human capabilities. This challenge has motivated us to consider a game theoretic model under conditions of uncertainty.
Game theory problems under uncertainty have been considered by many researchers, e.g., Nayak and Pal [1 3] , Narayanan [4] , and Nishizaki [5] . Narayanan [4] solved a 2 × 2 interval game using the probability and possibility approach but no certain distribution function has been used. Nayak and Pal [3] established a method of solution of a matrix game using interval numbers. But the solution has been considered under a certain condition which has been obviated by the authors in their work [6] . Biswas and Bose [7] constructed a quadratic programming model under fuzzily described system constraints on the basis of degree of satisfaction. Veeramani and Duraisamy [8] suggested a new approach for solving fuzzy linear programming problem using the concept of nearest symmetric triangular fuzzy number approximations with preserve expected interval. But this approach is not efficient when a primal basic feasible solution is not in hand. Ebrahimnejad and Nasseri [9] overcame this shortcoming using a new algorithm. Iskander [10] proposed a new approach for solving stochastic fuzzy linear programming problem using triangular fuzzy probabilities. Apart from this Kumar and Kar [11] , Marbini and Tavana [12] , Ebrahemnijad and Nasseri [13] have contributed substantially to the application of fuzzy mathematics in operations research. However, there are many shortcomings in the above-mentioned techniques for solving game problems in uncertain situations. These can be categorized as follows:
  • We consider the theory of fuzzy sets to address the uncertainty that occurs in industrial problems or machine learning. It is unlikely that we always have complete knowledge about the domain set. For example, when we try to address an object as ‘beautiful,’ we may not have complete knowledge about the parameters by which the beauty of an object can be explained, or the parameters, if assumed, may not match other people’s choices, i.e., they may not be unique. However, the reliability of the information must be taken into consideration, and this aspect is lacking in the fuzzy set description.
  • We use soft set theory or rough set theory to describe uncertain situations. However, we may not know the complete set of parametersEin the case of a soft set, i.e., there is an issue regarding the reliability of the information concerned. In the rough set description, the lower or upper approximation or boundary region may not always be described, because knowledge or information about the equivalence relationRor domainUare only approximations or assumptions, and may not be known in advance. Additionally, it is not guaranteed that the desired optimal solution will be compatible with industrial applications.
  • Different optimization techniques give rise to different optimal results, and we can only compare these results with those obtained by existing techniques. We cannot, however, ensure that a technique gives an actual optimal solution that will be universally accepted. The main reason behind this is that we cannot address the uncertainty properly, and thus use a number of assumptions.
Thus, we need to develop the mathematical structures that provide a generalization of uncertain situations and that consider the reliability of available information. Zadeh [14] has made an attempt towards such a generalization by proposing Z-numbers. A Z-number is an ordered pair ( A , B ) in which A represents the restriction on a real-valued uncertain variable X and B is the measure of sureness, reliability, or certainty about A . However, the Z-number lacks informativeness when first introduced. Their composition in a summation has been defined [14] , but the ranking or ordering of Z-numbers has not yet been considered. In this paper, we introduce a ranking of Z-numbers in the specific case where A represents a restriction on the measure of the possibility distribution with a Gaussian membership function, and B is a restriction on the probability measure with a normal density function. This ranking method is then used to solve a two-person zero-sum game using min − max and max − min principles [6] and the multi-section technique.
The remainder of this paper is organized as follows. In Section 2, we discuss the concept of Z-numbers, before presenting some basic definitions, notation, and comparisons related to interval numbers in Section 3. In Section 4, a matrix game with Z-numbers is proposed, and then Section 5 discusses the interval approximation of Z-numbers, and introduces some definitions, explanations, theorems about matrix games with Z-numbers, pure and mixed strategies, and saddle points. In section 6, a computational procedure is proposed along with a min-max algorithm, max-min algorithm, and multi-section technique. Section 7 presents an example in support of the proposed method, along with discussions related to our algorithm and the results of the numerical example. Brief conclusions are given in Section 8.
