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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

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Volume 15, Issue 1

# A simulation based research on chance constrained programming in robust facility location problem

Leng Kaijun
/ Shi Wen
/ Song Guanghua
/ Pan Lin
Published Online: 2017-03-24 | DOI: https://doi.org/10.1515/phys-2017-0010

## Abstract

Since facility location decisions problem include long-term character and potential parameter variations, it is important to consider uncertainty in its modeling. This paper examines robust facility location problem considering supply uncertainty, in which we assume the supply of the facility in the actual operation is not equal to the supply initially established, the supply is subject to random fluctuation. The chance constraints are introduced when formulating the robust facility location model to make sure the system operate properly with a certain probability while the supply fluctuates. The chance constraints are approximated safely by using Hoeffding’s inequality and the problem is transformed to a general deterministic linear programming. Furthermore, how the facility location cost change with confidence level is investigated through a numerical example. The sensitivity analysis is conducted for important parameters of the model and we get the main factors that affect the facility location cost.

PACS: 05.45.-a

## 1 Introduction

Many important facilities such as filling stations, factories, banks, retail stores, warehouses, etc are indispensible in daily life. Since those facilities will need to run for a long time once established and will significantly impact on decisions of daily operation, the site selection for them should be based on consideration of long-term factors, which might be complicated. However, due to human factors or disasters such as snowstorms, typhoons and earthquakes, which may badly damage those facilities in their operation, people begin to recognize the serious consequences of accidents, particularly in recent years when accidental disasters frequently occur. The normal function of facilities will be interrupted at the time of disasters, which influence the normal order of social production and life, the supply of products and service. Therefore, the immense loss resulting from the uncertain supply of facilities should be highly valued at the time of facility location. There are many causes for the undulant supply of facilities, such as low production efficiency due to harsh climates, increasing defective percentage as a result of machinery depreciation or natural loss of supply owing to maintenance. On account of this, validity and reliability should be the main consideration in location decisions. In previous facility location process, cost was the only concern so that the designed network was too sparse and the system became fragile in the event of capacity loss or congestion due to link failure and maintenance. Thus, the decision maker must design the maintain facility operation under the budget to ensure that the system can run normally even with disturbance or parameter fluctuation. Only in case uncertain elements in facility location are given sufficient attention can the system get favorable robustness and a reliable system is designed, that is to say, a system insensitive to external disturbance.

In recent years, facility location based on uncertain factors have attracted increasing attention. Snyder and Daskin [1] have applied a-robust solution in P-meso-position and facility location without capacity limitation. They put forward facility location of random a-robust and worked out the a-robust solution which can minimize the expected cost. In the first place, a-robust solution is the feasible solution of problems, the relative regret of which in each scenario (in contrast to the optimal value of certainty in the scenario) should be no more than a. Hu Dandan and Yangchao [2] have used the concept of a-robust solution in flow capturing location model of uncertain parameters based on scenario to maximize the expected value of captured flow under the constraint of a-robust solution. Tian Junfeng et al. (2012) [3] have made the assumption that parameters such as manufacturers’ output, warehouse capacity, demand and transportation costs are established on the scenario to apply the concept of a-robust solution in the network design of supply chain and come up with the network design model of robust supply chain with the constraint of regret. They designed the tabu search algorithm to solve the model with the numerical examples illustrating that designing the supply chain network by robust optimization model can avoid investment risks. Zhong Huiling et al. (2013) [4] have proposed the arc-covering model with a-robust of grade goals to maximize the weighting coverage rate of the expectation and make the weighting coverage rate of each scenario greater than or equal to 100(1-a)% of the maximum coverage rate. The arc-covering model is established on two targets: minimizing the amount of facilities need to be constructed and maximizing the weighting coverage rate of the expectation. Daskin et al. [5] have raised the concept of a-reliability and explored for the optimal solution of P-meso-position to minimize the maximum regret in some scenarios by the given condition of 0 < a < 1 and the sum of occurrence probability of some scenarios employed in the objective function is lower than a. Chen, et al. [6] have developed on Daskin et al’s study of minimizing the maximum regret in some scenarios to put forward the concept of mean excess regret and presented that the mean excess regret model is easily-solved, thus it can better settle practical problems. Wang et al. (2011) [7] have taken uncertain demand and supply uncertainty resulting from facility failure due to accidents into account and proposed the facility location model based on two uncertain factors. Thapalia et al. (2012) [8] have established the network design model of single-commodity flow under random edge capacity. The minimum expectation for cost for network construction and operation as well as unsatisfied demand and punishment is the objective function. Xu Jiwang and Huang Xiaoyuan (2006) [9] have described the uncertainty of demand in consumer markets and supply in raw materials market by discrete scenarios of known probabilistic and put forward multi-objective robust operation model of multi-product and multi-stage supply chain constituted by one manufacturer and one supplier. After reviewing the research progress in robustness of supply chain, Huang Xiaoyuan and Yan Nina (2007) [10] have indicated that the robustness of supply chain network will be one research direction in future.

