A multiperiod distribution network design model under demand uncertainty
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Abstract
Supply chain management is taken into account as an inseparable component in satisfying customers' requirements. This paper deals with the distribution network design (DND) problem which is a critical issue in achieving supply chain accomplishments. A capable DND can guarantee the success of the entire network performance. However, there are many factors that can cause fluctuations in input data determining market treatment, with respect to shortterm planning, on the one hand. On the other hand, network performance may be threatened by the changes that take place within practicing periods, with respect to longterm planning. Thus, in order to bring both kinds of changes under control, we considered a new multiperiod, multicommodity, multisource DND problem in circumstances where the network encounters uncertain demands. The fuzzy logic is applied here as an efficient tool for controlling the potential customers' demand risk. The defuzzifying framework leads the practitioners and decisionmakers to interact with the solution procedure continuously. The fuzzy model is then validated by a sensitivity analysis test, and a typical problem is solved in order to illustrate the implementation steps. Finally, the formulation is tested by some differentsized problems to show its total performance.
Keywords
Supply chain Distribution network design Uncertain demand Fuzzy logicIntroduction
Moving towards competitive markets compels the manufacturers to increase their qualifications so that they can best fulfill the customers' needs. The supply chain management (SCM) concept arose as an effective managerial tool to enhance the customers' satisfaction within the last 20 years in particular (Azaron et al. 2008;Jayaraman and Ross 2003). Generally, a supply chain (SC) can be defined as a set of plants, distribution centers, and customers in which the products are completed once they are transmitted downstream the chain and delivered to the customers (Altiparmak et al. 2006;Blackhurst et al. 2004).
Distribution network design (DND) is one of the primitive principles in establishing a successful SC. It is associated with the category of productiondistribution and facility locationallocation problems (Altiparmak et al. 2009). The researchers investigate the distribution networks under two distinctive categories including strategic and tactical levels. The first level concerns with whatever affects the entire network configuration (e.g., the numbers, capacities, and locations), and the second level deals with whatever affects the aggregate quantities (e.g., material handling, processing, and distribution) (Santoso et al. 2005). Consequently, considering the two planning levels simultaneously is very fruitful in enhancing the network performance.
A DND problem can also incorporate five aspects according to the necessity (please refer to Tang (2006) for more description):

Network configuration (NC)

Product assignment (PA)

Customer assignment (CA)

Production planning (PP)

Transportation planning (TP)
Regarding the above, the proposed model covers the first four aspects at the same time.
Some other issues are expected in the model structure likewise (e.g., the number of consideration periods, sources, echelons, and commodities). For example, Amiri (2006) suggested a multicapacity distribution network in deterministic environments so that both strategic and tactical levels were included. Gupta and Maranas (2003) concentrated on the tactical level in a multicommodity, multiechelon, multiperiod problem. The authors considered demand as the uncertain parameter. Selim and Ozkarahan (2006) developed a fuzzy multiechelon, multicommodity, multicapacity problem by considering both levels. They introduced the demand as the uncertain part likewise. Liu et al. (2006) introduced a multicommodity DND problem and dealt with the potential uncertainties by the combination of gray and fuzzy factors. Azaron et al. (2008) proposed a multicommodity, multisource model and solved it by the goal attainment technique. However, they extended the uncertainty and vagueness to all three demand, supply, and processside risks. Gumus et al. (2009) applied a neurofuzzy technique for demand uncertainty in a multiechelon condition. Xu et al. (2009) developed a multiechelon, multisource problem where the demand, supply, and process follow a fuzzy nature. They applied a heuristic solution approach called spanning tree technique that originated from genetic algorithm. Peidro et al. (2010) proposed a multiechelon, multiperiod, multicommodity, multisource mathematical modeling under fuzzy programming so that all three mentioned sides were uncertain. Compared with the mentioned researches, our developed model lies in a threeechelon, multisource, multiperiod, multicommodity, singlecapacity category in which the two levels are planned. The environment under consideration faces a demandside risk which is tackled with by the fuzzy mathematical modeling. A brief overview on DND literature is depicted in Table 1.
The rest of the paper is organized as follows: model formulation is presented in the next section. Solution methodology is described in the third section, which also demonstrates how the fuzzy steps can be implemented. The fourth section pertains to the computational study, and finally, the conclusions and future research interests are discussed in the ‘Conclusions’ section.
Model formulation
 1.
Assumptions

Demand parameter follows a fuzzy nature.

