Application of mathematical modeling valueatrisk (VaR) to optimize decision making in distribution networks
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Abstract
Managers and capital masters of companies and factories try to adopt methods that maximize their profit and minimize their costs. A way for increasing profit is risk management. The risk management and the type of risk are defined in the literature of financial management. For measurement of the risk, many methods are defined which are all created in the recent century and it means the risk and risk management are rather new concepts. Among the newest tools for measuring the risk is valueatrisk (VaR), which was modified by Morgan (Riskmetrics technical document, Morgan Guaranty Trust Company, New York, 1996). VaR represents the maximum expected loss over a certain period of time and at a given confidence level. Many parametric, semiparametric and nonparametric methods for VaR estimation have been developed. In this article using the mean–variance method one of the parametric techniques of VaR, has been tried to minimize the cost arisen due to locating the supply chain and minimize the maximum level of capital losses, to optimize decision making in the distribution network. The calculated model is tested with numerical examples by MATLAB and Lingo software, and this example supported the resulting model.
Keywords
Value at risk Supply chain Locationallocation Location Integer programming1 Introduction
Site selection is one of the most important issues in industrial engineering, which can lead to the reduction in costs and successfulness of industrial units. Site selection for service centers is known as a selection of a position for one or more service centers considering other centers and also existing limitations so that a particular goal would be optimized. Site selection problem is one of the network design problems which has been solved as one of the strategic decision making processes. In site selection problems, there are a set of nodes that certain demands and they must be fulfilled by a set of service centers through some transportation modes. The term site selection refers to modeling, formulation and solving those problems which are seeking the best location for the establishment of the service center and facilities [1]. The first scientific framework of this theory was introduced formally by Weber [2]. He was one of the theorists who theorized the site selection and cost minimization which constitute major transportation costs [3]. The site selectionallocation problem is also used as a combination of supply chain problems. A provided model which is a combination of the siteselectionallocation, path determination, and control problems developed by Kheybari et al. [4] is one of these problems.
 1.
Customers’ demands must be met.
 2.
Cost of satisfying these needs.
 1.
The role of facilities: What is the role of each facility? In addition, what is processed and run in each facility?
 2.
Location of facilities: Where should be the location of each facility?
 3.
Capacity allocation: How much capacity should be allocated to each facility?
Among the effective factors on the distribution networks, one can refer to taxes, tariffs, currency rate, demand risk etc. This research is going to consider the effect of the risk factors on the demand parameter. Site selection of the facilities was gradually proposed to supply chain which investigates the models of site selection in the supply chain [5]. Usually, in management of a supply chain, there are three levels of planning depending on time horizon: strategic, tactical and operational. SimchiLevi et al. [6] stated that the strategic level deals with decisions which would have longterm impacts on the firm.
Sometimes, the meaning of application of supply chain network design is the same as strategic planning [7, 8]. Designing a bilevel and single period distribution network is also studied for managing a supply chain. This problem includes site selection for factories and distributing stockpiles. Moreover, the capacity of these facilities is determined from the possible selected choices, the related decisions on how to distribute products from a factory to distribution piles and also from stockpiles to customers [9].
Lejarza and Baldea [10] considered the issue of the supply chain in which the number and site of the distribution center are determined. Customers are facing random demands and each distribution center maintains a certain amount of guaranteed storage in order to achieve a certain service level for customers.