The roubst optimization approach was proposed independently by El-Ghaoui and Lebert (1997, 1998) [11, 12]. Ben-Tal and Nemirovski (1998) develpo an approach for robust convex optimization that solves the robust counterpart of the uncertain problem. Bertsimas and Sim (2004) [13] improve former approach by making a trade-off between the probability of a constraint violation and the effect this has on the objective function. Marianov and Fresard (2005) [14] prove better results can be acquired for any scenario than in equally likely different scenarios by introducing a robust model. It is worthwhile to mention that there is also a flood of literature on robust facility location problem. For example, we obtain robust solutions by applying the concept of robust optimization as presented by Mulvey et al. (1995) [15] on our capacitated facility location problem. Then, Daskin et al. (1997) [2] introduce the robustness conception to discuss the facility location. Carrizosa and Nickel (2003) [16] present a robust location of a facility on the plane. In this paper, they define the robustness as the minimal change in the uncertain parameters to make a solution location becomes inadmissible under a total cost constraint. Baron et al. (2011) [17] employ a robust optimization method to solve the problem of multiple periods locating facilities in a network facing uncertain demand. They construct two models of demand uncertainty: demand within a bounded and symmetric multi-dimensional box, and demand within a multi-dimensional ellipsoid. The robust optimization approach is considered to evaluate the potential performance by using an extensive numerical study. The results indicate that the solutions given by the robust model with ellipsoidal uncertainty set address the need for robustness directly by considering alternate links between facilities and nodes to accommodate the uncertain demand.

