analysIs and deVelopment of CompromIse solUtIons In mUltICrIterIa transport tasKs

The object of research is the multicriteria transport problem of linear programming. Simultaneous consideration of several criteria is a problematic problem, since the optimal solutions for different criteria do not coincide. The possible solution of the problem is investigated – finding a way to obtain a compromise solution. Based on the results of the analysis of known methods for solving multicriteria problems (Pareto-set formation, scalarization of the vector criterion, concessions method), the last is justified. To implement the method, an iterative procedure is suggested, in which the initial plan is optimal according to the main criterion. At subsequent iterations, an assignment is made to the main criterion in order to improve the values of the additional criteria. The solution of the problem is continued until a compromise solution is obtained, ensuring the best value for the main criterion, provided that the values for the remaining criteria are no worse than those given. Important advantages of the proposed method: the simplicity of the computational procedure, the grounded technology of forming a new solution at each iteration, realizing the concept of assignment, quality control of the solution obtained at each step. The application of the proposed method opens the prospect of its generalization to the case when the initial data for the solution of the problem contain uncertainty.


Introduction
In the practice of planning and organization of transportation of goods, two different mathematical models are traditionally used: -the transport task by the cost criterion (at the same time the average total cost of transportation is minimized); -the transport task by the time criterion (the maximum of the traffic durations is minimized). These tasks are alternative in the sense that their optimal plans, as a rule, do not coincide (the shortest route is not necessarily the cheapest one). The technologies for solving these problems have been well worked out [1][2][3] and constructively take into account the different specifics and features of the productions of each of them. For this reason, they are fundamentally different and their integration into a single computational procedure is very problematic. At the same time, when solving the practical problems of transport logistics, there is a need to solve compromise problems, for example, such: a) to find a transportation plan that minimizes the average total cost of transport, provided that the longest of them does not exceed the prescribed one; b) to find a transportation plan that minimizes the maximum of the transportation duration, provided that their average total cost does not exceed the specified value.
It should be noted that when solving practical tasks of transportation planning, it is expedient to take into account one more criterion -the probability of successful implementation of the transportation plan for the aggregate of routes from suppliers to consumers, which are determined by the selected plan. The development of a method for solving this problem is of theoretical and practical interest.

the object of research and its technological audit
The object of research is the multicriteria transport problem of linear programming.
Simultaneous consideration of several criteria for optimal solutions is a problem. The fact is that the optimal solutions for different criteria do not coincide as a rule. This circumstance motivates the search for compromise solutions to the multicriteria problem. In this connection, the subject of research is the method of constructing a compromise solution.

the aim and objectives of research
The aim of research is development of a technology for finding compromise solutions for linear programming transport problems by the criteria: average total cost, the probability of implementing a transportation plan, the maximum duration of transportation.
To achieve this aim, it is necessary to solve the following tasks: 1. To analyze the general principles of compromise for a set of criteria and the choice of rational.
2. Development of a computational procedure for constructing a compromise solution.

research of existing solutions of the problem
Several different approaches to the solution of multicriteria problems are known and used in practice. One of the most commonly used is the formation of the Pareto optimal set of solutions [4][5][6][7][8]. Let X j n j , , ,.., = 1 2 be the set of possible solutions of the problem, and let F i m i , , ,.., = 1 2 be the set of criteria for evaluating the quality of the solution. Let's introduce the concept of a dominant solution. The decision is dominant if: , , ,.., , and for each i at least one of these inequalities holds strictly. A Pareto-optimal set is a subset of the set of all solutions X X X n 1 2 , ,.., , such that none of its elements is dominant. The shortcomings of this method are ob vious. First, there is no constructive way to form a Paretoset (except for the search of all solutions). Secondly, it is not clear how to choose the best from the set of unmodified decisions.
Another frequently used method is scalarization [9][10][11]. The simplest way to implement the basic idea is additive convolution. At the same time, for a set of criteria F F F m 1 2 , ,.., , its scalar mapping is formed by the formula: Another way is to convolve using the «ideal point» [11]. In this case, for each criterion F i , the best value F i ( ) 0 is chosen, and then the quality level of a particular solution X j is estimated by a number: ( ) The general disadvantage of this approach is the arbitrariness in the choice of numerical values of weighting coefficients a i i m , , ,.., . = 1 2 Finally, when solving multicriteria problems, the socalled method of concessions is often used [12,13]. Thus from a set of criteria the basic criterion is chosen, the others are defined as additional. Then the problems are solved according to the main criterion and then the values for all additional criteria are calculated for the obtained solution. If these values are satisfied by the person making the decision, then the solution of the problem is over. Otherwise, a concession is made: a new solution is found that will be worse than the previous one by the main criterion, but somewhat better by additional ones. The procedure continues until a compromise solution is found that satisfies all the criteria. The advantage of this approach: simplicity, interactivity and ability to control the breakpoint. The disadvantage -there is no general method that describes the procedure for moving from an existing solution to an alternative one. At the same time, if in accordance with the features and nature of the problem this transition is easily realized, then the proposed method is expedient to use.