The concept of risk is arisen from uncertainty for input parameters in a decision making process and shows the possible loss in the problem. Taymaz et al. [11] introduced the risk as an unwanted result for a decision making problem. Xu et al. [12] introduced the sustainability of risks in the logistic network design. Value at risk is a risk assessment and diagnosis that uses standard statistical techniques that are used every day in many tactical and operational problems. Value at risk indicates the maximum expected loss in a given planning horizon. In addition, there are other valueatrisk metrics that have been published independently and have been introduced as tools to maximize the portfolio of capital in order to maximize returns for a given level of risk [13]. VaR applied the confidence α in which the most values of VaR would not be exceeded to lose during the next h days [14, 15]. In the framework of parametric approaches, the first model is proposed by Morgan [16]. One major disadvantage of the mentioned model is the assumption of the financial yield normality while the experimental evidence indicates that the financial yield does not follow a normal distribution. The second disadvantage is that in order to estimate the fluctuation, the financial yield should be used; and the third one is that the parametric approach assumes independency of the yields. There are significant experimental evident on the normality of financial yield distribution [17, 18, 19, 20, 21]. The VaR also includes point estimation, but deviates from the median that requires some probability level details, and then provides a better expected point to occur in a better probability. VaR calculation method is divided into parametric and nonparametric approaches [22].
The reason that we considered risk in our model is uncertainty in the environment of decision space. In a decision making problem, there are some input parameters that fluctuates the results and they are not under decision maker control such as customers’ demand. Therefore, the decision maker should create the mathematical model in which the model is able to handle such uncertainty due to prevent future costs. Thus, we considered value at risk (VaR) as a new framework of controlling the risk in our problem.
In this research, the implementation of VaR regarded with the demands of customers for the distribution network site selection model under uncertainty is performed and the parametric method is used to solve this problem. In this paper, the parametric type of VaR (mean–variance) methods are applied to formulate the uncertainty.
2 Problem definition
In this research, the problem of the supply chain is considered regarding the demand under uncertainty with a normal distribution with minimization of costs such as fixed costs of locating the facility in a candidate site and costs of the services delivered to the customers by the facility as well as minimization of value at risk. The problem faced here includes a bilevel supply chain considering suppliers level and customers level. In different periods, the demand for different products is delivered probabilistically to the suppliers in most scenarios. The supplier is responsible for meeting the demands and he should minimize the costs of fulfilling and services to the customer.
VaR is a tool for controlling the risk in financial aspects. There are other tools such as robust optimization for controlling the risk but an expert tool for financial risk is VaR. Since in our paper, the total cost is based on the financial risk, therefore, we used VaR in this way. It should be noted this tool very recently is used in the optimization problems such as facility location. In the literature, we focused on the application of VaR in optimization problems due to its major impact in our model [23].
2.1 Problem assumptions
 1.
In site selection for a supply chain network, nodes are candidates for the locations of both customers and facilities.
 2.
Locating a facility in a candidate site has fixed and certain costs.
 3.
Only one system delivers services to each customer.
 4.
The amount of flow of each facility depends on its capacity.
 5.
The cost of delivering a service to a customer is fixed.
 6.
The amount of flow between a facility and each customer is determined by a standard normal distribution using variance and mean indexes.
 7.
The sum of demands is considered as the lost value which is not supplied considering the capacity of a facility and demands of a customer.
2.2 Proposed mathematical model
The site selection model of the supply chain is defined by implementing the VaR.
2.2.1 Indexes, parameters and decision variables
 I