The closest work to this paper in using robustness optimization to facility location under uncertainty is Gülpinar et al. (2013) [18]. Gülpinar et al. (2013) considers a stochastic facility location problem in which multiple capacitated facilities serve customers with a single product, and a stockout probabilistic requirement is stated as a chance constraint. They discuss robust approximations to the problem by taking into account the maker’s risk preferences in order to incorporate information about the random demand distribution in the best possible, computationally tractable way. The results show that robust optimization strategies for facility location appear to have better worst-case performance than nonrobust strategies. The value of the price of robustness parameter appears to have a significant impact on the performance of robust facility location models. Rosa et al. (2014) [19] develop a new robust formulation for strategic location and capacity planning considering potential company acquisitions under uncertainty. Long-term logistics network planning is among the most difficult decisions for supply-chain managers. Álvarez-Miranda et al. (2015) [20] deals with a recoverable robust facility location problem. They develop a sophisticated algorithmic framework based on a Benders decomposition approach. The computational study illustrates how robustness and recoverability are expressed in the structure of optimal solutions, and it demonstrates the benefts of recoverable robust optimization (RRO) when compared to a robust optimization (RO) model without recovery. Mišković et al. (2016) [21] study a multi-source variant of the two-stage capacitated facility location problem (TSCFLP) and propose a robust optimization model of the problem that involves the uncertainty of transportation costs. The results demonstrate that the deviations of the total cost are increasing as the protection level increases up to certain level, and then take constant values as protection level further increases. Uncertainty in facility supply includes facility failure, production uncertainty and lead time uncertainty. This paper will study production uncertainty of facilities, that is to say, the uncertainty in facility capacity and supply. Since facility supply is not equal to the initial value in actual operation or facility supply is fluctuating randomly in facility location, there are many issues leading to the uncertainty in facility supply, such as partial failure due to natural disasters, influence of weather on the production capability of facility, gradual decreasing capacity during the maintenance or natural losses in facility application, etc. Facility location is a long-term strategic problem of decision and is affected by various uncertain factors. Therefore, supply uncertainty should be considered in facility location so as to design a system which can run normally with fluctuating parameters, or a system with strong anti-interference or robustness. It is difficult to predict the fluctuation in facility supply, which will be assumed as a random variable in this paper. When the facility location model is established, the chance constrained is introduced. Charnes and Cooper Chance (1957) [22] have established chance constrained programming, which is for the optimization under certain probability and a random programming approach specific to the necessity of making decision before the realization of random variables is detected as random variables are contained in the constraint condition. Then more and more scholars began to study this problem. Iwamura and Liu (1999) [23] have applied chance-dependent programming in the entire programming field and provided the corresponding heuristic algorithm. Liu (1996) [24] have resolved network optimization of chance-dependent programming by genetic algorithm and obtained good results. In this paper, facility location model based on uncertain supply with chance constraint is established and chance constrained programming is securely and approximately conversed into certain mathematical programming by Hoeffding’s inequation (Hoeffding, 1963) [25] Finally, mixed integer programming is resolved by CPLEX, and the influence of different confidence levels on cost and increasing cost of the system for improving the robustness of network are analyzed by random-generating numerical examples. Meanwhile, important parameters impacting on the model are conducted with sensitivity analysis.

## 2.1 Basic model

It is assumed there are m alternative establishment points and n demand points, and then the question to be considered: to satisfy demand points, a supplier need to establish production facilities to manufacture certain commodities. Each facility provides products for several demand points, and each demand point can be divided to several facilities. It is assumed that products can be subdivided infinitely. The studied issues of this paper include: how to select positions of production facilities and ensure their supply (capacity), how to distribute products to demand points and minimize sum of fixed cost of facility construction, variable cost of supply setting, and the cost of distributing products to demand points.

To facilitate expression, the author firstly defines used parameters and decision variables in this paper as follows:

Sets and parameters

M refers to set of alternative production facilities, M = {1, 2, ..., m};

N refers to set of demand points, N = {1, 2, ...,n};

fi refers to fixed cost for constructing production facilities in i position, iM;

qi refers to variable cost for setting unit supply in i position, iM;

cij refers to unit distribution cost of distributing demand point j to production facility i, iM, jN;

Bi refers to maximal supply of production facility which can be set in I position, iM;

dj refers to demand in demand point j, jN;

Decision variables $yi=1,Construct facility iniposition0,Or,i∈M,$

ui refers to supply of production facility in I position, iM;

xij refers to demand ratio of demand point j distributed to facility i, iM, jN.

When parameters in model are confirmed, location of production facility can be shown in following mathematical model: $(DF)min∑i=1mfiyi+qiui+∑i=1m∑j=1ncijdjxij$(1) $s.t.ui≤Biyi∀i∈M$(2) $∑i=1mxij=1 ∀j∈N$(3) $xij≤yi∀i∈M,∀j∈N$(4) $∑j=1ndjxij≤ui∀i∈M$(5) $yi∈{0,1}∀i∈M$(6) $xij≥0∀i∈M,∀j∈N$(7) $ui≥0 ∀i∈M$(8)

Objective function (1) refers to cost of minimized system, including fixed cost of facility construction, variable cost of supply setting, and the cost of distributing products to demand points. Constraint function (2) means capacity of production facility cannot exceed maximal supply which can be provided by the facility, and supply can only be set in production facilities with station. Constraint function (3) means the total ratio of demand of demand point j(jN) being distributed to all facilities is 1. Constraint function (4) means only production facilities with station can provide products for demand points. Constraint function (5) means total demand distributed to production facility should not exceed supply of this facility. Constraint function (6), (7), and (8) restrict decision variables.