2. Z-number
Let us consider a fuzzy set
PPT Slide
Lager Image
defined over a universe of discourse X . How will we define
PPT Slide
Lager Image
if X is not known in advance? Again, if it is assumed that X is known with certain parameters, then what is the reliability of such an information? We may say that we can define
PPT Slide
Lager Image
as type−2 fuzzy set where, X is a type−1 fuzzy set. In that case, X is also defined over some domain set
PPT Slide
Lager Image
. However, what will happen if
PPT Slide
Lager Image
is also not known? In Z-number representation, we may always construct X with some arbitrary element x X with some probability or possibility p . In that case, we can always associate a statement x with some statement G as
x is G is λ
where, λ acts as a fuzzy quantifier and we consider it as probability induced by some possibility membership and we write
prob( x is G ) = λ (fuzzy granularity [15] ).
This situation, inspires us to define a Z -number formally. A Z-number Z is defined [14] as an ordered pair ( A , B ) where A and B are two fuzzy numbers and a Z-valuation is defined as
PPT Slide
Lager Image
where, X is a real-valued uncertain variable, A is a restriction on the real values of the uncertain variable X and B is the reliability or certainty of A . Here, we consider X to be a random variable, AX to be a restriction on the measure of a possibility distribution with membership function µAX and pX is the restriction on probability distribution(density) function of X . The scalar product µAX . pX gives the probability measure, PAX of AX and it is given as
PPT Slide
Lager Image
Here, we recall that B is a restriction on the probability measure of A and it is not a restriction on the probability of A . Here, we attempt to ordering of the Z-numbers. For that purpose, here we consider the membership function as gaussian membership function. The reason behind taking such a function as membership function is that it is non-linear in nature and assume the situation which occurs generally in industrial applications. Unless otherwise stated by Z-number we here understand Z-valuation. For the sake of computation here we will consider a Z-number as ( X ; AX , PAX ) or ( X ; µAX , PAX ) or( X ; < c , σ >, PAX ) where c and σ are parameters of the Gaussian membership function. Here we may consider some examples like (Population of India, about 1200 million, very likely ),(degree of satisfaction, very high, not sure) which are considered as Z-number.
To consider an arithmetic operation, let ZX = ( X ; µAX , PAX ) and ZY = ( Y ; µAY , PAY ) be two Z-numbers. Using the extension principle as described by [14] , we obtain
ZX + ZY = Z X+Y
PPT Slide
Lager Image
3. Interval-number
Let us consider that ℜ represents the set of all real numbers. We define an interval, Moore [16] , as
PPT Slide
Lager Image
where aL and aR are said to be the lower and upper limits of the interval
PPT Slide
Lager Image
, respectively. If aL = aR then
PPT Slide
Lager Image
is reduced to a real number a , where a = aL = aR . Corresponding interval arithmetic is given by 3.
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
For,
PPT Slide
Lager Image
PPT Slide
Lager Image
The order relation of interval numbers is discussed in several literature [16 , 17] . Recently Chakrabortty et al. [18] proposed a revised definition of order relations between interval costs(or times) for minimization problems and interval profits for maximization problems for optimistic and pessimistic decision making. Let us suppose, the intervals
PPT Slide
Lager Image
and
PPT Slide
Lager Image
represent the uncertain interval costs (or times) or profits in center-radius form.
For minimization problems the order relation ‘ ≤ o min ’ between the intervals
PPT Slide
Lager Image
and
PPT Slide
Lager Image
is
PPT Slide
Lager Image
This implies that
PPT Slide
Lager Image
is superior to
PPT Slide
Lager Image
and
PPT Slide
Lager Image
is accepted. This order relation is not symmetric.
In pessimistic decision making, the decision maker expects the minimum cost/time for minimization problems according to the principle ‘Less uncertainty is better than more uncertainty’.
For minimization problems, the order relation ‘ < p min ’ between the intervals
PPT Slide
Lager Image
and
PPT Slide
Lager Image
is
  • (i)iff, for type-I and type-II intervals,
  • (ii)iffand, for type-III intervals.
PPT Slide
Lager Image
which is said to be matrix game
PPT Slide
Lager Image
with Z-numbers.