Set of customers’ sites; i = 1, 2,…, m
 J

Set of candidate sites for facilities; j = 1, 2, …, n
 h _{ i }

Average of demand for custome i
 σ _{i}

Amount of variance of demand for customer i
 σ _{ j }

Total Amount of variance of demands assigned to candidate facility j
 f _{ j }

Fixed cost of locating facility in candidate site j
 b _{ j }

Total Amount of mean of demands assigned to candidate facility j
 c _{ ij }

Cost of each transportation unit between customer i and candidate site j
 p _{ j }

Amount of capacity for candidate facility j
 Y _{ ij }

A binary variable that equals 1, if demand of customer i is supplied by candidate facility j, otherwise it is equal to zero
 X _{ j }

A binary variable that equals 1, if a facility is located in site j otherwise it is equal to zero
 VaR _{ j }

The highest expected loss in a given time horizon with the level of reliability at site j
2.2.2 Objective functions and limitations of the model
The first objective function minimizes the total amount of VaR for all activated facilities. The second objective function minimizes total costs including the cost of establishing facilities and also cost of transportation between facilities and customers.
Above procedure says that in order to compute linear approximation of \(w_{j}\), firstly, It should be clear that the point is like \(w_{j}\), which lies between two consecutive extremes of defined distances.
Then, the value of \(\sqrt {w_{j} }\) is computed exactly according to the same convex linear combination which was used for \(w_{j}\) (for more information, we can refer to Ghezavati et. al. [24]).
2.3 Implementation of genetic algorithm for the proposed model
In this paper, we use Lingo solver for solving the proposed model for smallsized examples and we aim to use a genetic algorithm for solving largesized examples. For this purpose, we need to code the genetic algorithm in a language programming. In this way, we choose MATLAB software to code the genetic algorithm.
The steps of the proposed genetic algorithm are described as follows:
Step 1 Initialize the input parameters of the algorithm including: the probability of the crossover and mutation, population size, the maximum number of iterations.
Step 2 The required number of initial solutions are generated randomly.
Step 3 Feasibility of the initial solutions is checked.
Step 4 The required solutions (called parents) are selected for crossover operation according to the crossover probability.
Step 5 The crossover operation is performed on parents to generate new solutions (called offspring).
Step 6 Feasibility of the offspring solutions is checked.
Step 7 The required solutions (called parents) are selected for mutation operation according to the mutation probability.
Step 8 The mutation operation is performed on parents to generate new neighborhood solutions.
Step 9 Feasibility of the neighborhood solutions is checked.
Step 10 Create the pool of solutions including the current solutions, offspring solutions, and neighborhood solutions.
Step 11 The fitness function for all solutions in the pool is computed according to the objective function in the Lpmetric method.
Step 12 According to the population size, the required number of solutions are selected according to the roulette cycle.
Step 13 Check the stopping criterion. If it is satisfied to stop and report the best solution, otherwise, go to step 4.
2.4 Implementation of Lpmetric approach
The system by which code writing and implementing is done is involves 4 GB RAM, CPU: Core i5, 230 GHz. Once the proposed model is transferred to a single objective function, it is solved by Lingo 8 software for small sized examples and also it is solved by the proposed genetic algorithm for large sized examples that are coded by MATLAB 2011.
3 Numerical example
In order to show the effect of VaR on the mathematical model, a problem with 10 customers and 8 facility sites from 5 data sets and different input sets are considered. Later, a problem with 20 customers sites, 15 facilities sites and 5 data sets and different input sets, in addition to a problem with 50 customers sites and 40 facilities sites, and finally a problem with 70 customers site and 50 facilities sites are considered. Furthermore, it is attempted to consider the cost of each transportation unit between location j and customer i, fixed cost of locating the facility for site j (f), mean customer’s demand (h) and capacity of facility j (p) as being different. Therefore, in all problems \(\sigma_{i}\) is assumed to be the normal distribution with a mean of 12 and variance of 2. Other parameters are also considered as \(\alpha = 95\%\) and a = 120 and L = 12.
Input data
Data  I  J  c  f  h  P 

Sets and inputs  
1  10  8  [5–15]  [10–20]  [20–25]  [20–40] 
2  10  8  [5–15]  [15–50]  [100–250]  [150–300] 
3  10  8  [5]  [10]  [10–25]  [15–35] 
4  10  8  [5]  [10]  [10–25]  [20–50] 
5  10  8  [5]  [10–20]  [20–25]  [20–40] 
6  20  15  [5–15]  [10–20]  [20–25]  [20–40] 
7  20  15  [5–15]  [10–20]  [20–25]  [20–50] 
8  20  15  [5–15]  [10–20]  [20–25]  [40–50] 
9  20  15  [5–15]  [10–20]  [20–30]  [20–60] 
10  20  15  [5–15]  [10–20]  [20–30]  [20–50] 
11  50  40  [5–15]  [10–20]  [10–30]  [20–60] 
12  70  60  [5–15]  [10–20]  [20–30]  [20–50] 
Comparison between results of MATLAB and LINGO
MATLAB  LINGO  

Data  Result  Time (s)  Result  Time (s) 
In the first function  
1  9.057  137  9.041503  945 
2  9.6325  88  9.6288  38 
3  56.5394  110  56.5213  192 
4  3.1173  156  3.1088  199 
5  34.3619  62  34.3438  465 
6  92.221  151  92.4037  6480 
7  49.0423  248  49.041  7800 
8  36.7346  257  36.724  8640 
9  1.4371  142  1.4213  7560 
10  99.9337  196  99.9124  5200 
11  39.144  355  221.617  34 h 
12  51.6566  450  Infeasible  – 
Comparison between results of MATLAB and LINGO
MATLAB  LINGO  