A given plant or distributor can be open or close in each horizon.

The number of practicing facilities cannot exceed the predefined upper bound.

The processing capacity of each plant and distributor is known with respect to the commodities.

The facility capacity is fixed during the total horizon periods.

Each tier can supply its needs from more than one member of the upper tier.
 2.
Indices

C: set of commodities (c &z.epsi; C)

P: set of plants (p &z.epsi; P)

J: set of distributors (j &z.epsi; J)

I: set of customers (i &z.epsi; I)

H: set of planning horizons (h &z.epsi; H)
 3.
Parameters

FC_{ ph }: annual fixed cost of plant p in period h

FC'_{ jh }: annual fixed cost of distributor j in period h

LC_{ pchj }: logistics cost for supplying commodity c in period h from plant p to distributor j

LC'_{ jchi }: logistics cost for supplying commodity c in period h from distributor j to customer i

d_{ ich }: demand of customer i for commodity c in period h

α_{ pc }: capacity of plant p for commodity c

α'_{ jc }: capacity of distributor j for commodity c

P_{Upper}: the upper bound for the number of open plants

J_{Upper}: the upper bound for the number of open distributors
 4.
Decision variables

x_{ pchj }: the proportion of commodity c in period h transported from plant p to distributor j

y_{ jchi }: the proportion of commodity c in period h transported from distributor j to customer i

Z_{ ph }: the binary variable that takes 1 if plant p is open in period h and 0, otherwise

Z'_{ jh }: the binary variable that takes 1 if distributor j is open in period h and 0, otherwise

LQ_{ chj }: the left quantity of commodity c in period h for distributor j
An overview on DND literature
Articles  Aspects  Features  Nature  Uncertain parameter(s)  Solution approach  

