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In this paper, we consider multiobjective two-person zero-sum games with vector payoffs and vector fuzzy payoffs. We translate such games into the corresponding multiobjective programming problems and introduce the pessimistic Pareto optimal solution concept by assuming that a player supposes the opponent adopts the most disadvantage strategy for the self. It is shown that any pessimistic Pareto optimal solution can be obtained on the basis of linear programming techniques even if the membership functions for the objective functions are nonlinear. Moreover, we propose interactive algorithms based on the bisection method to obtain a pessimistic compromise solution from among the set of all pessimistic Pareto optimal solutions. In order to show the efficiency of the proposed method, we illustrate interactive processes of an application to a vegetable shipment problem.

In this paper, we propose interactive algorithms for multiobjectve two-person zero-sum games with vector payoffs and vector fuzzy payoffs under the assumption that each player has fuzzy goals for his/her multiple expected payoffs.

Shapley [

On the other hand, Campos [

As a natural extension to multiobjective programming problems, Nishizaki and Sakawa [

In such situations, in this paper, we focus on two-person zero-sum games with vector fuzzy payoffs under the assumption that a player has fuzzy goals for the expected payoffs which are defined as nonlinear membership functions. In Section 2, introducing the pessimistic Pareto optimal solution concept by assuming that a player supposes the opponent adopts the most disadvantage strategy for the self, we translate two-person zero-sum games with vector payoffs into the corresponding multiobjective programming problems. We propose an inter- active algorithm based on the bisection method and linear programming techniques to obtain a pessimistic com- promise solution from among the set of all pessimistic Pareto optimal solutions. In Section 3, we also consider multiobjectve two-person zero-sum games with vector fuzzy payoffs, and propose an extended interactive algo- rithm to obtain a pessimistic compromise solution from among the pessimistic Pareto optimal solution set on the basis of the possibility measure [

We consider two-person zero-sum games with multiple payoffs which are defined by

In this section, we assume that each player has fuzzy goals for his/her expected payoffs

Assumption 1. Let

Similarly, the nonlinear membership functions

Then, we can formulate the following multiobjective programming problem for Player 1 under the assumption that Player 1 supposes Player 2 adopts the most disadvantage strategy for the self.

To deal with the multiobjective minimax problem (1), the following Pareto optimal solution concept can be defined.

Definition 1.

with strict inequality holding for at least one k.

We assume that Player 1 can find a pessimistic compromise solution from among the pessimistic Pareto optimal solution set. It should be noted here that a pessimistic compromise solution concept is different from a satisfactory solution concept employed in usual multiobjective programming problems. A pessimistic com- promise solution can be interpreted as a most better solution among the pessimistic Pareto optimal solution set in his/her preference.

For generating a candidate of a pessimistic compromise solution, Player 1 is asked to specify the reference membership values [

By introducing auxiliary variable

Since the inverse functions

As a result, the problem (3) is expressed as the following problem:

It should be noted here that the problem (5) can be easily solved by combined use of the bisection method and the first-phase of the two-phase simplex method of linear programming.

The relationship between the optimal solution

Theorem 1.

(i) If

(ii) If

Proof:

(i) Since

Assume that

with strict inequality holding for at least one

This contradicts the fact that

(ii) Assume that

Then, there exists some

From Assumption 1 and the fact that

This contradict that the fact that

Unfortunately, from Theorem 1, it is not guaranteed that the optimal solution

simultaneously hold. For the optimal solution

Test problem 1:

Theorem 2. For the optimal solution

Now, from the above discussions, we can present an interactive algorithm for deriving a pessimistic compromise solution from among the pessimistic Pareto optimal solution set.

Interactive algorithm 1:

Step 1: Player 1 sets his/her membership functions

Step 2: Set the initial reference membership values as

Step 3: Solve the problem (5) by combined use of the bisection method and the first-phase of the two-phase simplex method of linear programming. For an optimal solution

Step 4: If Player 1 agrees to the current pessimistic Pareto optimal solution, then stop. Otherwise, Player 1 updates his/her reference membership values

In this section, we consider two-person zero-sum games with vector fuzzy payoffs which are defined by

where the function

In this section, we assume that Player 1 has fuzzy goals for his/her fuzzy expected payoffs

Assumption 2. Let

where

Using the concept of the possibility measure [

where

In order to deal with the multiobjective maximin problem (9), we introduce the pessimistic Pareto optimality concept.

Definition 2.

with strict inequality holding for at least one k.

The constraints (10) are transformed into the following forms, where

It should be noted here that the decision vector

Similar to the previous section, we assume that Player 1 can find a pessimistic compromise solution from among the pessimistic Pareto optimal solution set.