There are many deficiencies in selected facility location model because many uncertain factors may happen during the running of facilities. Normal running of facilities will be influenced and great inconvenience will be brought to social life once being influenced by these uncertain factors. Uncertain facility supply is one of major factors which affect normal running of facilities. Therefore, the uncertainty of facility supply must be considered when selecting address for facilities. This paper assumes facility supply not equal to initially set supply, which means there is fluctuation in facility supply. Fluctuation of facility supply is assumed to be a random variable. Some scholars consider uncertainty of supply when designing logistics transportation network. For example, Thapalia et al. (2012) [8] mentioned design problem of single commodity flow transportation network in random edge capacity with target functions of network constructing and running cost and maximal expectation for not satisfying demands. Thapalia et al. (2012) [8] described the possible appearing of uncertain parameters by scenario analysis. However, it is difficult to set scenes or happening ratio of scenes. When there are many scenes, the problem scale will increase, and solving of model will cost more time. Then the model cannot be applied in practical problem. Aiming at deficiencies in above research, in the following text, this paper assumes the fluctuation of facility supply as random variables with incomplete probability distribution information, and comes up with robust location model based on uncertain facility supply. The advantage of this model is that there is no need to know specific probability distribution of fluctuation of facility supply, but only needs to estimate its interval.

## 2.2 Robust model of uncertain supply

This section assumes the random fluctuation of facility supply based on robust location model of uncertain facility supply. The optimal solution of model (DF) is ${x}_{1}^{\ast }$ if the location decision is designed according to confirmed model (DF). Obviously, ${x}_{1}^{\ast }$ might be non-feasible solution when there is fluctuation in facility supply. At this moment, production facility cannot provide enough products to satisfy demand point, which will affect normal running of system and bring poor robustness to system. Model in this section is: there is certain probability for system to run normally even it is affected by fluctuation of facility supply during running. That means, the robust model in this section makes system, with certain probability, able to run normally even parameters fluctuate randomly. The system owns certain robustness or anti-interference ability.

With incomplete probability distribution information, the fluctuation of facility supply is an uncertain random variable. Making ũi(iM) as random supply fluctuation, it is assumed that: $u~i=uiβiη~∀i∈M$

Among which, βi ≥ 0 can be used to measure the uncertainty level of fluctuation of facility supply. Obviously, bigger βi means higher uncertainty level of supply. $\stackrel{~}{\eta }$ means random variables which symmetrically distribute in the interval [-1, 1] (the distributions of $\stackrel{~}{\eta }$ and − $\stackrel{~}{\eta }$ are the same). E[Z] refers to the mathematical expectation of random variable Z. It is assumed E[ $\stackrel{~}{\eta }$ ] = 0, and get E[ũi] = 0(iM) from definition, which means the expected value of random fluctuation of supply is 0. Ng and Waller(2010) [15] showed random fluctuation of demand and supply with above form, studied personnel evacuating issue based on demand inflation and supply tightening. To make evacuation planning more reliable, he considered more evacuating personnel and lesser supply.

At this moment, the supply of facility i(iM) is uiũi (uiũi and ui + ũi means same random variable for $\stackrel{~}{\eta }$ distribute symmetrically), which is ${u}_{i}-{u}_{i}{\beta }_{i}\stackrel{~}{\eta }=\left(1-{\beta }_{i}\stackrel{~}{\eta }\right){u}_{i}.$ When βi = 0.1, the supply of facility i(iM) is (1 − 0.1 $\stackrel{~}{\eta }$)ui, which is between the interval [0.9ui, 1.1 ui].