Data  Result  Time (s)  Result  Time (s) 
In second function  
1  1286  92  1286  1 
2  10640  68  10,640  1 
3  985  95  985  1 
4  875  91  875  1 
5  1297  53  1297  1 
6  2504  71  2504  1 
7  2628  46  2628  1 
8  2509  48  2509  1 
9  2737  62  2737  1 
10  2856  58  2856  1 
11  4615  150  4615  43 
12  6663  310  6663  190 
Comparison between results of MATLAB and LINGO
MATLAB  LINGO  

Data  Result  Time (s)  Result  Time (s) 
After implementing LPmetric  
1  0.1844  115  0.184307  1983 
2  0.0918  12  0.0918  106 
3  0.0355  65  0.0332  1690 
4  0.04  134  0.04  394 
5  0.01045  118  1034  186 
6  0.2015  137  0.2011  7095 
7  0.2159  243  0.2153  9300 
8  0.2758  218  0.2742  9660 
9  1.174  155  1.1622  9480 
10  0.1459  130  0.145  6900 
11  0.4516  328  Not computed  – 
12  0.3498  1537  Infeasible  – 
Now testing will be done to show the effect of increasing each input on the value of objective functions. The first 5 numerical examples of Table 4 are used in the test. Considering the importance of the objective function results, it is decided to ignore the calculated time. It should be noted that each test has done by Lingo.
3.1 First example
Model inputs by increase “c”
Data  I  J  f  h  p 

1  10  8  [10–20]  [20–25]  [20–40] 
2  10  8  [15–50]  [100–250]  [150–300] 
3  10  8  [10]  [10–25]  [15–35] 
4  10  8  [10]  [10–25]  [20–50] 
5  10  8  [10–20]  [20–25]  [20–40] 
Result of increase “c”
Data  New first objective function  Old first objective function  New second objective function  Old second objective function  New general objective function  Old general objective function 

1  9.0415  9.057  2276  1286  0.121  0.1844 
2  9.6288  9.6325  18,685  10,640  0.0522  0.0918 
3  56.5213  56.5394  1960  985  0.18293  0.0355 
4  3.1088  3.1173  1740  875  0.0201  0.04 
5  34.3619  34.3619  2238  1297  0.0138  0.01045 
3.2 Second example
Model inputs by increase “f”
Data  I  J  c  h  P 

1  10  8  [5–15]  [20–25]  [20–40] 
2  10  8  [5–15]  [100–250]  [150–300] 
3  10  8  [5]  [10–25]  [15–35] 
4  10  8  [5]  [10–25]  [20–50] 
5  10  8  [5]  [20–25]  [20–40] 
Result of increase “f”
Data  New first objective function  Old first objective function  New second objective function  Old second objective function  New general objective function  Old general objective function 

1  9.0415  9.057  1296  1286  0.1852  0.1844 
2  9.6288  9.6325  10,652  10,640  0.0919  0.0918 
3  56.5121  56.5394  987  985  0.0325  0.0355 
4  3.1088  3.1173  877  875  0.0478  0.04 
5  34.2438  34.3619  1307  1297  0.1038  0.01045 
As can be seen, the increasing the value of parameter “f” does not show a distinguished effect on the first objective function and causes just an insignificant reduction. However, its effect on the second objective function is obvious and increases the cost, but it is still not as much effective as parameter “c”.
3.3 Third example
Model inputs by increase “h”
Data  I  J  c  f  p 

1  10  8  [5–15]  [10–20]  [20–40] 
2  10  8  [5–15]  [15–50]  [150–300] 
3  10  8  [5]  [10]  [15–35] 
4  10  8  [5]  [10]  [20–50] 
5  10  8  [5]  [10–20]  [20–40] 
Result of increase “h”
Data  New first objective function  Old first objective function  New second objective function  Old second objective function  New general objective function  Old general objective function 