In this paper, arithmetic operations on interval [ a , b ] and their ranking as proposed in [6] serves as a level−1 computation [14] and the same is used in ranking of Z-numbers. Gregorzewski [19] proposed a method for interval approximation of fuzzy number. Here, in same way, we will approximate a Gaussian fuzzy number to an interval number. For that, let us consider a Gaussian fuzzy number < x , µ ( x ; c , σ )| x X > where the membership function is given as
PPT Slide
Lager Image
Now, we define an α -cut set Aα as Aα = { x : µ ( x ; c , σ ) ≥ α }. Then,
PPT Slide
Lager Image
and let us consider that
PPT Slide
Lager Image
and
PPT Slide
Lager Image
. Let [ a , b ] be the corresponding interval approximation. Then
PPT Slide
Lager Image
Therefore, if Z = ( X ; AX , PAX ) be a Z -number with Gaussian membership function µ ( x ; c , σ ) then the corresponding interval approximation is
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Here, we get the probability density function pXij as the normal density function N ( cij , σ ). Hence,
PPT Slide
Lager Image
where
PPT Slide
Lager Image
. Then, we obtain from Eq. (9) that
PPT Slide
Lager Image
Now, let us consider two Z -numbers Z 1 , Z 2 and corresponding interval approximations as
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Using the interval arithmetic, we propose the ranking of Z -numbers as
PPT Slide
Lager Image
4. Solution of Matrix Game
Suppose, the pay-off for player A in a matrix game with Z-number be represented as ( Xij ; AXij , PAXij ). Then, the corresponding interval approximation will be given by
PPT Slide
Lager Image
. The pay-off matrix with elements as interval approximation of Z-number can then be represented as
PPT Slide
Lager Image
where
PPT Slide
Lager Image
, i = 1, 2 · · · m , j = 1, 2, · · · n and
PPT Slide
Lager Image
Theorem 4.1. If Z 1 ( X 1 ; A X1 , P AX1 ) ≤ Z 2 ( X 2 ; A X2 , P AX2 ) then P AX1 P AX2 for optimistic decision maker and P AX1 P AX2 for pessimistic decision maker .
Proof . From the expression in (13) we see that PAXij depend only on σij and not on cij . Using this fact and combining the relation in (14) we can easily construct the proof of the theorem.
Notes : Here it should be noted that for pessimistic decision maker the degree of certainty PAX is lesser iff the membership value or the interval approximation is lesser and for optimistic decision maker the degree of certainty does not matter at all, it only gives the degree of reliability of the information.
- 4.1 Pure Strategy
In the context of Z-number, a pure strategy may be considered as a decision making rule in which one particular course of action is selected with some degree of reliability or certainty for the pay off considered. Actually, Z-number gives higher level of generality compared to interval numbers where length of the interval actually measures the certainty. For lack of informativeness of Z-number, we develop the concept of pure strategy in the domain of interval numbers with parallel computation. For matrix game with Z-number, we define the min max and max min as
PPT Slide
Lager Image
where ‘∨ ' and ‘∧ ' the max and min operators for two Z-number in the domain of Z-numbers Z respectively. In accordance with ranking of Z-numbers in (14), for games such as
PPT Slide
Lager Image
with pure strategy, we define the concept of saddle point solution.
Definition 4.1. (Saddle Point) The concept of saddle point in classical form was proposed by Von Neumann and Morgenstern [20] . The ( k , r ) th position of the pay-off matrix
PPT Slide
Lager Image
with Z-numbers is said to be a saddle point of the matrix game
PPT Slide
Lager Image
, if and only if,
PPT Slide
Lager Image
The position ( k , r ) is said to be a saddle point, the entry itself [ akr , bkr ] represents the value of the game (denoted by
PPT Slide
Lager Image
) and the pair of pure strategies leading to it are optimal pure strategies. Now we have to confirm that the relation as defined here for the saddle point solution actually exists for the matrix game with Z-numbers. For that purpose we must consider the following theorems.
Theorem 4.2. Let
PPT Slide
Lager Image
; i = 1, 2, ..., m ; j = 1, 2, ..., n be the m × n pay-off matrix for a two-person matrix game Γ with Z-numbers . Suppose
PPT Slide
Lager Image
and
PPT Slide
Lager Image
both exist . Then
PPT Slide
Lager Image
Proof . For some fixed i , we have, by using the order relation on Z ,
PPT Slide
Lager Image
PPT Slide
Lager Image
From (19) and (20) we have,
PPT Slide
Lager Image
Here, we see that
PPT Slide
Lager Image
is independent of j , since ( Xij ; AXij , PXij ) has obtained minimum value for some fixed value of j . Hence we write
PPT Slide
Lager Image
Again, the right-hand side of (21) is independent of i , hence, we obtain
PPT Slide
Lager Image
Hence the theorem.
