Diagnosis and Resolution of Infeasibility in the Constraint Method for Solving Multi Objective Linear Programming Problems

In this paper we discuss about infeasibility diagnosis and infeasibility resolution, when the constraint method is used for solving multi objective linear programming problems. We propose an algorithm for resolution of infeasibility, which is a combination of interactive, weighting and constraint methods. Numerical examples are provided to illustrate the techniques developed.


Introduction
Almost every important real world problem involves more than one objective.The multi objective linear programming (MOLP) problem can be formulated as follows: . .: , 0 POSs and for improving the previous methods [5][6][7]. c x (1) where j are n-dimensional vectors, x is an n-dimensional vector, b is an m-dimensional vector and A is an m × n matrix.
Excluding the trivial case in which a point exists in the feasible region which maximizes all objectives simultaneously, we must often propose a compromise solution to decision maker (DM).In the special case, the point that simultaneously maximizes all objectives is called complete optimal solution.In general, such point rarely exists.Thus, instead of complete optimal solution, Pareto optimal solution (POS) is introduced.x * is a POS for Problem (1.1) if there does not exist another x such that for all i and for at least one .There are several methods for solving the MOLP Problem (1).By the utility function method [1] we obtain a compromise solution.The weighting method proposed by Kuhn and tucker [2], the constraint method proposed by Haimes et al. [3] and the weighted minimax method proposed by Bowman [4], characterize POSs.Since then, many different approaches are developed for obtaining Also, there are several fuzzy approaches for solving MOLP problems such as [7][8][9].
In all of the methods and ap M) has an essential role.However, choosing unsuitable bounds in the constraint method by DM may be occur infeasibility in the problem.
In this paper we use th ter algorithms for isolating IIS (irreducible infeasible subsystems) in order to diagnose infeasibility.For resolution of infeasibility and obtaining a POS, we propose analgorithm which is a combination of interactive, weighting and constraint methods.Also we recall the fuzzy method of Leon and Liern [14] for repairing the constraint, in order to attain a feasible space.
Section 2 recalls the constraint method S for the multi objective linear programming problem.Section 3 discuses about infeasibility analysis and recall two useful filters for isolating IIS.The resolution of infeasibility is discussed in Section 4. When the smallest cardinality set of constraints to cover all IISs is singleton, we have a special case of infeasibility.An interactive approach for this case is proposed in Section 4. For the other cases we propose a combination of the weighting method and the constraint method.Finally we recall the approach of Leon and Liern for repairing infeasibility.Numerical examples are provided to illustrate the technique developed in this chapter.

Solving MOLP Problems
ing MOLP problems g the many possible ble bounds is a pr terizing POS is to solve d by taking one objec-, 1, , : There are several methods for solv and obtaining the POSs [1].Amon ways of scalarizing the MOLP, the weighting and the constraint methods are the most famous methods.In these methods DM should determine the weights and the bounds for every objective, respectively.
In this section we recall the constraint method.However, determining good weights and suita oblem for DM.Therefore, we propose a combination method of the weighting and the constraint method.

The Constraint Method
The constraint method for charac the constraint problem formulate tive as the objective function and letting all the other objective functions be inequality constraints.The constraint problem is defined by max . .ε whe n by DM.If is a unique optimal solution to Problem (2) k; i to Proble 1).C tion o r some inconsistency quent is x X is a POS of Problem (1), then x * is an optimal solu f the Problem (2) for some for some objectives, ctives he or she hasn't ining   S we have    S X .

A C ination Me
Sometimes, DM has suitable bounds tentatively, but for the other obje any assessment.Similarly, in weighting method, it's possible, DM has some weights for comparison of some objectives together, but he or she doesn't know how to appropriate some weights for the other objectives.In this case the combination of the two methods may be useful.Consider Problem (1).Suppose DM has some good weights 1 0, and some suitable lower bounds where  1 2  , J J k .We S: 2 . ., nique optimal so-Proof: Let (contrary) x * be inefficient, then there exist x for all x , for some contradiction to unique optimality of x * . irreducible in- [18].An IIS has till infeasible, go to step 1).

3.
The deletion filter proposed by Chinneck and Dravnieks ation of exactly one IIS after he set.

Diagnosis of the Infeasibility
All infeasible systems have one or more feasible subsystems (IISs) of constraints the property that it is itself infeasible, but any proper subsystem is feasible.An infeasible set of constraints can be rendered feasible by deleting or repairing at least one member of every IIS it contains.Finding the smallest cardinality set of constraints to cover all IISs is known as the minimum-cardinality IIS set-covering problem (MIN IIS COVER) [19].
There are several practical issues related to IIS isolation.An excellent summary of the recently developed algorithmic methods has appeared in [17].Here we recall the deletion filter and the elastic filter.The deletion filter guarantees the identification of exactly one IIS, but the exiting of the elastic filter is a small infeasible set that is not necessarily an IIS, but it has at least one IIS.
If the number of objectives in Problem ( 1) is small, it's better we use deletion filter.Else, we use elastic filter and then use deletion filter on the exiting of elastic filter for obtaining an IIS.
It may be that there are multiple infeasibilities in the model, hence IIS isolation typically is used in a cyclic manner 1) Isolate an IIS; 2) Determine a repair for this IIS; 3) If the model is s 1.The Deletion Filter [18] guarantees the identific a single pass through the set of constraints.This is an essential property possessed by very few of the IIS isolation methods.In the following algorithm, for diagnosis infeasibility of the problems, we use the phase I of the simplex method.