Because ũi(iM) is a random variable, constraint function (5) cannot be simply turned to be (5) at this moment: $∑j=1ndjxij≤ui−u~i∀i∈M$(9)

Chance constraint is introduced here. Considering the violated probability of constraint $\sum _{j=1}^{n}{d}_{j}{x}_{ij}\le {u}_{i}-{\stackrel{~}{u}}_{i}\left(i\in M\right),$ which is the establishing probability of $\sum _{j=1}^{n}{d}_{j}{x}_{ij}>{u}_{i}-{\stackrel{~}{u}}_{i}$ (iM), the violated probability of constraint function (5) is controlled within a satisfied range pi(0 < pi ≤ 1). In consideration of: $P∑j=1ndjxij>ui−u~i≤pi∀i∈M$(10)

In the left of inequation (9), the probability $P\left\{\sum _{j=1}^{n}{d}_{j}{x}_{ij}>{u}_{i}-{\stackrel{~}{u}}_{i}\right\}$ means the probability that demand distributed to production facility is larger than facility supply, which is the probability that facility i cannot satisfy distributed demand. Obviously, decision maker does not want this condition happen. We can assume the probability is small enough. This probability is assumed to be smaller than pi, then the establishing probability of constraint function (5) is $P∑j=1ndjxij≤ui−u~i≥1−pi∀i∈M$

1 − pi is also called confidence level. Confidence level is also called reliability degree or confidence coefficient, which refers to degree of trust of specific individuals on authenticity of particular propositions.

To sum up, robust facility location issue based on uncertain facility supply can be shown in following mathematical model: $(S-DF)min∑i=1mfiyi+qiui+∑i=1m∑j=1ncijdjxijs.t.ui≤Biyi∀i∈M∑i=1mxij=1∀j∈Nxij≤yi∀i∈M,∀j∈NP∑j=1ndjxij>ui−u~i≤pi∀i∈Myi∈{0,1}∀i∈Mxij≥0∀i∈M,∀j∈Nui≥0 ∀i∈M$

Obviously, smaller value of pi means more reliable system and stronger robustness of system. We only need to know interval of random variable but no need to know specific probability distribution when using model (S-DF). This method is frequently used in robust optimization. Considering of constraint condition not being satisfied by decision under adverse condition, chance constraint rule adopts one principle: decision is allowed to not satisfy constraint condition to some extent, while the establishing probability shall not be smaller than a certain level.

## 2.3 Switching of robust model

Model (S-DF) contains chance constraint. There are about two ways to deal with rule of chance constraint. The first one is to translate chance constraint rule into confirmed rule with some switching methods, and then solve problems by using of theories of confirmed rules. The second one is to deal with chance constraint by random simulation technique, select the superior and eliminate the inferior with genetic algorithm, and get objective function’s optimal value and decision variable’s optimal solution set of chance constraint rules. According to special form of random variables, this section uses Safety Approximate chance constraint of Hoeffding’s inequation.

Let’s first introduce Hoeffding’s inequation below before disposing chance constraint (9).

#### Lemma 1

(Hoeffding’s inequation) It is assumed X1, X2, …, XK are mutually independent variables, and Xk is bounded almost everywhere, which is $P\left\{{X}_{k}\in \left[{a}_{k},{b}_{k}\right]\phantom{\rule{1em}{0ex}}\right\}=1$ (ak, bk is a known parameter), and 1 ≤ kK. Make $s=\sum _{k=1}^{K}{X}_{k}.$ Then the inequation $P\left\{s-E\left(s\right)\ge t\right\}\le \mathrm{exp}\left(-2{t}^{2}/\sum _{i=1}^{n}\left({b}_{i}-{a}_{i}{\right)}^{2}\right)$ is true to any positive number t.

#### Deduction 1

X is assumed to be a random variable, P{X ∈ [−1, 1] } = 1, and E[X] = 0. Then inequation P{Xt} ≤ exp (−t2/2) is true to any positive number t.