Theorem 4.3. Let both
PPT Slide
Lager Image
and
PPT Slide
Lager Image
exist . Then a necessary and sufficient condition that ( Xij ; AXij , PXij ) will be a saddle point at i = k , j = r is
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Proof . Condition is necessary : Let
PPT Slide
Lager Image
Let i = k make
PPT Slide
Lager Image
a maximum and let j = r make
PPT Slide
Lager Image
a minimum. Then, we write
PPT Slide
Lager Image
As,
PPT Slide
Lager Image
, we have
PPT Slide
Lager Image
. Also, using the order relation over Z-numbers in Z ,
PPT Slide
Lager Image
This is one of the conditions for ( Xij ; AXij , PAXij ) to have a saddle point. The other condition can similarly be deduced.
Condition is sufficient : Let ( Xkr ; AXkr , PAXkr ) be the saddle point of the pay-off matrix
PPT Slide
Lager Image
, then for i = k , j = r we have, by definition of saddle point
PPT Slide
Lager Image
or,
PPT Slide
Lager Image
or,
PPT Slide
Lager Image
or,
PPT Slide
Lager Image
as
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Using the above Theorem 4.2 we have,
PPT Slide
Lager Image
Hence the necessary and sufficient condition for the existence of a saddle point is proved.
Example 4.1. Let us consider the 2 × 2 matrix game with Z-number having the pay-off matrix as in the following:
PPT Slide
Lager Image
It can be easily verified that
PPT Slide
Lager Image
Therefore, the matrix game with Z-number
PPT Slide
Lager Image
has a saddle point at (1, 1) and the optimal strategies for players A and B are the pure strategies A 1 and B 1 , respectively. The value of the matrix game
PPT Slide
Lager Image
is
PPT Slide
Lager Image
- 4.2 Mixed Strategy
In a situation where the saddle point of a pay off matrix does not exist we allow mixed strategies to get a solution. In mixed strategies, the probability with which a player chooses a particular strategy is considered. In the context of Z-number, we can say that in mixed strategy game we find an expected pay off with some reliability or certainty of the pay off obtained. Suppose
PPT Slide
Lager Image
and
PPT Slide
Lager Image
be the m and n dimensional vector spaces, respectively. We denote x = ( x 1 , x 2 , ··· , xm ) T and y = ( y 1 , y 2 , ..., yn ) T , respectively, where the symbol ‘ T' denotes the transpose of a vector. The strategy spaces for players A and B are denoted as
PPT Slide
Lager Image
respectively. Vectors x ∈ SA , y ∈ SB are called mixed strategies of players A and B , respectively. Now, we should remember that Z-number is a higher(level 3) level of generality [14] and all the operational rules like multiplication, division are not known. In that case, it is better idea to find the mixed strategy solution using the interval approximation. Interval is a particular case of a Z-number and it is level−1 domain of computation. Now, the question may arise: Does an optimal mixed strategy solution with interval numbers actually correspond to an optimal mixed strategy solution with Z-numbers? To get an answer to this question we must consider the following theorem where, a Z-number is modelled with a gaussian membership function and normal probability density function.
Theorem 4.4. An optimal solution of the matrix game with pay-off elements as interval approximation of some Z-number corresponds to the optimal solution of the matrix game with pay-off elements the concerned Z-number .
Proof . Let us consider a maximization problem where, the elements of the pay off matrix are interval approximation of Z-numbers. Let us construct an interval approximation function ϕ : Z I (ℜ). Now, the author’s [19] approach of interval approximation assures that the set of such functions is nonempty. We first establish that ϕ is a bijective mapping. Let Z 1 , Z 2 Z . We can then find c 1 , c 2 , σ 1 , σ 2 ∈ ℜ such that
PPT Slide
Lager Image
Using interval arithmetic, we can easily verify that ϕ ( Z 1 ) = ϕ ( Z 2 ) ⇒ c 1 = c 2 and σ 1 = σ 2 . Hence Z 1 = Z 2 . Therefore, ϕ is injective.