methods of research
Let there be m centers -suppliers of cargo and n centers of its consumption. In this case: a i -volume of cargo to be transported from the i-th supplier; b j -volume of cargo to be transported to the j-th customer; c ij -average cost of transporting a unit of cargo from the i-th supplier to the j-th consumer; p ij -probability of successful realization of cargo transportation from i-th supplier to j-th consumer; t ij -duration of the corresponding transportation. Let's introduce the set: where x ij -the volume of cargo planned for transport from the i-th supplier to the j-th consumer, i m = 1 2 , ,..., , j n = 1 2 , ,..., . Let's formulate the criteria for the effectiveness of the transportation plan X: where d x x The required transportation plan must satisfy the following constraints: ISSN 2226-3780 , ,..., , , ,..., .
It is assumed that the balance condition is met, and in addition, the duration of transportation from the i-th supplier to the j-th consumer does not depend on the volume of transportation x ij , i m = 1 2 , ,..., , j n = 1 2 , ,..., . Let's set the task of developing a method for finding a transportation plan X x ij = ( ) that satisfies the constraints (4)-(6) and optimizing the criteria (1)-(3) optimally.

research results
First let's consider the two-criterion problem by the criteria (1), (3), using the following iterative procedure. Iteration 1. Using standard methods, the transport problem is solved by the cost criterion: to find a plan X x ij = ( ) that minimizes (1), satisfying constraints (4)-(6).
It is clear that the route from the supplier i 1 to the consumer j 1 is the least reliable of the plan's routes X 1 ( ) . Let's call it critical. Now let's modify the initial matrices C and P, as follows: Operations (8) and (9) exclude the possibility of using routes whose reliability is worse than critical. Iteration 1 is complete. The first stage of the assignment procedure is carried out. Iteration 2. Using the resulting matrix C c ij = be the solution of this problem. It is clear that the plan X ( ) 2 will not be better than the plan X ( ) 1 by criterion (1), but it is certainly better than this plan by criterion (3). Thus, the assignment by criterion (1)   The values of the criteria (1) and (3) for the plan are: . .
In this case, L X L X . , L X L X P X P X . , L X L X P X P X . , L X L X P X P X A joint analysis of all the results of solving the problem allows to draw the following conclusions.
1. The realized iterative procedure of successive concessions ensured the receipt of a set of plans, none of which is majorizing, that is, for no pair: the inequalities: aren't satisfied simultaneously. 2. The obtained results make it possible to choose a compromise solution of the problem. Let the criterion (1) be chosen as the main one, and it must be minimized, and the criterion (3) is constrained (for example, P x P all ( ) . . ≥ =0 05 Then the plan X 4 ( ) is a compromise, since this plan minimizes (1), satisfying the constraint P X 4 0 084 0 05 . . Let's note now that the formulated multicriteria transport problem can be solved in a similar way in the other case, when the main criterion is the reliability of the transportation plan implementation, and the average total cost should not exceed a given threshold. At the same time, as the initial transportation plan, the resulting plan of the previous task can be chosen, for which the maximum probability of its realization is achieved. Iteration 1. So the initial plan has the form: The values of the criteria (1) and (3) for the plan X 2 ( ) are: . ; L X L X P X P X Then the set is formed: The values of the criteria (1) and (3) for the plan X 3 ( ) are: L X P X L X L X P X P X Let's note that an independent solution of the problem formed in the example using two different basic criteria yields partially coincident partial solutions, but this, apparently, is an effect of small dimension of the original problem. On the other hand, the joint use of all the particular solutions obtained allows to realize important advantages of the resulting Pareto-set solutions, since these solutions do not majorize each other. Analysis of this set gives a wide scope for choosing a compromise solution.
The proposed method is easily generalized to the case of a larger number of criteria. Let the duration of transportation be chosen as the third criterion. Let's now construct the matrix T t ij = ( ), as before, the problem is solved according to the basic, cost criterion (1). Let X Xij

  
These operations exclude the possibility of further using routes that are worse than critical ones.
Further actions are performed as described above. The direction of further research is connected with the consideration of the situation when the initial data of the problem are not defined exactly. The real variants of the emerging uncertainty are described in [14][15][16][17][18]. Possible approaches to solving transport problems with the uncertainty of the initial ones are discussed in [19][20][21].

sWot analysis of research results
Strengths. The proposed method for solving the transport problem of linear programming, in contrast to the known ones, allows one to obtain a compromise solution that ISSN 2226-3780 provides the best value for the main criterion, provided that the additional criteria take values not worse than those given. The achievement of this effect by traditional methods is not feasible.
Weaknesses. The application of this method increases the total time for solving the problem.
Opportunities. Application of the proposed method opens up the prospects for solving transportation problems in conditions when the initial data on the cost of transportation, the likelihood of the plan and the duration of its implementation contain uncertainty.
Threats. The application of the method of concessions provides the possibility of obtaining a compromise solution that will inevitably be worse than the optimal solution by the main criterion. At the same time, the more stringent the requirements will be on additional criteria, the more tangible the deterioration of the decision will be on the basis of the main criterion.

Conclusions
1. Based on the results of the analysis of known methods for solving multicriteria problems (Pareto-set formation, scalarization of the vector criterion, concession method), the last is justified. Important advantages of the proposed method: simplicity of the computational procedure, grounded technology of forming a new solution at each iteration, realizing the concept of assignment, quality control of the solution obtained at each step.
2. To implement the method, an iterative procedure is proposed in which the initial plan is optimal by the main criterion. At subsequent iterations, an assignment is made to the main criterion in order to improve the values of the additional criteria. The solution of the problem is continued until a compromise solution is obtained, ensuring the best value for the main criterion, provided that the values for the remaining criteria are no worse than those given. The application of the proposed method opens the prospect of its generalization to the case when the initial data for the solution of the problem contain uncertainty. references