UNIFORM WORKLOAD DISTRIBUTION PROBLEMS UNIFORM WORKLOAD DISTRIBUTION PROBLEMS

real matrix with daily records of vehicles as columns, we need to minimize some irregularity measure of row sums (workloads) by permuting matrix columns. The problem has various practical modifications, such as the weighted rows of matrix, workload uncertainty variance, graph approach of exchange of elements and generalized inverse formulations. Some of them are presented in the paper.


Introduction
In this paper we study problems which occur in uniform scheduling of workload distribution. The uniformity is expressed by an irregularity measure and the goal is to minimize the irregularity measure of parts of the schedule. This problem is motivated by the following job scheduling problem: Let n vehicles and m×n jobs be given. Every job has assigned to it the value ai,j which represents its quantity. These values form the m×n matrix A. We need to find the most regular m-days job schedule, which gives minimal difference between the sums of rows of the permuted version of the matrix A.
The problem is also called the Matrix Permutation Problem (MPP) which was first mentioned in [1]. Further investigation of MPP can be found in [2], where this problem was solved for garbage trucks.
The mathematical formulation of the problem is the following: Let a nonnegative real m×n matrix A=(a ij ) be given. Let I={1,2,…,m}, J={1,2,…,n} be sets of row and column indices. For each column j ! J of matrix A we will use the notation π j for the permutations of elements in that column. Let π=(π 1 ,π 2 ,… ,π n ) denote the vector of permutations of all columns of A and let A π denote the permuted matrix. Let P mn be the set of all such permutation vectors. Let S π =(s 1 π , s 2 π ,…,s m π ) denote the vector of row sums of permuted matrix A π . Let f be a function called the irregularity measure. The following optimization problem will be called the uniform workload distribution problem (UWDP) min{ f (S π ); π!P mn }.
The irregularity measure f : J m " 0,∞) is any Schur-convex function, where J=(a,b) is an interval. More on irregularity measures and Schur-convex functions can be found in [3]. The most used irregularity measures are:

Computational complexity of UWDP
In [4] it was proved that UWDP (MPP) is an NP-hard problem. In [2] it was shown that even a 2-row version of UWDP is NP-hard. The proof of this fact was established using transformation from the Set partition problem (definition of SPP can be found in [5]).
We get optimal column permutations , 1 2 : Update π by applying permutation 1 Tests on real data give promising results. The authors Š. Peško and M. Kaukič state conjecture that this algorithm can also be (with a suitable iteration count and sufficient number of restarts) the exact algorithm (at least for irregularity measure f dif ).

Model of linear programming
In the mentioned work [7] the following model of mixed, integer, linear programming (MILP) was introduced. Objective function is f dif .
The value of x ijk is equal to one if π j (i)=k and otherwise zero. The real variables z i i!I are the i-th row sums of permuted matrix A π and the variables z L , z U are variables for lower and upper bounds S L , S U of row sums.

Model of quadratic programming
In [7] the model of mixed, integer, quadratic programming (MIQP) was introduced. Objective function is fsqr d .
On the other hand, the two-column case is solvable in polynomial time. A simple polynomial algorithm can be found in [4]. It is enough to order the elements of the first column in a descending order and the elements of the second column in an ascending order. It is possible to show that we obtain the optimal solution for any irregularity measure mentioned above. The complexity of the algorithm based on this approach is O(m log m).

Solutions of UWDP
In this section we describe two heuristics for the solution of general UWDP. Both algorithms are based on the fact that the two-column case is solvable in polynomial time. We also consider the representation of UWDP by two models of mathematical programming.

Decomposition method
In the paper [2] the following heuristic was introduced: Input: Matrix A=A 1 of the type m×n. For i =1 to n-1 do: Create submatrix B m×2 from the first pair of columns of A i . Solve the UWDP for matrix B m×2 . (The row-sum vector of the solution is denoted by S π .) Create matrix A i+1 by replacing the first two columns of A i with column S π .
Output: Solution for the matrix A n-1 .
Tests showed that this method is not very successful since the set of possible column permutations is very restricted.

Stochastic decomposition method
This method was introduced and studied in [6 and 7]. Let J 1 be the nonempty proper subset of column indices J and J 2 =J-J 1 . The set {J 1 ,J 2 } will be called the admissible decomposition of the columns of matrix A. S1: Let π=(π 1 ,π 2 ,…,π n ) be arbitrary permutations of index set I.

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The last sum is taken over all vertices of G and hence it is a constant. The consequence is that δ is a constant for given graph G and mapping x. Then

Conclusions and further research
There are several approaches how to represent uncertainty. In [12] the UWDP in interval arithmetic was introduced. It means that elements of matrix A are not exact values, but we have intervals, to which these values belong. It was proved that this problem is NP-complete and the two-column case is solvable in polynomial time.
The most common approach in systems with uncertainty is the fuzzy arithmetic. Our future plan is to define the UWDP in fuzzy arithmetic. We suppose that the complexity results will not change but it remains an open problem.
The weighted version of the problem was studied in [13]. It is called the weighted uniform workload distribution problem Variables x ijk and z i mean the same as in the MILP model. Tests show that the MILP model is more effective than the MIQP model, but the mathematical programming solvers (even of such quality as Gurobi) have sometimes difficulties with the solving of not very large UWDP instances in reasonable time (since the model has large number of bivalent variables).

Restricted sets of permutations
In [8] the generalization of UWDP was suggested in which the set of permutations is restricted. The permitted permutations are represented by graphs. Paper [9] deals with conditions which allow to represent the set of permitted permutations by graphs. The two-column case of this generalization is studied in [10]. It was shown that this problem can be solved as the most regular perfect matching in a graph. In the mentioned paper [10], there are introduced exact polynomial algorithms for finding the most regular perfect matching in graphs for irregularity measures We will show that there is also polynomial algorithm for irregularity measure fsqr d , but the weights of edges must satisfy a special property (which is fulfilled by graphs that arise from the mentioned generalization of UWDP This leads to the unsolvable system of linear equations: x 1 + x 2 = 2 x 1 + x 3 = 2 x 1 + x 4 = 2 x 2 + x 3 = 4 x 2 + x 4 = 2 x 3 + x 4 = 4