International Journal of Production Economics 218 (2019) 96–105
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International Journal of Production Economics journal homepage: www.elsevier.com/locate/ijpe
Design of robust distribution network under demand uncertainty: A case study in the pulp and paper
T
Mustapha Ouhimmoua,∗, Mustapha Nourelfathb, Mathieu Bouchardb, Naji Brichab a b
Department of Systems Engineering, École de Technologie Supérieure, Montréal, QC, H3C 1K3, Canada Department of Mechanical Engineering, Université Laval, Québec, QC, G1V 0A6, Canada
ARTICLE INFO
ABSTRACT
Keywords: Robust optimization Distribution network design Benders decomposition Demand uncertainty
The design of a supply chain network helps companies in dealing with variability and uncertain evolution of demand over time. An efficient supply chain network may contribute to fulfill the customers’ demands in a quick and least cost manner. Therefore, it is important to solve the problem dealt with in this article concerning the design of the distribution network under demand uncertainty. The problem is to determine which warehouses to open and how much space to rent (outsource) in warehouses owned by third-party logistics providers. This paper presents the development and application of the robust optimization methodology to distribution network design problem under demand uncertainty. The proposed method allows the designer to find a network configuration having a total cost that is robust to typical changes in the geographical distribution of the demand. The Algorithm is an iterative process based on Benders decomposition. At each iteration, the following two steps are performed. In the first step, the global design problem (master problem) is solved to decide on the best use of warehouses according to the information provided by the previous iterations. For a given warehouse configuration and under some restrictions on demand variations, the second step determines the demand that incurred the largest transportation cost, granted that the transportation cost is optimal. These steps are repeated until finding the warehouses configuration that gives the smallest worst-case transportation cost. At each iteration the worst-case transportation cost sub-problem provides new information to the global design problem, such that the latter can improve its robustness. We report numerical results for real size network problems. The main results show that a high level of robustness of the distribution network can be achieved at a relatively low cost.
1. Introduction Generally, there are two main categories of risk that affect the design of the supply chain: (1) the risk of bad coordination of supply and demand; (2) the risk of events such as natural disasters, economic disruption, political situation, labor disputes or terrorist attacks (Klibi et al., 2010). The present paper deals with the context of the first risk kind. In this context, customer's demand is for the most part variable, uncertain, complex and subject to several influential factors such as competition, changes in consumer habits, regulation, crises, etc. Customer's demand is difficult to predict and manage. The logistics network can become dynamically unstable when faced with disruptions in production systems by endogenous factors due to variations in customer's demand. These disruptions occur not only because of the time needed to adapt stock production, but also through feedback decisions along the chain of customer demand. Due to variable customer demand,
the design of a robust logistics network has become crucial. Two major fields of research in mathematical programming have been developed to deal with uncertainty. Stochastic programming (Dantzig, 1955; Shapiro, 2005) and robust optimization (Kouvelis and Yu, 1997; Bertsimas and Sim, 2004). A recent and comprehensive literature review on supply chain (SC) network design under uncertainty can be found in (Govindan et al., 2017), where the authors have identified robust optimization as a suitable tool to deal with real-world applications. After highlighting missing aspects of the existing literature, they recommended a list of potential issues for future research directions. Among these issues, designing a SC network in which customers' demand is sensitive to SC's responsiveness has been identified as a valuable future research. Furthermore, Govindan et al. (2017) pointed out that only a few existing papers deal with real-world situations, concluding that it is important to perform studies based on SC networks defined for real-life industrial contexts. Regarding these research gaps,
Corresponding author. E-mail addresses:
[email protected] (M. Ouhimmou),
[email protected] (M. Nourelfath),
[email protected] (M. Bouchard),
[email protected] (N. Bricha). ∗
https://doi.org/10.1016/j.ijpe.2019.04.026 Received 23 October 2017; Received in revised form 27 February 2019; Accepted 22 April 2019 Available online 26 April 2019 0925-5273/ © 2019 Elsevier B.V. All rights reserved.