NC  PA  CA  PP  TP  Product  Stage  Period  Source  Capacity  
Jayaraman and Ross (2003)  ***  ***  ***  Multiple  Three  Single  Single  Single  Deterministic    Simulated annealing  
Santoso et al. (2005)  ***  ***  ***  ***  Multiple  Two  Single  Multiple  Single  Stochastic  Demand, supply, and process  Sample average approximation scheme and accelerated Benders' decomposition  
Altiparmak et al. (2006)  ***  ***  ***  Single  Multiple  Single  Single  Single  Deterministic    Genetic algorithm  
Selim and Ozkarahan (2006)  ***  ***  ***  Multiple  Three  Single  Single  Multiple  Fuzzy  Demand  Fuzzy multiobjective programming  
Goh et al. (2007)  ***  ***  Single  Two  Single  Multiple  Single  Stochastic  Demand and supply  Heuristic algorithm  
Chen et al. (2007)  ***  ***  ***  Multiple  Three  Multiple  Multiple  Single  Fuzzy  Demand  Fuzzy programming  
You and Grossmann (2008)  ***  ***  ***  Multiple  Multiple  Multiple  Single  Single  Probabilistic  Demand  ϵConstraint  
Cakir (2009)  ***  ***  ***  ***  Multiple  Two  Single  Multiple  Single  Deterministic    Benders' decomposition  
Altiparmak et al. (2009)  ***  ***  ***  ***  Multiple  Multiple  Single  Single  Single  Deterministic    Genetic algorithm  
Georgiadis et al. (2011)  ***  ***  ***  ***  Multiple  Three  Single  Multiple  Single  Stochastic  Demand  Branchandbound  
Cintron et al. (2010)  ***  ***  Single  Three  Single  Single  Multiple  Stochastic  Demand  Goal programming  
HajiaghaeiKeshteli (2011)  ***  ***  Single  Two  Single  Multiple  Single  Deterministic    Genetic algorithm and artificial immune algorithm  
CardonaValdes et al. (2011)  ***  ***  ***  ***  Single  Three  Single  Multiple  Single  Stochastic  Demand  Lshaped algorithm  
Rezapour and Farahani (2010)  ***  ***  Single  Three  Single  Multiple  Multiple  Deterministic    A modified projection method  
Park et al. (2010)  ***  ***  Single  Three  Single  Single  Single  Deterministic    A twophase heuristic algorithm  
Our proposed model  ***  ***  ***  ***  Multiple  Three  Multiple  Multiple  Single  Fuzzy  Demand, supply, and process  Fuzzy programming 
Considering the presented formulation, Equation 1 is the objective function (OF) which consisted of logistic and annual fixed costs. In other words, the first two parts deal with the total sum of logistic costs (i.e., variable costs including processing, transportation, etc.). It is calculated for both the practicing plants and distributors separately. This type of cost is obtained from the commodity volume multiplied by the associated capacity. For example, the logistic cost of plants is determined by the corresponding unit logistic cost, to supply a given commodity from a plant to a distributor in a specific period, multiplied by the satisfied quantities. The values of responded needs are obtained from the multiplication of the given capacity by the satisfied proportion. The same calculations can be presented for distributors likewise. Furthermore, the fixed cost is realized for each of the plants and distributors with respect to the extant cost once the facility is practicing in the corresponding period. Equation 2 shows the fraction of satisfied demands for each of the commodities in each planning horizon with respect to the existing customers. However, a potential customer can acquire its demanded commodities from a set of distributors instead of focusing on a single one. Equation 3 guarantees that the transportation quantity from plants to distributors cannot surpass the given plant capacity, if it is open in the mentioned period. Equation 4 denotes that the left quantity for each commodity and distributor in each period can be calculated by the sum of transported quantities to customers subtracted from the sum of transported quantities to a distributor added to the left quantity in the previous period. Equation 5 is the same as Equation 3 but concerns the distributors and affirms that the output of each distribution center does not exceed its extant capacity, restricted by the remaining commodities. Moreover, the volume of the left commodities cannot exceed the distributor capacity. Equations 6 and 7 state that the number of open plants and distribution centers in each of the planning horizons should not exceed their upper bounds (i.e., the maximum allowed numbers), respectively. Furthermore, Equation 8 deals with the nature of existing variables.
Solution methodology
Since we presumed that the developed distribution network practices in a vague environment, demand uncertainty can be represented by fuzzy numbers efficiently. In order to solve the model, we applied the fuzzy approach suggested by Jimenez et al. (2007) which consists of a threephase interaction algorithm.
Converting the fuzzy model into its crisp equivalent
The uncertainty in the nature of fuzzy problems makes the decisionmakers (DMs) find a solution so that both feasibility and optimality conditions can be satisfied efficiently. Consequently, there has been a great effort on determining different methods which could fulfill the two above characteristics (e.g., refer to Rommelfanger and Slowinski (1998)). Some papers have also discussed the ranking methods besides their justifications. We applied the method introduced by Jimenez (1996) to our problem.
Interaction resolution method
In this section, we explain the three levels of the resolution method in which the DM is involved interactively. Firstly, the linear fuzzy model transformed into its crisp mode should be solved regarding each different value of αcuts independently. In order to make the comparison more clearly, the solutions are compared against three quantitative measures as follows:

Fixed cost: it can be calculated by the sum of annual fixed costs of practicing facilities in planning horizons.