Similarly, using interval arithmetic, we can easily verify that for every interval c 1 , c 2 , σ 1 , σ 2 ∈ ℜ we can find Z 1 , Z 2 Z , the set of solutions with Z -numbers, assuring that ϕ is surjective. Hence, ϕ is bijective. This property assures that every solution in I (ℜ) corresponds exactly to one solution in Z . Let I be the optimal solution of the maximization problem. Then, for every solution I , we must have,
PPT Slide
Lager Image
Now, let us construct I and I as
PPT Slide
Lager Image
Using equation (4.2) we have either
PPT Slide
Lager Image
Since, ϕ is bijective we find Z such that Z Z Z Z . Therefore, Z is the optimal solution of the maximization problem modelled by Z-numbers. Similar approach can be made for minimization problem. Hence the theorem is proved.
Note: We compute the inverse function
PPT Slide
Lager Image
, a Z-number ∀[ a , b ] ∈ ℜ which gives the optimal solution of the matrix game with Z-number corresponding to the optimal solution with interval number.
Definition 4.2. (Interval expected pay-off ): If the mixed strategies x = ( x 1 , x 2 , ..., xm ) and y = ( y 1 , y 2 , ..., yn ) are proposed by players A and B respectively, then the expected pay-off of the player A by player B is defined by
PPT Slide
Lager Image
where,
  • CL= (a11+a22−b12−b21)x1y1+ (a12−b22)x1+ (a21−b22)y1+a22
  • CU= (b11+b22−a12−a21)x1y1+ (b12−a22)x1+ (b21−a22)y1+b22
The composition rules on interval numbers [6] are used in this definition (3) of expected pay-offs.
Definition 4.3. Suppose,
PPT Slide
Lager Image
and
PPT Slide
Lager Image
be two intervals defined over ℜ. Let us consider that there exist strategies x S A , y SB . If, for any strategy x ∈ S A , y ∈ SB ,
PPT Slide
Lager Image
satisfies both
PPT Slide
Lager Image
then, x ∈ S A , y ∈ SB ,
PPT Slide
Lager Image
is said to be a reasonable solution to the interval matrix game
PPT Slide
Lager Image
and
PPT Slide
Lager Image
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 for players A and B , respectively.
Definition 4.4. Let us consider that there exist two reasonable values
PPT Slide
Lager Image
and
PPT Slide
Lager Image
If there do not exist reasonable values
PPT Slide
Lager Image
and
PPT Slide
Lager Image
such that they satisfy both
PPT Slide
Lager Image
and
PPT Slide
Lager Image
, then
PPT Slide
Lager Image
is said to be a solution of the interval matrix game
PPT Slide
Lager Image
; x is called an optimal (or a maximin) strategy for player A and y is called an optimal (or a minimax) strategy for player
PPT Slide
Lager Image
and
PPT Slide
Lager Image
are called Player A ’s gain-floor and B ’s loss-celling, respectively.
5. Computational Methods
In this section, we discuss the computation procedure to find out the solution to a matrix game with Z-number. We first transform (through approximation) the pay off matrix with Z-numbers to a pay off matrix with corresponding interval-number. We then use multisection technique and min max algorithm [6] to solve the matrix game. The multisection algorithm is formulated according to the approach given in the work of Chakrabortty et al. [18] . The concept of multisection is inspired by the concept of multiple bi-section, where more than one bi-section is made at a single iteration cycle. The basis of this method is the comparison of intervals (as described in Section 3 of this paper) according to the decision makers point of view.
Algorithm for multisection technique
Input: λ (number of divisions), y , l (lower bound)and u (upper bound) of x .
Output: Probability x
Step 1://calculation of step lengths//calculate step length h = ( u l )/ λ
end for
Step 2://Division of concerned region into equal subregions //
Step 2.1: For j = 0 to λ − 1 Calculate l 0 = l + j h
Step 2.2: //Call the function CL and CU //.
Section 3,
Calculate CL = lower value of the interval number
PPT Slide
Lager Image
,
obtained by as in Eq.