International Journal of Production Economics 218 (2019) 96–105
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the present paper deals with a complex situation of demand randomness that has been encountered when dealing with an industrial real case. Considering the customer's demand as uncertain, we propose a method for planning storage capacity in a context where total demand varies very little and, on the other hand, certain regional demands can significantly change. This problem has been motivated by an industrial real case from the pulp and paper industry. Although the company manages to keep the demand for its products constant over the years, the loss of a key customer in a given geographical region, even if it is compensated by a demand gain elsewhere, may cause important perturbations in the quantity of products needed in this specific region. That is, the global demand for a given product is assumed to be constant. However, this demand could vary to a great extent inside a particular market zone. The problem addressed in this paper is to decide which warehouses to open or lease and how much storage capacity to use (strategic decisions), so as to minimize the overall cost of transportation and storage. The remainder of the article is organized as follows. To highlight our contribution, Section 2 presents a literature review related to robust network design under uncertainty. Section 3 presents the problem statement and the motivating real industrial case. Section 4 develops our methodology to solve the newly formulated problem. Section 5 details the numerical results, and Section 6 concludes the paper.
the minimum-expected-cost solution to facility location problems. Daskin et al. (2005) reported that robust facility location problems tend to be more difficult computationally than stochastic problems because of their minimax structure. They have also observed in robust network design literature that researchers developed two main categories of methodologies: analytical and polynomial time algorithms for restricted problems and heuristics for more general problems. For a more complete literature review on supply chain network design under uncertainty, the reader is referred to Daskin et al. (2005), Snyder (2006), Klibi et al. (2010) or Govindan et al. (2017). Supply chain network design deals with long-term strategic decisions involving, among others, the definition of numbers, locations and capacities of facilities, the selection of suppliers and transportation modes, etc. These decisions must be made now, in an uncertain environment, and take place in the future, but in the meantime, parameters such as demand and costs may significantly change which can lead to suboptimal, or worse, infeasible solutions. The observation that large improvements in reliability and robustness of network can often be attained with only small increases in the cost of the supply chain network (Snyder and Daskin, 2006) is encouraging an increasing number of decision makers to consider robustness criteria at the network design. The supply chain design robustness concept studied in this paper aims to propose a network design that performs well in any scenario of the data and hedges against the worst-case scenario that may occur as defined in Daskin et al. (2005). Thiele et al. (2009) proposed an approach to linear optimization with recourse and no probability information about uncertainty that preserves the model linear properties and can be solved by any commercial linear optimization software. They applied their approach to a multi-item news vendor problem and a production planning problem with demand uncertainties, and reported good computational results. Our proposed approach is based on the work of Thiele et al. (2009) and broaden its use to include situations where the uncertainty does not essentially lie in the value of the coefficients, but rather in their distribution across a given set, in our case the set of customers. Pan et al. (2010) considered a robust optimization model with three components in the objective function: expected total costs, cost variability due to demand uncertainty, and expected penalty for demand unmet at the end of the planning horizon. For designing a network of multi-product comprising several capacitated production facilities, distribution centers and retailers in markets under uncertainty, Baghalian et al. (2013) developed a stochastic mathematical model witch considers demandside and supply-side uncertainties simultaneously, which makes it more realistic. Hatefi and Jolai (2014) developed a robust model for an integrated forward-reverse logistics network design. A mixed integer linear programming model with augmented p-robust constraints was proposed to control the reliability of the network among disruption scenarios. The objective function of this model was minimizing the nominal