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We consider a geographical region with spatially separated customers, whose demand is currently served by some pre-existing facilities owned by different firms. An entering firm wants to compete for this market locating some new facilities. Trying to guarantee a future satisfactory captured demand for each new facility, the firm imposes a constraint over its possible locations (a finite set of candidates): a new facility will be opened only if a minimal market share is captured in the short-term. To check that, it is necessary to know the exact captured demand by each new facility. It is supposed that customers follow the partially binary choice rule to satisfy its demand. If there are several new facilities with maximal attraction for a customer, we consider that the proportion of demand captured by the entering firm will be equally distributed among such facilities (equity-based rule). This ties breaking rule involves that we will deal with a nonlinear constrained discrete competitive facility location problem. Moreover, minimal attraction conditions for customers and distances approximated by intervals have been incorporated to deal with a more realistic model. To solve this nonlinear model, we first linearize the model, which allows to solve small size problems because of its complexity, and then, for bigger size problems, a heuristic algorithm is proposed, which could also be used to solve other constrained problems.

A new heuristic algorithm for solution of bi-objective discrete competitive facility location problems is developed and experimentally investigated by solving different instances of a facility location problem for firm expansion. The proposed algorithm is based on ranking of candidate locations for the new facilities, where rank values are dynamically adjusted with respect to behaviour of the algorithm. Results of the experimental investigation show that the proposed algorithm is suitable for the latter facility location problems and provides good results in sense of accuracy of the approximation of the true Pareto front.

The analysis of the method for multiple criteria optimization problems applying a computer network has been presented in the paper. The essence of the proposed method is the distribution of the concrete optimization problem into the network rather than the parallelization of some optimization method. The aim of the authors is to design and investigate the interactive strategies to solve complex multiple criteria problems by applying a computer network. The optimized objective function is the weight sum of the criteria. The multiple criteria problem is iterated by selecting interactively different weight coefficients of the criteria. Therefore, the process is organized by designating the computers as the master (that coordinates the process of other computers) and the slaves (that execute different tasks). In the beginning of the process the researcher allocates a certain number of optimization problems to the network. The objective function optimization problems differ only in weight coefficients of the criteria. As soon as the task of a slave has been executed, the result is sent to the master. Every computer of the network behaves in analogous way. Whenever the researcher receives an immediate result from one of the computers, he gives a decision taking into consideration the latter and all the previous results, i.e., he selects new weight coefficients for the criteria and assigns a new task to the network. Likewise the multiple criteria problem is solved until the result is acceptable for the researcher. The application of the proposed method is illustrated on the basis of the problem for the selection of the optimal nutritive value. Message Passing Interface (MPI) software has been used. The trials have been carried out with the network of computers under the operation system Windows NT.

We consider a possibility of automating the analysis of a computer program realizing the objective function of an extremal problem, and of distributing the calculation of the function value into parallel processes on the basis of results of the analysis. The first problem is to recognize the constituent parts of the function. The next one is to determine their computing times. The third problem is to distribute the calculation of these parts among independent processes. A special language similar to PASCAL has been used to describe the objective function. A new scheduling algorithm, seeking to minimize the maximal finishing time of processing units, was proposed and investigated. Experiments are performed using a computer network.