The subject of the paper is multicriteria problems that arise when modeling the complex diversification of a centralized pharmacy network. The purpose of the work is to analyze the peculiarities of solving the three-criteria problem of pharmacy network diversification by the method of successive concessions in the MATLAB package. The paper solves the following problems: research of the advantages of the proposed three-criteria model of pharmacy network diversification in relation to the classical two-criteria model of portfolio theory; construction of the relation of dominance on a set of criteria; determination of the area of stability in the space of the parameters of the concessions method; evaluating the effectiveness of the method for problems of different sizes. The following methods are used: classical portfolio theory, multicriteria optimization, the successive concessions method, computer modeling of the Pareto set. The results obtained: a study of the processes of complex diversification of the pharmacy network by building portfolio models and solving the relevant multicriteria problems by the successive concessions method. Acceptable sets and sets of pareto-optimal portfolios for the risk management are graphically found, taking into account the activity of the network itself and the client portfolio. Conclusions: The results of computer modeling and numerical analysis of solutions by sequential concessions will be useful for automating the business processes of pharmacy networks, risk management, analysis of market data to improve their efficiency.
The successive concessions method is used to solve multicriteria problems, in which all partial
criteria are arranged in the order of importance. It is considered that each criterion is much more
important than the next that it is possible to be limited to consideration of only pairwise connection of
criteria and to choose the acceptable concession for the next criterion taking into account behavior of
only one following criterion. The disadvantages of the method include the complexity of the
appointment and coordination of the size of concessions, the need for a priori ranking of criteria. The
complexity increases with the dimension of the problem and the number of criteria. The result does
not necessarily belong to a subset of Pareto-efficient solutions. Therefore, the work of the method of
concessions is combined with dialogue - by multiple solutions of the optimization problem, graphs of
the dependences of the solution at the next stage from the concession of the previous stage obtained.
The expert selects the assignments based on the analysis of such diagrams and the assessment of the
gains or losses appropriate to the assignment. This paper considers the problem of the effectiveness of
the concessions method to solve the problems of portfolio optimization, which are known due to the
research of H.M. Markowitz [
When applying the methods of multicriteria optimization, traditional definitions of the dominance
relation, effective boundary, Pareto set, etc. were used. In particular, in the work of M.Ehrgott [
The group of publications of the short literature review devoted to portfolio optimization unites the
use of evolutionary methods in research, which is a promising area because the tasks of portfolio
optimization belong to the class of complex NP tasks. In [
Another group of publications of the literature review consists of works devoted to the study of the
peculiarities of the methods of multicriteria optimization, especially the method of successive
concessions. [
The variety of subject areas in which researchers solved multi-criteria problems by classical and
modern methods indicates the relevance of scientific research in this direction. The authors of this
paper in [
• Research of the advantages of the pro-pcroisterdia mthordeel of pharmacy network diversification over the classical two-criteria model of portfolio theory.
• Construction of the dominance relation on a set of criteria. • Determiantion the area of stability in the space of the parameters of the method of concessions. • Evaluaotin the effectiveness of the method for tasks (networks) of different sizes.
The paper [
We present one of the tasks of complex diversification, which is formalized as a portfolio model for optimizing the distribution of finances between the outlets of the centralized pharmacy network. Optimally distributing goods among outlets, the network gets the maximum profit in time to respond to the circulation of medications in a certain period, because the demand for medications in a particular pharmacy is not constant and goods not sold in one outlet can be sold on time in another and bring profit without increasing the length of the composition.
The turnover of the pharmacy network is the sum of the turnover of its pharmacy establishments. The turnover of an individual pharmacy is estimated to be sufficient at the level of UAH 2 million, which defines additional limitations in Model 1.
Let, - the share of the -th pharmacy in the turnover of the pharmacy network, which is equal to the share of distributed financial resources for the -th pharmacy; - the expected turnover of the pharmacy (UAH); - number of pharmacies in the network.
The model of the optimal portfolio considered in the paper has a composition of the vector objective function, consisting of three criteria. The first two criteria correspond to the classical portfolio theory: risk criterion - Risk, which decreases; Sum criterion - the profitability on the portfolio, which is desirable to increase.
The third criterion - entropy - Entropy, is introduced by us as a value that characterizes the level of diversification of the portfolio and the assessment of the diversity of its components, which the pharmacy network is trying to increase in its activities. In addition, the model has its own specific system of constraints, which is determined by the business process of the pharmacy network in the structure of its activities.
In this model, the Risk criterion - structural risk - is the risk of irrational distribution of financial resources of the centralized pharmacy network between outlets. Structural risk is defined as the covariance of the turnover of -th and -th pharmacies.