1  12.0415  9.057  1391  1286  0.1866  0.1844 
2  11.6288  9.6325  10,770  10,640  0.0948  0.0918 
3  75.8962  56.5394  1085  985  0.0360  0.0355 
4  17.6843  3.1173  975  875  0.0416  0.04 
5  49.6171  34.3619  1403  1297  0.1066  0.01045 
According to the obtained results, increasing “h” has a significant impact on increasing the first and second objective functions and therefore increases the general objective function.
3.4 Forth example
Model inputs by increasing “p”
Data  I  J  c  f  h 

1  10  8  [5–15]  [10–20]  [20–25] 
2  10  8  [5–15]  [15–50]  [100–250] 
3  10  8  [5]  [10]  [10–25] 
4  10  8  [5]  [10]  [10–25] 
5  10  8  [5]  [10–20]  [20–25] 
Result of increase “p”
Data  New first objective function  Old first objective function  New second objective function  Old second objective function  New general objective function  Old general objective function 

1  1.6284  9.057  1286  1286  0.1955  0.1844 
2  7.6288  9.6325  10,640  10,640  0.1859  0.0918 
3  43.3129  56.5394  985  985  0.0549  0.0355 
4  0.6886  3.1173  875  875  0.04  0.04 
5  26.3811  34.3619  1297  1297  0.1182  0.01045 
Above results show that increasing of parameter “p” does not influence the second objective function but considerably reduces the first objective function with an insignificant effect on the general objective function.
4 Results and discussion
It should be noted that increasing the capacity of facility j leads to a reduction of VaR in the first objective function. Considering that during solving the model by software, the objective function is simulated individually. Also, it can be observed that in the second objective function the values obtained from Lingo and MATLAB are the same, but the run time in Lingo for small dimensions is 1 s and as the dimensions increase, this time increases as well. Meanwhile, change of the value of parameters and problem inputs affects the amount of cost value (second objective function) more than the first one.
In addition, it can be seen that any increase in the value of fixed costs (f) leads to a small decrease in the first objective function of the proposed model. Once the value of capacities of the facilities increased in the model, the first objective function reduces more than previous satiations, however, such an event cannot change the global objective function. According to the definition of VaR in previous sections, the value of the objective function improves by the length of the interval of demand once the technique of VaR is applied.
Sensitivity analysis for the value of parameters on the VaR and costs
Parameter  Value at risk  Total cost  General 

c  Reduce  Exceeded  Indeterminate 
f  Reduce  Exceeded  Indeterminate 
h  Exceeded  Exceeded  Increase 
p  Reduce  Indifferent  Increase 
5 Conclusion
In this study, a framework for evaluating and managing the investment in facility location problem regarded with site selection by applying a general site selection problems framework with respect to risk concepts is presented using the VaR approach. The best advantage of VaR is the creation of a structured approach to making precise risk decisions. The process of achieving the right VaR depends on its numerical value. In this article, the decision maker tries to optimize the value of VaR in the site selection problem and the costs associated with fixed costs where the manager would be able to forecast the total risks which can occur in the future.
It was also worked in this research to create a certain and linear method by implementing the VaR and linearization of the model by fragmentation linear approximation method. Also, by considering the fact that the obtained mathematical model is biobjective, each objective function was solved separately by lingo and MATLAB using Lpmetric and the obtained results are compared. Meanwhile, it is proved that applying the VaR, one can minimize the costs and because of the profit by predicting the maximum loss. In this paper, demand was considered uncertain and it followed a normal distribution function. In this condition, even if the demand was considered as a risk factor and the amount of customer’s demand was not determined, one can manage the risk and control the maximum loss using the criteria of VaR measurement to be able to minimize the total costs.
For future researches, authors can consider the main parameters regarded to VaR as a fuzzy numbers since tactical financial parameters are almost uncertain. In addition, implementing the proposed method for designing distribution networks and reverse logistics is necessary. Furthermore, developing for VaR concept using the fuzzy theory can be a suitable research in this area. Finally, applying multi objective metaheuristic methods such as multi objective particle swarm optimization (MOPSO) method will be another interesting improvement.
Notes
Compliance with ethical standards
Conflict of interest
The author declares that they have no conflict of interest.
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