#### Proof

it can be known from Hoeffding’s inequation, $PX≥t=PX−E(X)≥t≤exp−2t2/(1−(−1))2=exp−t2/2$ □

#### Theorem 1

It is assumed that $\stackrel{~}{\eta }$ is random variable symmetrically distributed in interval [−1, 1], and E[$\stackrel{~}{\eta }$] = 0.

Make ${\rho }_{i}=\sqrt{-2\left(\mathrm{ln}{p}_{i}\right){\beta }_{i}^{2}}\phantom{\rule{1em}{0ex}}{u}_{i},$ and 0 < pi ≤ 1. If $\sum _{j=1}^{n}{d}_{j}{x}_{ij}\le {u}_{i}-{\rho }_{i},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{then}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}P\left\{\sum _{j=1}^{n}{d}_{j}{x}_{ij}>{u}_{i}-{\stackrel{~}{u}}_{i}\right\}\le {p}_{i}.$

#### Proof

if $\sum _{j=1}^{n}{d}_{j}{x}_{ij}\le {u}_{i}-{\rho }_{i},$ and then $P∑j=1ndjxij>ui−u~i≤Pui−u~iρiuiβi≤exp−ρi22uiβi2$

The reason the first inequation being true is: make A random event $\sum _{j=1}^{n}{d}_{j}{x}_{ij}>{u}_{i}-{\stackrel{~}{u}}_{i},$ and B random event uiρi >uiũi. If A happens, because of $\sum _{j=1}^{n}{d}_{j}{x}_{ij}\le {u}_{i}-{\rho }_{i},$ and then B happens, A ⊆ B. Therefore: $P∑j=1ndjxij>ui−u~i≤Pui−u~i

The reason the last equation being true is $\stackrel{~}{\eta }$ is random variable symmetrically distributed in interval [−1, 1]. The reason the last inequation being true is that, according to deduction 1, make $\mathrm{exp}\left(-\frac{{\rho }_{i}^{2}}{2{\left({u}_{i}{\beta }_{i}\right)}^{2}}\right)={p}_{i},$ and ρi = $\sqrt{-2\left(\mathrm{ln}{p}_{i}\right){\beta }_{i}^{2}}\phantom{\rule{1em}{0ex}}{u}_{i}.$

Therefore, when $\sum _{j=1}^{n}{d}_{j}{x}_{ij}\le {u}_{i}-{\rho }_{i},$ $P∑j=1ndjxij>ui−u~i≤Pui−u~i

And $\mathrm{exp}\left(-\frac{{\rho }_{i}^{2}}{2{\left({u}_{i}{\beta }_{i}\right)}^{2}}\right)\phantom{\rule{2em}{0ex}}=\phantom{\rule{2em}{0ex}}{p}_{i},$ and then $P\left\{\sum _{j=1}^{n}{d}_{j}{x}_{ij}>{u}_{i}-{\stackrel{~}{u}}_{i}\right\}\le {p}_{i}$ □

Make safety approximate of chance constraint (9) to be $\sum _{j=1}^{n}{d}_{j}{x}_{ij}\le {u}_{i}-\sqrt{-2\left(\mathrm{ln}{p}_{i}\right){\beta }_{i}^{2}}\text{\hspace{0.17em}}{u}_{i},$ and then get following formula about mixed interger programming: $(S-DF1)min∑i=1mfiyi+qiui+∑i=1m∑j=1ncijdjxijs.t.∑j=1ndjxij≤liui∀i∈Mui≤Biyi∀i∈M∑i=1mxij=1∀j∈Nxij≤yi∀i∈M,∀j∈Nyi∈{0,1}∀i∈Mxij≥0∀i∈M,∀j∈Nui≥0∀i∈M$(11)

Among which, ${l}_{i}=1-\sqrt{-2\left(\mathrm{ln}{p}_{i}\right){\beta }_{i}^{2}}.$

When constraint function (10) is true, constraint function (9) is true too. So (10) is the safety approximate of (9). Safety approximate is an important method to deal with chance constraint in robust optimization. It is known from theorem 1 that: the feasible solution of question (S-DF1) is must the feasible solution of question (S-DF).