cost, while reducing disruption risk using the p-robustness criterion. Christian et al. (2015), proposed to provide groundwork for an emerging theory of supply chain robustness which has been conceptualized as a dimension of supply chain resilience. Hasani et al. (2015) proposed a robust optimization model based on the uncertainty budget concept to consider uncertain parameters in the design of a global logistics network. More recently, Fattahi et al. (2017) developed a stochastic programming approach to address a multi-stage and multiperiod supply chain network design problem under stochastic and highly variable demands. Mohammed and Wang (2017) applied a fuzzy multi-objective programming model to a case study for product distribution planning in a three-echelon green meat supply chain design. Using a multi-objective approach, Mohammed et al. (2017) developed a cost-effective decision-making Algorithm for the design of a threeechelon Halal Meat Supply Chain network that is monitored by a radio frequency identification management system. Jabbarzadeh et al. (2017) presented a robustness approach for a realistic production–distribution planning model. The latter was shown to be robust to common supply
2. Literature review To cope with data uncertainty, two major domains of research in mathematical programming have been developed: stochastic programming and robust optimization. In stochastic programming, uncertainty is represented through scenarios where the data occur with a given probability (Dantzig, 1955) and the objective is to minimize the expected cost. In some robust optimization approaches, no probability information is available and the objective is to minimize the worst-case cost and immunize the model against constraints violations (Kouvelis and Yu, 1997; Mulvey et al., 1995; El Ghaoui et al., 1998; Ben-Tal and Nemirovski, 2000; Bertsimas and Sim, 2004). Both approaches have been widely applied to supply chain network design under uncertainty. Stochastic programming has been adopted to model network design using first stage variables corresponding to the design decisions (before the realization of the scenario), and second stage variables corresponding for example to flow decisions (after the realization of the scenario) (Birge and Louveaux, 2011; Ruszczynski and Shapiro, 2003; Shapiro, 2005, 2007; Villa et al., 2007). Other studies developed multistage stochastic programming (MSSP) models to deal with different kinds of uncertainties. Fattahi et al. (2018) proposed a MSSP model for simultaneous decisions at tactical and strategic levels. They addressed a multi-period supply chain network redesign problem in which customer zones have price-dependent stochastic demand for multiple products. Following the construction of a suitable scenario tree, Fattahi et al. (2018) applied Benders decomposition Algorithm for solving the resulting mixed-integer linear programming problem. Other papers developing MSSP models include Zanjani et al. (2013), which dealt with uncertainties in raw materials and demand for sawmills in the forest industry. Their industrial application has led to a huge combinatorial optimization problem solved using a scenario decomposition heuristic approach. On the other hand, robust optimization has been applied to different versions of the facility location problem under uncertainty (Gutierrez et al., 1996; Kouvelis and Yu, 1997; Yu and Li, 2000; Snyder and Daskin, 2006). In (Ukkusuri et al., 2007), instead of finding optimal network design solutions for a given future scenario, the authors address the problem of a traffic network design problem under demand uncertainty with solutions that are in some sense “good” for a variety of demand realizations. In (Mudchanatongsuket al., 2008), the authors present a robust optimization-based formulation for the network design problem under transportation cost and demand uncertainty. Snyder and Daskin (2006) combine the stochastic and robust approaches by finding 97
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interruptions and demand variations. Haddadsisakht and Ryan (2018) developed a three-stage hybrid robust/stochastic programming model to optimize the design of a closed-loop supply chain network that encompasses flows in both forward and reverse directions, while accommodating a carbon tax with tax rate uncertainty. Quddus et al. (2018) developed a two-stage chance-constrained stochastic programming model to deal with uncertainties caused by feedstock seasonality in a bio-fuel supply chain network. To the best of our knowledge, there is no existing work dealing with the case where the overall demand can be considered constant, but each customer demand is random. This situation has been encountered in the industrial case study described in the next section which also presents our methodology to deal with the newly formulated problem.