So, we have such multi-criteria problem of quadratic programming.
The solution of problem (1) is a vector X * x1, x2..xn - the optimal plan for the distribution of financial resources of the centralized pharmacy network between outlets.
Solving multicriteria problems of complex diversification will be considered in detail on the example of the successive concessions method [14, 15].
∑ ∑ { ∑( ∑ ̄ ) (
̄ ) ∑
( ) [ ] (1)
The successive concessions method for solving multicriteria problems is applied when partial criteria can be ordered in ecrease of their importance. To choose a diversification strategy, we first choose the following order: Entropy-Risk-Sum.
In the first stage, we determine the optimal value of the first most important Entropy criterion in the field of acceptable solutions.
In the second stage, we solve the conditional optimization problem by the next most important Risk criterion, adding to the conditions that determine the acceptable solutions, the conditions for the deviation of the first Entropy criterion from the found optimal value by no more than the acceptable concession . So we have the formalization of the second stage: ∑ ̄
The procedure is repeated for the next most important criterion adding to the conditions that determine the acceptable solutions, the conditions for the deviation of the first criterion and the second criterion from the found optimal values , ot more than the values of the acceptable concessions and . ( ) ( ) (2) (3) (4)
The solution obtained in the third stage is the solution of the three-criteria conditional optimization problem (1).
Experiments with the models were conducted on real data of one of the pharmacy chains operating in Zaporizhzhia. All calculations were performed in the MATLAB package [13].
Here are some useful definitions for analyzing the effectiveness of solving the three-criteria problem of diversification the pharmacy network by the successive concessions method in the MATLAB package, which are taken from sources [14, 15].
Definition 1. Stability of the algorithm - the ability to perform calculations and obtain the final result with a given accuracy when changing the parameters of the algorithm and input data in some area, which is called the area of stability.
Definition 2. Convergence is a property of an algorithm by changing its parameters to perform calculations with an arbitrarily small error for a given class of input data (i.e. when increasing the number of iterations for matching algorithms, the error will tend to zero). Moreover, the increase in accuracy is achieved by changing the internal parameters of the algorithm (for example, the maximum acceptable difference between the previous and next approximation).
Definition 3. The correctness of the computational method is a property of the indisputable existence of the solution the problem and ensuring stability of the computational algorithm that implements this method.
Let’s ocnsider the parameters that affect the results of solving the three-criteria problem of diversification by the successive concessions method.
The method of concessions requires the decision maker to determine the relation of dominance on multiple criteria.
The criteria are: total income of the pharmacy network for a certain period , the risk of income loss, which is defined as covariance on the set of income of pharmacies and the degree of diversification, which is estimated by the entropy of the investment portfolio of the pharmacy network .
Therefore, the first parameter of the method is the order of solving optimization problems according to one of the three criteria.
The vector of deviation values is the second parameter of the method. After solving the first onecriteria problem, the system of constraints is supplemented by a new criterion, which limits the deviation from the found of the optimal solution at the previous stages.
The number of pharmacies in the network, which determines the size of the problem, is the third parameter of the method.
The fourth parameter concerns the initial approximation and accuracy of the method. In this study, its characteristics are the accuracy of algorithms implemented by standard functions OPTIMIZATION TOOLBOX MATLAB, as the authors have developed a software product for solving research problems in the MATLAB package. Such function is the exitflag, which describes the conditions of the exit. If the value of exitflag is greater than zero, it means that the function matches the desired solution by X. If it is zero, then the maximum value of the function estimate or iteration has been exceeded. But if the value of exitflag is less than zero, then the function does not match to some solution.
Therefore, to analyze the effectiveness of the method of concessions on certain parameters, it is necessary to perform four steps that meet the objectives of this paper.
Step 1. Research of advantages of the offered three-criterion model of diversification of a pharmacy network concerning classical two-criterion model of the portfolio theory.
The initial approximation of the first stage is fixed as 0.1 from the lower limit of the admissibility interval. For each subsequent stage, we take the optimal value of the previous stage as the initial approximation.
Let's compare the solutions obtained using the classical two-criteria approach and the proposed three-criteria approach.
Consider the five modifications of problem (1), which are formed due to the different sequences of consideration of the criteria of profitability (SUM), risk (Risk) and the degree of diversification (Entropy) by the successive concessions method.
So we have two two-criteria problems, which we denote according to the sequence of criteria: and The results of solving these problems for a small network (the number of pharmacies in the network n = 5) with estimates of the main parameters are given in Table 1.
According to the data in Table 1, we can see that the best result among the two-criteria modifications is achieved for the model RISK_SUM, n = 5.