When pi = 1, that is $P\left\{\sum _{j=1}^{n}{d}_{j}{x}_{ij}>{u}_{i}-{\stackrel{~}{u}}_{i}\right\}\le 1\left(\mathrm{\forall }i\in M\right).$ At this moment, li = 1, constraint function (10) is equivalent to $\sum _{j=1}^{n}{d}_{j}{x}_{ij}\le$ ui, question (S-DF1) is translated to confirmed model (DF), which means the facility location of uncertain facility supply is not considered. When βi = 0, ũi = 0, the random fluctuation of facility supply is confirmed to be 0. Now li = 1. Obviously, (S-DF1) is translated to confirmed model (DF).

Question (S-DF1) is a question about mixed integer programming. Numerical examples in following section solve model (S-DF1) by using of CPLEX, mixed integer programming software.

## 3 Numerical examples and analysis

This section analyzes the change of total cost under different confidence level with random generated numerical examples, makes sensitivity analysis to important parameters which affect model, and finds influence of change of parameter on total cost. Randomly generate m potential demand points, which are also alternative facility points, on plane of [0, 100] × [0, 100]. The fixed cost for constructing stations of facilities is 1000, the variable cost for setting facility supply is 10, and the cost of providing service to demand point j by facility i = h × eij. eij means the distance from facility i to demand point j. Demand of demand point randomly generates in interval [500, 1000]. Find the optimal solution of question (S-DF1) with CPLEX 12.4, and make experiment in computer with CPU of Intel Core i5-2450M 2.50GHz and internal storage of 4.00GB.

## 3.1 Total cost under different confidence levels

In this section, 50 demand points are randomly generated in plane of [0,100] × [0,100]. Uncertainty level of random fluctuation of facility supply is randomly generated in plane of [0.05, 0.2]. By solving question (S-DF1), the total cost is shown in figure 1 under different confidence levels. It can be seen from figure 1 that, total cost increases along with the increasing of confidence level. When confidence level is higher, the establishing probability of constraint function (5) is larger, and value of li becomes smaller. That means more facilities or supply are needed to avoid insufficient supply brought by random fluctuation of facility supply. It can also be found from figure 1 that, the curve becomes steeper and steeper when confidence level increases. It shows when confidence level is in low level, slight increasing of system cost will largely increase confidence level. When confidence level is in high level, such as over 90%, it is hard to be improved even to improve system cost.

Figure 1

Total cost and confidence level

Obviously, high confidence level means stronger robustness of system. Strong robustness is always got at the cost of increased cost. Above analysis only directly shows relation between confidence level and total cost through changing trend of figure. In following text, we will quantize this relation and provide data support for decision makers. It can be easily found that when li ≤ 1, solution of constraint condition (10) must satisfy constraint condition (5). It is assumed the optimal target function value of confirmed question (DF) is P*, optimal target function value of robust question (S-DF1) is P1*, and obviously P* ≤ P1*, which means more facilities or supply are needed to ensure normal running of facility supply within certain probability under random fluctuation. Robustness of system is got at the cost of increased cost (cost opposite to confirmed question). How much does it increase, and how to confirm if this increasing is worthy? To study these questions, following measurement is defined: $cos⁡t1=P1∗−P∗cos⁡t2=P1∗−P∗P∗$

cos t1 is the absolute difference between optimal solution of robust question and optimal solution of confirmed question, and cos t2 is the relative difference between optimal solution of robust question and optimal solution of confirmed question. Two of them show that, to avoid fluctuation of facility supply in system, the cost of system will increase, which is the price decision makers have to pay.

As to above randomly emerging numerical examples, the optimal value of confirmed model is P* = 402339 when the supply of facility is confirmed. Table 1 shows absolute and relative values added of total cost under difference confidence levels. It can be seen from table 1 that how total cost increases along with the increasing of system robustness. After weighing relation between increased cost and confidence level in table 1, decision makers can: 1) calculate obtainable confidence level according to budget cost, or calculated required cost according to limitation of confidence level. 2) Get maximal confidence level increasing among increasing of marginal cost by marginal analysis, and design facility network.