(23)
Calculate CU = upper value of the interval number
PPT Slide
Lager Image
,
obtained by as in Eq.
(23)
Step 2.3: For j 1 = 0 to λ − 1 Calculate l 1 = l + j 1 ∗ h
Step 2.4: Calculate lmin = lower value of the interval number
PPT Slide
Lager Image
,
obtained by as in Eq.
(23) at l 0
Calculate umin upper value of the interval number
PPT Slide
Lager Image
,
obtained by as in Eq.
(23) at l 0
Calculate Cl = lower value of the interval number
PPT Slide
Lager Image
,
obtained by as in Eq.
(23) at l 1
Calculate Cu = upper value of the interval number
PPT Slide
Lager Image
,
obtained by as in Eq.
(23) at l 1
Step 2.5: Applying required order relation (defined in Section 3)
between any two interval numbers [ CL , CL ] and [ l min , u min ] choose the optimal interval number.
end j 1 loop
Step 2.6: Choose the subregion Eopt among Ej obtained in step 2.5 which has a better objective function value by comparing the interval values Ej to each other.
Step 3: //calculation of widths//.
Step 3.1: Calculate widths wj = uj lj of Ej where uj and lj are upper bounds and lower bounds of Ej
Step 3.2: While wj > ε
break
Step 3.3: Set Eopt Ej
Return to step 2.1
end for
endwhile.
end j loop
Output
END MULTISECTION
On the basis of this technique we have developed an algorithm for max min and min max solutions of a single objective interval game.
- 5.1 Min-Max Principle
Algorithm for min max principle
We conduct operations not on the degree of certainty PAXi j but on interval numbers [6] using the following steps:
Step 1: Put y 1 = nh ,where h = 1/ M and n = 0, 1, 2, 3 ···
M = Number of divisions of the interval [0, 1]
Step 2: For n = i
Find max/optimistic order relation of
PPT Slide
Lager Image
,
where 0 ≤ x 1 ≤ 1 by using multisection algorithm 4.1.
Step 3: Let the solution set for x is
PPT Slide
Lager Image
Using pessimistic order relation find minimum of
PPT Slide
Lager Image
Suppose it occurs at x 1 , which is a crisp number.
Step 4: Calculate
PPT Slide
Lager Image
Step 5: Using pessimistic order relation calculate
PPT Slide
Lager Image
for 0 ≤ y 1 ≤ 1
by multisection technique. Suppose, the minimum value is
PPT Slide
Lager Image
(say).
Then
PPT Slide
Lager Image
(by Theorem 4.2) and
PPT Slide
Lager Image
Therefore ( x 1 , y 1 ) is the optimal solution.
- 5.2 Max-Min Principle
Algorithm for max min principle 4.2.1
Step 1: Put x 1 = nk , where k = 1/ N and n = 0, 1, 2, 3, ···
N = number of divisions of the interval [0, 1]
Step 2: For n = i
Find min/pessimistic order relation of
PPT Slide
Lager Image
where 0 ≤ y 1 ≤ 1 by using multisection technique 4.1.
Step 3: Let the solution set for y is
PPT Slide
Lager Image
Using optimistic order relation find maximum of
PPT Slide
Lager Image
Suppose it occurs at y 1 , which is a crisp number.
Step 4: Calculate
PPT Slide
Lager Image
.
Step 5: Using optimistic order relation calculate
PPT Slide
Lager Image
for 0 ≤ x 1 ≤ 1
by multisection technique. Suppose the maximum value is
PPT Slide
Lager Image
(say).
Then
PPT Slide
Lager Image
(by Theorem 4.2) and
PPT Slide
Lager Image
Therefore e ( x 1 , y 1 ) is the optimal solution.
Thus
PPT Slide
Lager Image
is a reasonable solution of the interval matrix game
PPT Slide
Lager Image
,
PPT Slide
Lager Image
is player A ’s gain-floor and
PPT Slide
Lager Image
is player B ’s loss-ceiling.