min (i (Q) + max d
D
(Q, d ))
subject to
Q
(1)
Robust optimization models uncertain parameters using uncertainty sets. The objective is then to minimize the worst-case cost in that set. Here, the set of demands D is the uncertainty set. If D = {d }, where d is the demand mean value, the above problem is the ‘nominal’ problem. As D expands around d , the decision-maker protects the network against more realizations of the uncertain demands, and the solution becomes more robust, but also more conservative. If the decision-maker does not take uncertainty into account, he might incur very large costs once the uncertainty has been revealed. On the other hand, if he includes every possible outcome in his model, he will protect the system against realizations that would indeed be unfavourable to his cost, but are also very unlikely to happen. Selecting a good trade-off between performance and conservatism is central to robust optimization. Using the approach developed by Bertsimas and Sim (2004), we focus on polyhedral uncertainty sets and model the uncertain demand dk, k C (with C is the set of customers), as a parameter of known mean, or nominal demand dk , and belonging to the interval [dk dˆk , dk + dˆk]. Equivalently, dk = dk + dˆk z k , |z k | 1, k . To avoid overprotecting , which bounds the the network, we impose the constraint k C |z k | total scaled deviation of the demands from their nominal values. The parameter , which we assume to be integer, is called the budget of uncertainty. The proposed methodology for solving our practical robust optimization problem is detailed in the next section.
3. Problem statement In this paper, we propose a method for planning the storage capacity in an industrial case of the pulp and paper industry where the global demand varies very little, and demands in regional markets can significantly change. The company we worked with owns numerous mills scattered across a wide geographical area. A mill can produce many products and a product can be made by many mills. The demand for each product is divided into demand zones. All the products leaving a mill have to be temporarily stored in one of the warehouses owned or leased by the company before being shipped to the final customers. The problem is to decide which warehouses should be opened or leased and what storage capacity should be used to minimize the overall cost of transportation and storage. Since the global demand is known, the amount of storage capacity necessary to cover the demand is also known. A configuration is a set of open warehouses with specific capacities. An optimal flow cost Φ(Q,d) for a configuration Q and demand d is the minimal value of a transportation solution that satisfies all demands. The worst-case transportation cost Φ(Q) of a given configuration Q is the smallest value Z such that, for every possible demand d within a set of allowable demands, we have Z ≥ Φ(Q,d). For a distribution of regional demands to be in the set of allowable demands, we assume that it satisfies the three following requirements:
4. Methodology One important ingredient of our proposed methodology is Benders decomposition method. This method offers a solution approach for combinatorial optimization problems like mixed-integer programs based on the concept of delayed constraint generation (we generate the constraints dynamically one by one). Thus, it can solve problems with complicating variables (integer variables). When we fix these complicating variables, we obtain an easy-to-solve problem (generally a transportation problem or a linear program). The method consists in alternating the sequence of a linear program and a pure integer program. That is, in Benders decomposition, the variables of the original problem are divided into two subsets so that a first-stage master problem is solved over the first set of variables, and the values for the second set of variables are determined in a second-stage sub-problem for a given first-stage solution. If the sub-problem determines that the fixed first-stage decisions are in fact infeasible, then so-called Benders cuts are generated and added to the master problem, which is then resolved until no cuts can be generated. The main advantage of Benders decomposition involves the ability to find an optimal solution without generating all extreme points and extreme rays of the dual sub-problem. Appendix 1 presents the method of Benders decomposition. For more details about the Benders Algorithm, see Benders (1962) and Geoffrion and Graves (1974).