Logistics cost: it can be calculated by the sum of cost of supplying commodities from plants to customers in the given periods.
 Average customers' satisfaction: it can be determined by Equation 17 so that the average satisfied demands are determined as follows:$\begin{array}{cc}\hfill {\displaystyle \sum _{h=1}^{H}\frac{\left({\displaystyle \sum _{j=1}^{J}{y}_{\mathit{jchi}}{d}_{\mathit{ich}}}/{d}_{\mathit{ich}}\right)}{H}}\hfill & \hfill ;\forall c,i.\hfill \end{array}$(17)
Note that P is the joint acceptation index obtained by the linear combination of the two mentioned degrees and θ &z.epsi; [0,1] points to the relative importance the DM assigns to feasibility in comparison with global satisfaction.
Computational study
In this section, the developed fuzzy model is tested by the sensitivity analysis, and then a numerical example is solved to show its applicability and efficiency in realworld problems. Finally, the total performance of the model is investigated by differentsized problems. The calculations are run by Lingo 8 solver on a PC with characteristics of Intel®, Pentium® 4, central processing unit (CPU) 3.20 GHz, and 2,046 MB of RAM.
Sensitivity analysis
Applying the sensitivity analysis helps us understand how accurate the problem performance is. This is demonstrated by changes in model outputs according to manipulations in parameters. In other words, it is assumed that the input data may vary independently. Hence, we have to investigate the potential changes in values of model variables. To do so, a typical example whose data have been created randomly is considered here. The results are depicted in Table 2.
Implementation of interactive phases
This section is presented in order to show how the model can be implemented with respect to the three interactive solution steps. To do so, the DM is required to specify the aspiration level of each three introduced measures ${\mu}_{\tilde{z}}\left(\mathrm{OF}\right)$ as follows. It should be noted that the measures' quantity is calculated according to the lower and upper bound results, obtained from different cuts:

Fixed cost: z = US$17,650,000 and z + t = US$18,800,000.

Logistic cost: z = US$945,000 and z + t = US$1,060,010.

Average customers' satisfaction: z  t = 77.3% and z = 94.5%.
The global satisfaction degree can be determined by weighting the measures which is done here by considering equal importance (i.e., ω_{1} = ω_{2} = ω_{3}).
The results of fuzzy sensitivity analysis
Variation source  Fixed costs  Logistic costs  OF  Average customers' satisfaction 

(US$)  (US$)  (US$)  (%)  
Fixed costs  18,095,016  947,500  19,042,516  89.2 
Logistic costs  17,650,000  765,014  18,415,014  83.1 
Demand  17,923,000  1,100,540  19,023,540  85.7 
Capacities  17,025,000  945,000  17,970,000  86.2 
The comparison results between feasibility sets and deterministic state
αcut  OF (US$)  CPU time (s)  Iteration  Fixed cost (US$)  Logistic cost (US$)  Average customers' satisfaction (%) 

0  18,595,000  14  52,741  17,650,000  945,000  78.6 
0.1  18,602,400  13  47,856  17,665,500  952,400  81.4 
0.2  18,784,700  15  58,996  17,823,000  961,700  86.2 
0.3  18,937,300  14  44,258  17,965,000  972,300  85.7 
0.4  19,204,020  21  68,439  18,220,000  984,020  88.9 
0.5  19,558,500  26  89,477  18,575,000  983,500  94.5 
0.6  19,510,750  27  71,644  18,522,000  988,750  92.8 
0.7  19,810,600  34  83,355  18,800,000  1,010,600  92.1 
0.8  19,844,330  31  67,195  18,800,000  1,044,330  87.7 
0.9  19,844,050  50  85,008  18,800,000  1,044,050  85.5 
1  19,860,010  63  107,414  18,800,000  1,053,070  77.3 
Deterministic  19,789,840  19  104,220  18,800,000  1,060,010  81.2 
Referring to Table 3, it can be observed that the fuzzy formulation led to lower costs for all cuts compared with the deterministic mode. However, cuts greater than 0.6 resulted in an equal value for the annual fixed cost. In order to compare the results with respect to the elapsed solution time (i.e., the CPU time), it is apparent that the deterministic mode needs a rather small time although the lower cuts do the same. However, it indicates an approximately ascending order, so the last cuts required much more time that the first ones. On the other hand, the average customers' satisfaction measure showed the third lowest value for the deterministic mode. However, the measure increases as the cut magnitude goes up until it reaches the maximum value and then starts to decrease.
The best cut with respect to the DM's opinion
α  ρ _{α}  Ω  ρ _{Ω}  P 