6. Demonstration — An Example
Suppose, a company conducts an opinion pole about an election. They place some questions in front of the voters and get answers as ‘We are not very sure that the candidate A’s honesty is high’ or ‘It is very likely that inflation during the period of the present government is high’. In such cases, we can consider the statements as (A’s degree of honesty, high, not sure) or (price hike, high, very likely). These conditions are representation of Z-numbers. When this happens between two candidates in an election, then it forms a matrix game with Z-number. Suppose, the pay-off matrix is given by
PPT Slide
Lager Image
What are the optimal strategies and what is the value of game?
This is an example of 2 × 2 matrix game with Z-number which has no saddle point because
PPT Slide
Lager Image
Using the definition of interval expected pay off (4.2) When we run min max and max min programmes in TURBOC we get the reasonable solutions as
x = (0.0.9375, 0.0625), y = (0.5, 0.5),
PPT Slide
Lager Image
and
PPT Slide
Lager Image
- 6.1 Results and Discussions
In this example, we obtain a reasonable solution
PPT Slide
Lager Image
and
PPT Slide
Lager Image
as respectively gain-floor and loss-ceiling of players A and B with the probabilities x = (0.0.9375, 0.0625), y = (0.5, 0.5). There is a significance behind the result obtained and the technique adopted.
  • (i) Here the pay off actually means some restriction on the measure of possibility that one can gain or loose with some degree of certainty and value of the game actually means measure of possibility of a solution to be an optimum solution.
  • (ii) We have tried to arrive at a reasonable solution with some degree of certainty which is compatible with the real world situation as most of the optimization results obtained with other numbers or intervals lack some compatibility with the real world situation. For example, there are several kinds of imprecisions. What imprecision will then be modelled using a particular approach is a matter of concern as compared to the existing techniques[6]. In our approach, we have tried to consider a typical imprecision by using Z-number.
  • (iii) We have tried to arrive at a higher degree of generality in the decision making process by using Z-number.
7. Conclusions
In a decision-making process, the information is often found to be imprecise, incomplete, e.g., ‘about 5%’, ‘high price” etc. In such a situation it is unlikely that usual approach would give a desired result. Again, formalization of the imprecision hardly occurs in our optimization models and there is no universal model which can consider all types of imprecisions. We often model certain types of imprecise data with certain type of membership function or interval numbers. It does not, however, ensure the optimization universally, i.e., there may be a chance to arrive at a better optimal solution if we model it otherwise. So, there is a need for formalization of imprecision and for that purpose we have used Z-number as a pay off which actually gives the degree of certainty. On the restriction of the measure of possibility of pay off one would gain or loose. Though we have modelled a Z-number with a particular type of membership and density function there is scope of further generalization. There is also scope for using this procedure to solve multi-objective decision-making problems.
Conflict of InterestNo potential conflict of interest relevant to this article was reported.
BIO
Sibasis Bandyopadhyay is a Research scholar in Visva Bharati, Shantiniketan West Bengal, India-731235. His Research interest includes fuzzy logic, Game theory and uncertainty.
E-mail: sibasisbanerjee@rediffmail.com
Swapan Raha is a professor and Head of the Department in Department of Mathematics, Visva Bharati, Shantiniketan, India-731235. His research interest includes Cybernetics, Fuzzy logic and Approximate reasoning, Fuzzy pattern recognition, Fuzzy control.
E-mail: swapan.raha@visva-bharati.ac.in
Prasun Kumar Nayak is an Assistant Professor and Head, Department of Mathematics, Bankura Christian College, Bankura, West Bengal, India. His research interests include EOQ/EPQ inventory models, Game Theory in inexact environment, etc. He is a member of the editorial Board of International Journal of Fuzzy Mathematical Archive and Wesleyan Journal of Research.