1. the overall demand is equal to a given fixed value; 2. no zone will more than double its past demand; 3. the demand does not differ too much from the past demand, i.e. the sum, over all zones, of the absolute value of the variations does not exceed a fixed parameter. All the 3 above assumptions are derived from a real case study in the pulp and paper sector where there are many commodity and mature products sold on the market. In this context, the demand is constant over time and on global level (assumptions 1 and 3). On the other hand, we suppose that no market zone will double it past demand as the products are mature (assumption 2). A worst-case transportation cost is robust to typical demand change in the sense that it will never underestimate the transportation cost of any realistic demand change. The storage cost of any distribution network's configuration depends mainly on the number, size and type of the warehouses that are open and used to serve the customers. Since the overall demand is fixed and known, every configuration with enough storage capacity (to fulfill the overall demand) is a feasible solution, regardless of what the specific regional demands are. Only the transportation cost is affected by the distribution of regional demands. This paper presents a method to find the configuration that minimizes the sum of the storage cost i(Q) and the ‘stable’ transportation cost for that configuration. More formally, considering the set of all possible demands D and all configurations Δ, our case corresponds to a robust problem with recourse, or adjustable robust problem, which is formulated as:
4.1. Main steps Let us first look at a global overview of the developed methodology from the algorithmic point of view. The Algorithm is an iterative process in which, at each iteration, the following two steps are performed: Step 1. Solving the global design problem (master problem) Decide on the best use of warehouses according to the information provided by the previous iterations. Step 2. The worst-case transportation cost sub-problem For a given warehouse configuration and under some restrictions on demand variations, this step determines the demand that incurred the largest transportation cost, granted that the transportation cost is 98
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optimal. The above steps are repeated until we find the warehouses configuration that gives the smallest worst-case transportation cost. At each iteration the worst-case transportation cost sub-problem provides new information to the global design problem, such that the latter can improve its robustness. The worst-case transportation cost sub-problem is itself an iterative process repeating the two following steps:
warehouse j per period; it represents the warehouse capacity. – dks, demand of product s for customer k. – dkt, total demand for customer k ( s S dks ) in period t. – cijs, the cost for transiting one sqf of product s from mill i to warehouse j. – cjks, the cost for transiting one sqf of product s from warehouse j to customer k. 4.3. The optimal flow sub-problem
1. Considering the partial information provided by the previous transportation solutions, choose the demand that has the best chance of increasing the transportation cost. 2. For a given configuration and a given demand, find the optimal transportation solution.
The flow of products is defined by the two following sets of variables: – lijs, the quantity of product s (in sqf) transiting from mill i to warehouse j. – fjks, the quantity of product s (in sqf) transiting from warehouse j to customer k.
These two steps are repeated until we find the demand for the given configuration that has the highest transportation cost. The overall Algorithm can be summarized as follows:
The optimization flow problem can be expressed as follows:
Master problem - Global design problem: Decide the best use of the warehouses
min
( s S
cijs lijs +
cjks f jks )
i M ,j W
(1a)
j W ,k C
subject to
Sub-problem
lijs
Choose the demand that has the best chance of increasing the transportation cost
f jks = 0,
i M
j
W, s
S
(2)
k C
lijs
Qis,
i
M, s
S
(3)
j W
For a given configuration and a given demand, find the optimal transportation solution
f jks
Qj ,
j
W
(4)
s S k C
f jks
Highest transportation cost
dks ,
k
C, s
S
(5)
j W
lijs , f jks
Smallest worst-case transportation cost
0
(6)
Constraint (2) indicates that the total product quantity transiting from mills to warehouses is equal to the total product quantity transiting from warehouses to customers (flow conservation). Constraints (3) and (4) specifies the capacity limits of warehouses, while Constraint (5) states that the total product quantity transiting from warehouses to customers is higher or equal than the demand. Constraint (6) state that the decision variables are positive. To find a demand that maximizes the cost of the flow problem, it is useful to consider the dual formulation to have a maximization objective function:
To be more specific, the approach used to solve the network design problem is based on a Benders' decomposition method where the subproblem is a mixed integer programming problem. The master problem consists of choosing the best configuration Q given the current set of constraints, where Q is the warehouse capacity vector. Once this configuration is found, a cut is generated by searching for the worst-case transportation cost for this configuration. Furthermore, the problem of finding the worst-case transportation cost is itself solved by Benders’ decomposition where the master problem looks for the worst-case demand for the fixed configuration, and the cut generating sub-problems are simple flow problems using fixed configuration and demands. In what follows, we first present some useful definitions, then we successively propose the formulations for the flow sub-problem, the worst-case transportation cost sub-problem, and finally the global design problem.
max
ks dks
is Qis
k C,s S
i M,s S
µ j Qj j W
subject to js
ks
4.2. Definitions
is ,
The distribution network is composed of mills, warehouses and customer zones. The problem, for every period, consists of transiting manufactured products through warehouses to customers. The following notations are used:
cijs ,
is
µj
µj ,
js
ks
i
M, j
cjks ,
j
W, s
W, k
S
C, s
S
0
(7)
where λjs, αis, uj and σks are, respectively, the dual variables of constraints (2)–(5). We call Φ(Q,d) the optimal value of the objective function for the linear program (7).