0  0  0.02  0  0 
0.1  0.05  0.09  0.1  0.075 
0.2  0.15  0.26  0.33  0.24 
0.3  0.25  0.32  0.4  0.325 
0.4  0.4  0.47  0.55  0.475 
0.5  0.55  0.71  0.78  0.765 
0.6  0.7  0.65  0.7  0.7 
0.7  0.85  0.81  0.93  0.89 
0.8  0.9  0.82  0.95  0.925 
0.9  0.95  0.77  0.86  0.905 
1  1  0.66  0.72  0.86 
Total performance of the model
The results when considering differentsized problems
Problem size p× c× j× i× h  CPU time (s)  OF (US$)  Average customers' satisfaction (%)  Variables  Constraints 

2 × 2 × 4 × 6 × 2  68  16,855,025  81  418  420 
3 × 2 × 5 × 8 × 2  217  17,120,250  75  622  754 
3 × 2 × 5 × 9 × 3  873  23,864,700  86  1,216  1,436 
3 × 3 × 6 × 9 × 3  1,307  25,004,610  79  1,484  1,668 
4 × 3 × 6 × 10 × 2  4,876  93,324,080  80  1,560  1,610 
5 × 3 × 10 × 12 × 3  9,654  26,080,950  76  3,044  3,152 
5 × 4 × 10 × 15 × 4  14,662  35,775,940  78  5,696  5,842 
It is clear that because of intrinsic intricacy, it takes a long time for the model to respond as the dimension enlarges. These problems belong to the NPhard category, and the solution time increases exponentially as the problem size rises, in other words. For example, in Table 5, there is a noticeable difference in CPU time while even one of the problem elements changes to a higher value. This can be easily observed as the required solution time gets multifold once the considered problems become larger. For instance, the second problem is larger than the first one just as one unit for the number of plants and two units for the number of distribution centers, but its elapsed time is 3.19 times as much as that of the first one. Like the previous comparison, the third problem needed a more remarkable time than the second one while the number of considered customers and planning horizons has been added by one unit. That same analysis can be carried out for the rest of the problems as well. Consequently, it is suggested that intelligent computation approaches be applied for largesized cases in particular. This makes the mathematical model be applied efficiently for any real market conditions.
Conclusions
The noticeable increase in competition amongst the market holders makes the SCM a necessary tool in fulfilling the customers' needs. DND is of great importance as it establishes the first steps in handling the network. Consequently, a multiperiod, multicommodity, multisource DND problem was considered in this paper. We extended our formulation to an uncertain environment, due to receiving demands, by fuzzy mathematical programming. The fuzzy model consisted of a threeinteraction resolution method, so it included the DMs throughout the solution procedure. Afterwards, it was validated by a sensitivity analysis, and then a numerical example was solved in order to give a picture of how the method steps can be implemented. The consideration of differentsized problems was also carried out and indicated to the intrinsic complexity.
The presented model focused on increasing the network design efficiency so that it can help more to the drawn decisions. Considering that demand uncertainty causes the disruptions, it threatens the network (e.g., the Bullwhip effect) to decrease to a great extent as the practitioners are well equipped by demand fluctuations a priori. In fact, this issue helps in the necessity of the least changes in executive plans.
For future studies, it is suggested that the presented model be solved by more effective tolls like metaheuristic or exact methods and even compared with each other. The model can also be enriched by added concepts like the inventory control management. Furthermore, the problem can be considered by other backgrounds like economic viewpoints at the same time.
Notes
Acknowledgments
We would like to express our appreciation to the University of Tehran for the financial support for this study (grant number 8108023/1/18). The authors are also grateful to all persons who helped them in developing this paper.
Supplementary material
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