INPUT: an infeasible set of constraints.
For each constraint in the set: Proof: See [18].

3.2.
e method origin [20].A fully elastic program ables e i to every constraint of these variables as an elastic objective function.This allows finding a feasible solution for the original infeasible model.Namely we solve the following problem: , ; m are shown in ing: INPUT: an infeasible set of constraints tive elastic variables e i .tive function.

4.
asibility analysis is the lem to make it ches and pracl prob IIS COVER).There are several algohe number of co ve the Problem, (4): The details of the algorith the follow-.

1) Make all constraints elastic by incorporating nonnega
2) Solve the model using the elastic objec 3) IF feasible THEN enforce the constraints in which any e i > 0 by permanently rem ELSE (infeasible) Exit.OUTPUT: the set of de-elasticized enforced constraints contains at least one IIS.

Resolution of the Infeasibility
The second major aspect of infe infeasibility resolution to repair the prob feasible.However, most published resear tice results in recent years have focused on the diagnosis side.Little investigation has been made in infeasibility resolution [13,14,16].

Resolution of Infeasibility in the Constraint Method
The smallest cardinality set of constraints to cover al IISs is known as the minimum-cardinality IIS set-covering lem (MIN rithms for finding MIN IIS COVER [17]. In this paper we concentrate on infeasibility in constraint objectives.In almost all of real problems, the number of objectives is very less than t nstraints.It is usual that in many problems, MIN IIS COVER for the set of constraint objectives be singleton.For this special case our algorithm proposes an interactive method, and for the other cases we use the combination method mentioned in Section 2.2 or the approach of Leon and Liern (Section 4.2) in order to repair the constraints and resolve infeasibility.Besides, a POS is obtained.

The Combination Algorithm
Consider Problem (2): Let DM chooses the first objective as the main and ε 2 = 18, ε 3 = 6, ε 4 = 14 as the upper b und of the other objectives.So the problem of findi POS changes to so o ng a lving the following problem: The last problem is infeasible.By solving Problem ( 4 5 that is a r the main pr llowing probl , , 0. x x x  objective as the main and ε 2 = 6, e bounds of the other objectives.Let DM choose = 0.5, have th ing num w 2 = 0.2, w 4 erical form of P optimal solution is agai

ty by Fuz
osed by in order to resolve infeasibility.

Resolution of Infeasibili zy Approach
In this section we recall the fuzzy method prop Leon and Liern [14] for repairing the constraints The main idea is that the fuzzy membership function expresses the degree to which a particular point satisfies a given constraint.The membership function makes use of Roodman's limits [1 ws: So as to obtain a feasible solution, let us reformulate the system , 1, , ; , 0 As the fuzzy case Assume that denote the fuzzy constraints b C y , 1, , ; d their membership functions by , 1, , ; , respectively.The con the membership function by Roodman's ba (5) and its dual (DPI).The ε , 1, , ; , , , . .0 struction of approach is sed on Phase I problem (PI) associated System se problems can be stated as follows: (PI) , where s and r are surplus an l vectors, respectively. (DPI) where , 1, , are the elements of T i c .Let z * be the optimal value of (PI) and * π , 1, , ; , be the optimal solution of (DPI).Since system (5) is infeasible, then * 0 z  .For . Define the fuzzy set of feasible solutions (4.1) as lu n s en by

  
The so tio with the highe t degree of membership     1.
According to Table 1, the summation of satisfaction degree in our algorithm is better than the Leon and Liern method, however, the minimum degree in the Leon and Liern method is better.

Conclusion
In this paper we discussed about infeasibility, diagnosis The sum of satis-3 8706 nd resolution, e used t meth astic filter and filter for fi osed a combin terest of rep POSs.constraint resolution of infeasibility, we proposed an algorithm, which was a combination of interactive, weighting sum and constraint method.We solved some numerical examples and compared our method with Leon and Liern method.

5 
) and (DPI) lead to z = 4.25, 2  = 0.25.Therefore, p 2 = 4.25, p 3 = 0, p 4 = 4.25, p 5 = 17.So we must solve the following problem: In order to compare the solutions by two methods described in Examples 4.2 and 4.3, we compute the satisfaction degree of the objectives with th ns (4.3).The results a