Table 1

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The relation between relative increasing value of system cost and confidence level is shown in figure 2. It is shown by figure 2 that, curve becomes more abrupt when confidence level improves, which means system cost increases faster and faster along with the improving of confidence level.

Figure 2

Relative increased cost and confidence level

## 3.2 Sensitivity analysis of model parameters

(1) Sensitivity analysis of uncertainty level of facility supply

Make sensitivity analysis for uncertainty level of facility supply, which is value of βi. Figure 3 shows the change of total cost when confidence level changes in three uncertain levels of supply when βi is separately 0.10, 0.15 and 0.20. It can be seen from figure 3 that, in same confidence level, cost becomes larger when uncertainty level improves, which means total cost is sensitive to uncertainty level. As confidence level improves, gaps between three curves become larger and larger, which means the sensitivity of total cost on uncertainty level of supply increases when confidence level improves.

Figure 3

Total cost and confidence level in three uncertainty levels

(2) Sensitivity analysis of demand

Make sensitivity analysis for demand, which is value of dj. Figure 4 shows the change of total cost along with the change of confidence level when demand separately increases 10% and decreases 10% and uncertainty level of supply βi = 0.10. It can be seen from figure 4 that, in same confidence level, cost becomes larger when demand improves. Curve translates upward or downward when demand increases or decreases, which means total cost is sensitive to demand, and the sensitivity does not change when confidence level changes.

Figure 4

Total cost when demand changes

(3) Sensitivity analysis of fixed cost

Make sensitivity analysis for fixed constructing cost, which is value of fi. Make uncertainty level of facility supply βi = 0.10. figure 5 shows the change of total cost along with the change of confidence level when fixed constructing cost separately increases 10% and decreases 10%. It can be seen from figure 5 that, in same confidence level, cost becomes larger when fixed constructing cost improves. In figure 5, three curves do not have large gaps, which means total cost is not that sensitive to fixed constructing cost. Especially when confidence level is relatively high, three curves almost coincide with each other.

Figure 5

Total cost when fixed cost changes

## 4 Conclusions

Facilities like warehouse, retail store, gas station, and parts manufacturing plant, whose normal running is crucial to daily life of people, are indispensable part in daily life. In recent years, because of natural disaster and deliberate sabotaging, facilities may be affected by uncertain events during running process, which prevents their normal running upon original plan. Reasons like natural hazards and weather factors make manufacturers fail to provide sufficient products to satisfy consumer demand, and result in phenomenon like low service efficiency of production facilities and failing to satisfy demand. This paper studies location issue of robust facilities when supply of production facilities fluctuates randomly. Firstly, this paper comes up with robust location model based on chance constraint. Then, according to features of random variables, it translate safety approximate of chance constraint into general linear constraint by using of Hoeffding’s inequation, translate robust model into linear model, and make model easy to solve. Obviously, the robustness of system is got at the price of increased cost, which means decision makers must properly increase system cost to avoid the influence from random fluctuation of facility supply to normal running of system. Numerical examples randomly generated shows that total cost changes along with the change of confidence level. The result shows that, when confidence level is relatively low, slight increasing of cost will largely improve confidence level. Robust location model adopted in this paper definite relation between the increasing of system cost and its robustness, and provides quantitative data support for decision makers.

## Acknowledgement

This paper is sponsored by the National Science Foundation (Grant No. 71402048), Hubei Provincial Department of Education(Grant No. D20162204), National Social Science Fund of China (Grant No. 16CXW019) and Fundamental Research Funds for the Central Universities (Grant No. 31541411307)

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Accepted: 2016-09-26

Published Online: 2017-03-24

Citation Information: Open Physics, Volume 15, Issue 1, Pages 87–96, ISSN (Online) 2391-5471,

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