E-mail: nayak prasun@rediffmail.com
References
Nayak P. K. , Pal M. 2007 Solutions of rectangular fuzzy games OPSEARCH 44 (3) 211 - 226
Nayak P. K. , Pal M. 2006 Solution of rectangular interval games using graphical method Tamsui Oxford Journal of Mathematical Sciences 22 (1) 95 - 115
Nayak P. K. , Pal M. 2009 Linear programming technique to solve two person matrix games with interval pay-offs Asia-Pacific Journal of Operational Research http://dx.doi.org/10.1142/S0217595909002201 26 (2) 285 - 305    DOI : 10.1142/S0217595909002201
Narayanan A. L. , Meenakshi A. R. , Ramasamy A. M. S. 2002 Fuzzy games Journal of Fuzzy Mathematics 10 (4) 817 - 829
Nishizaki I. , Sakawa M. 1995 Equilibrium solutions for multiobjective bimatrix games incorporating fuzzy goals Journal of Optimization theory and Applications http://dx.doi.org/10.1007/BF02192089 86 (2) 433 - 457    DOI : 10.1007/BF02192089
Nayak P. K. , Bandyopadhyay S. , Pal M. 2014 An algorithm for solution of interval games International Journal of Operational Research http://dx.doi.org/10.1504/IJOR.2014.061772 20 (2) 207 - 225    DOI : 10.1504/IJOR.2014.061772
Biswas A. , Bose K. 2011 A fuzzy programming approach for solving quadratic bilevel programming problems with fuzzy resource constraints International Journal of Operational Research http://dx.doi.org/10.1504/IJOR.2011.042503 12 (2) 142 - 156    DOI : 10.1504/IJOR.2011.042503
Veeramani C. , Duraisamy C. 2012 Solving fuzzy linear programming problem using symmetric fuzzy number approximation International Journal of Operational Research http://dx.doi.org/10.1504/IJOR.2012.049485 15 (3) 321 - 336    DOI : 10.1504/IJOR.2012.049485
Ebrahimnejad A. , Nasseri S. H. 2012 Linear programmes with trapezoidal fuzzy numbers: a duality approach International Journal of Operational Research http://dx.doi.org/10.1504/IJOR.2012.044028 13 (1) 67 - 89    DOI : 10.1504/IJOR.2012.044028
Iskander M. G. 2012 An approach for linear programming under randomness and fuzziness: a case of discrete random variables with fuzzy probabilities International Journal of Operational Research http://dx.doi.org/10.1504/IJOR.2012.048868 15 (2) 215 - 225    DOI : 10.1504/IJOR.2012.048868
Kumar A. , Kaur P. 2011 A new approach for fuzzy critical path analysis International Journal of Mathematics in Operational Research http://dx.doi.org/10.1504/ijmor.2011.040030 3 (3) 341 - 357    DOI : 10.1504/ijmor.2011.040030
Hatami-Marbini A. , Tavana M. 2011 An extension of the linear programming method with fuzzy parameters International Journal of Mathematics in Operational Research http://dx.doi.org/10.1504/IJMOR.2011.037312 3 (1) 44 - 55    DOI : 10.1504/IJMOR.2011.037312
Ebrahimnejad A. , Nasseri S. H. 2010 A dual simplex method for bounded linear programmes with fuzzy numbers International Journal of Mathematics in Operational Research http://dx.doi.org/10.1504/IJMOR.2010.035498 2 (6) 762 - 779    DOI : 10.1504/IJMOR.2010.035498
Zadeh L. A. 2011 A note on Z-numbers Information Sciences http://dx.doi.org/10.1016/j.ins.2011.02.022 181 (14) 2923 - 2932    DOI : 10.1016/j.ins.2011.02.022
Klir G. , Yuan B. 1996 Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems: Selected Papers by LA Zadeh World Scientific Publishing Company Singapore
Moore R. E. , Bierbaum F. 1979 Methods and Applications of Interval Analysis SIAM Philadelphia
Ishibuchi H. , Tanaka H. 1990 Multiobjective programming in optimization of the interval objective function European Journal of Operational Research http://dx.doi.org/10.1016/0377-2217(90)90375-l 48 (2) 219 - 225    DOI : 10.1016/0377-2217(90)90375-l
Chakrabortty S. , Pal M. , Nayak P. K. 2010 Solution of Interval-valued manufacturing inventory models with shortage International Journal of Engineering and Applied Sciences 4 (2) 89 - 94
Grzegorzewski P. 2002 Nearest interval approximation of a fuzzy number Fuzzy Sets and Systems http://dx.doi.org/10.1016/s0165-0114(02)00098-2 130 (3) 321 - 330    DOI : 10.1016/s0165-0114(02)00098-2
Von Neumann J. , Morgenstern O. 1947 Theory of Games and Economic Behavior Princeton University Press Princeton, NJ