For generating a candidate of a pessimistic compromise solution, Player 1 is asked to specify the reference membership values [

This problem can be equivalently transformed into the following form:

where

From the above discussion, the problem (13) for Player 1 can be expressed as

It should be noted here that the problem (14) can be easily solved by combined use of the bisection method with respect to

The relationship between the optimal solution

Theorem 3.

(i) If

(ii) If

is an optimal solution of (14) for some reference membership values

Proof:

(i) Since

Since the constraints of (13) are equivalent to those of (14), the following relations hold.

Assume that

with strict inequality holding for at least one k. Therefore, it holds that

This contradicts the fact that

(ii) Assume that

Then, there exists some

Because of

This contradict that the fact that

Unfortunately, from Theorem 3, it is not guaranteed that the optimal solution

simultaneously hold. For the optimal solution

Test problem 2:

Theorem 4. For the optimal solution

Now, from the above discussions, we can present an interactive algorithm for deriving a pessimistic compromise solution from among the pessimistic Pareto optimal solution set to (9).

Interactive algorithm 2:

Step 1: Player 1 sets his/her membership functions

Step 2: Set the initial reference membership values as

Step 3: For the reference membership values

Step 4: If Player 1 agrees to the current pessimistic Pareto optimal solution, then stop. Otherwise, Player 1 updates his/her reference membership values

In this section, we apply the proposed method to multi-variety vegetable shipment planning problems. We assume that a farmer (Player 1) must decide a ratio of the shipment amount between tomato and cucumber.

year | 2009 | 2010 | 2011 | 2012 | 2013 |
---|---|---|---|---|---|

January | 323 | 306 | 293 | 371 | 317 |

February | 316 | 349 | 285 | 444 | 361 |

March | 423 | 385 | 296 | 500 | 383 |

April | 377 | 415 | 268 | 448 | 356 |

May | 281 | 249 | 183 | 329 | 217 |

June | 216 | 226 | 259 | 263 | 221 |

July | 225 | 226 | 305 | 272 | 314 |

August | 303 | 302 | 364 | 247 | 277 |

September | 377 | 555 | 424 | 415 | 412 |

October | 278 | 458 | 446 | 555 | 440 |

November | 212 | 433 | 383 | 518 | 455 |

December | 259 | 277 | 413 | 389 | 402 |

year | 2009 | 2010 | 2011 | 2012 | 2013 |
---|---|---|---|---|---|

January | 340 | 320 | 293 | 423 | 437 |

February | 318 | 371 | 285 | 421 | 292 |

March | 368 | 375 | 296 | 412 | 208 |

April | 206 | 295 | 268 | 232 | 222 |

May | 156 | 172 | 183 | 206 | 145 |

June | 162 | 196 | 259 | 169 | 220 |

July | 149 | 165 | 305 | 168 | 205 |

August | 234 | 195 | 364 | 136 | 165 |

September | 160 | 307 | 424 | 194 | 370 |

October | 221 | 289 | 446 | 256 | 313 |

November | 355 | 331 | 383 | 335 | 423 |

December | 371 | 326 | 413 | 510 | 360 |

We assume that some column of the price lists arises in the future (in other words, Nature (Player 2) selects some year between 2009 to 2013). We also assume that miscellaneous costs to cultivate vegetables with manure can be ignored. Utilizing the

where

strategy of Player 2 (Nature). For example, if

According to Interactive algorithm 1, Player 1 updates his/her reference membership values to obtain a candidate of the pessimistic compromise solution from among the pessimistic Pareto optimal solution set. The interactive process with a hypothetical Player 1 is summarized in

1 | 2 | 3 | |
---|---|---|---|

1. | 0.45 | 0.5 | |

1. | 0.4 | 0.35 | |

0.40911 | 0.43406 | 0.47474 | |

0.40911 | 0.38406 | 0.32474 | |

208.47 | 213.42 | 222.80 | |

161.61 | 157.19 | 148.81 | |

0.14460 | 0.14803 | 0.15454 | |

0.39073 | 0.40001 | 0.41759 | |

0.04932 | 0.04797 | 0.04541 | |

0.13249 | 0.12887 | 0.12200 | |

0.28286 | 0.27512 | 0.26045 |

In this paper, we propose interactive algorithms for multiobjectve two-person zero-sum games with vector payoffs and vector fuzzy payoffs under the assumption that each player has fuzzy goals for his/her multiple expected payoffs. In the proposed method, we translate multiobjective two-person zero-sum games with fuzzy goals into the corresponding multiobjective programming problems and introduce the pessimistic Pareto optimal solution concept. The player can adopt nonlinear membership functions for fuzzy goals, and he/she can be guaranteed to obtain multiple expected payoffs, which are better than a pessimistic Pareto optimal solution whatever the other player does.

HitoshiYano,IchiroNishizaki, (2016) Interactive Fuzzy Approaches for Solving Multiobjective Two-Person Zero-Sum Games. Applied Mathematics,07,387-398. doi: 10.4236/am.2016.75036