When adding the third criterion the vector objective function, you can get a total of 3! = 6 combinations to consider the sequence of criteria by the method of concessions. But the purpose of the third criterion is to increase the degree of diversification of the portfolio, so we consider it the most important and put in the first place (in addition, a number of experiments have confirmed that such arrangement gives better results than others).
Therefore, we will consider two three-criteria problems, which we will also denote according to the sequence of consideration of criteria by the method of concessions: and . The results of solving three-criteria problems (n = 5) with estimates of the main parameters are given in Table 2, where an inefficient modification is also given for comparison.
According to the estimates given in Table 2, it can be stated that the ENTROPY_RISK_SUM model n = 5 has the best result. Comparing the results of calculations from tables 1 and 2, we obtain experimental confirmation that the introduction of the entropy criterion allows to increase the income of the pharmacy network, but also increases the risk in different proportions.
Step 2. Construction of the dominance relation on the set of criteria.
The dominance ratio on the set of criteria, based on the significance of a particular criterion for the optimal decision-maker solution and the results of the experiments in step 1, is most successful in the ENTROPY_RISK_SUM model, where the criteria are ranked in descending order of importance. So this is the best result recommended for decision-maker, so Figures 1, 2, and 3 below show the computer simulation results for the the results of computer modeling of the domain of acceptable solutions, Pareto set and effective solution (red triangle) ENTROPY_RISK_SUM model.
First Step Entropy 0 0
As can be seen from Figures 1, 2 and 3, the visualization of solutions using computer modeling eliminates the previously mentioned disadvantage of the concessions method that the result does not necessarily belong to a subset of Pareto-efficient solutions, and therefore requires verification of its membership compromises. Thanks to the graphical interpretation, the check is performed automatically.
Step 3. Determination the area of stability in the space of the parameters of the concessions method
Construct n area of stability in the space of parameters of the concessions method and . The color will indicate the end conditions of the algorithm. Red color indicates the successful completion of the algorithm with the value exitflag=1 (First order optimality conditions satisfied). Marked in green Exitflag = 4 (Computed search direction too small). Marked in blue exitflag = 5 (Predicted change in objective function too small). and for the model ENTROPY_RISK_SUM, n=5 and for the model ENTROPY_SUM_RISK, n=5
For the ENTROPY_SUM_RISK model, experiments have shown that the concessions method is resistant to parameter changes for both small and medium networks (n <33) and large networks with n = 65 objects.
Step 4. Evaluation the effectiveness of the method for tasks (networks) of different sizes.
Let us evaluate the efficiency of the method application for networks of different size n = 5, 33, 65 and problems with different composition of the vector objective function.
Previously, Table 1 and 2 presented the results of solving two-criteria and three-criteria problems for small networks n = 5. Similar experiments were performed with medium networks, the results of which are presented in Table 3 and 4.
The results of experiments with large networks (n = 65) are shown in Table 5 and 6. SUM Risk ENTR exitflag
Evaluation of the effectiveness of the method of concessions for networks of different sizes is performed by the value of the exitflag function of the MATLAB package, which describes the exit conditions. Therefore, pay attention to the resulting column of each SOLUTION table, which corresponds to the last stage of the method of concessions.
For all problems of dimension n = 5 the value of exitflag is greater than zero, ie this function coincides with the desired solution by X. For the problem with the objective function of the model ENTROPY_RISK_SUM at n = 33 and 65 the value of exitflag is zero. This indicates that the maximum value of the function or iteration estimate was exceeded. But in no case was the exitflag value less than zero when the function did not match some solution.
Therefore, there is always a solution for the considered problems, and its belonging to the acceptable area is checked graphically thanks to the developed software in the MATLAB package.
This work continues the study diversification models of the centralized pharmacy network, which
was started in [
In contrast to [
Thanks to the developed software in the MATLAB package it is solved the task of studying the advantages of the proposed three-criteria model of pharmacy diversification networks relative to the classical two-criteria model of portfolio theory, built the relation of dominance on a set of criteria; determining the area of stability in the space of the parameters of the concessions method - which is the scientific novelty of this paper.
The results of computer simulation and numerical analysis performed in this paper successive concessions will provide investors with an appropriate tool decision support will be useful for automating pharmacy business processes networks, risk management, analysis of market data to improve their efficiency functioning.
Among the areas of further research - a comparative analysis of the effectiveness of other methods solving diversification problems and taking into account the dynamic factors that affect the degree of the risk.
The work was carried out as the part of the research work “Mathematical modeling of socioeconomic processes and systems”, the registration number DB05038, at the Department of System Analysis and Computational Mathematics of National University “Zaporizhzhia Polytechnic”.
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