– – – –
S, the set of products sold. M, the set of mills. W, the set of warehouses. C, the set of customers (aggregated demand of all customers in a geographical area). – Qis, the quantity (in sqf) of product s available at mill i. – Qj, the maximal quantity (in sqf) of product that could pass through
4.4. The worst-case transportation cost sub-problem Now we consider, given a fixed Q, the problem of finding the value of the demand d that will maximize Φ(Q,d). Since the total demand is considered constant, every unit of past demand lost by a customer is gained by another customer. The worst that could happen is that, at most, Γ units of demands can be subtracted from the past demand for all 99
International Journal of Production Economics 218 (2019) 96–105
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50
100
Customer 1
w1
5
100
150
75
Customer 2
w2
100
doubles his past demand, and the binary variables wk− that specifies whether customer k has a null demand. We also add the binary variables fk+ and fk− identifying, respectively, which customer is the positive and which customer is the negative fractional customer. Let g be d such that every dk is a multiple of g and let g (k ) = gk . In order to model the added or subtracted demand of the fractional customers, we introduce the binary variables i+ and i with i {0, ..., gmax } and gmax = [log2 (max k K (g (k )))] 1. Here, i+ (respectively i ) correspond to an increase (respectively decrease) of 2ig units of demand for the positive (respectively negative) fractional customer. Now we define the variables z k+ and z k equal to s S ks dks if, respectively, wk+ = 1 and wk− = 1 and 0 otherwise. Problem (8) can be formulated as follows:
10
75
Mill 200
Fig. 1. An illustrative example.
customers and no customer will more than double his past demand. Only the total demand of a customer changes; the relative proportion of this demand across all products remains the same for every customer. One might think that an optimal solution to this problem would be to double the demand of customers which are the farthest away from the nearest warehouses and assign null demand to those which are the nearest. Fig. 1 presents a simple example where such a solution would not be optimal. In the example there is one mill m of capacity 200 producing one product, the customers c1, c2 have both 100 units of past demand and warehouses w1, w2 have capacities of 50 and 150 respectively. The distances are expressed near the edges between business units and the cost of moving one unit of product on unit of distance is 1. Assuming Γ ≥ 100, the solution giving a demand of 200 to c2, the farthest customer, and 0 to c1, the nearest customer, results in a transportation cost of 37000, while a null demand for c2 and a demand of 200 for c1 costs 46000. If we define δk+ and δk− to be respectively the positive and negative change to the demand of customer k, we can modify the dual of the flow sub-problem (7) to include this varying demand, but doing so results in a nonlinear objective function:
max
(d k C , s S ks ks
+(
+ k
k
d
) dks ) k
i M,s S
is Qis
j W
max
j W
dk ,
k
k
dk ,
k
k C k C is,
µj ,
k + k
=
js
ks ,
k C + k, k
(z i+
zi )
i=0
k C
is Qis i M, s S
µj Qj
(9)
cijs ,
is
ks
µj
js
wk dk +
gmax
S
(11)
2ig
(12)
+ i i 2g
=
wk dk + k C
gmax i
2ig
(13)
i=0
(15)
1
wk ,
k
C
(16)
1
wk+,
k
C
(17)
2ig
i
One way around nonlinear objective function is to use a mixed integer formulation similar to the one proposed in Thiele et al. (2009). It can be shown that there always exists an optimal solution where, except for two customers, every customer either doubles his past demand, does not change his past demand, or have null demand. In fact, fixing all variables except k+ and k , Problem (8) is linear with (2n+2) constraints, and any extremal optimal solution will have at most (2n+2) non-zero variables (including the slack variables). Let Xs the set of variables k+ and k that are neither 0 or the maximum; X0 the set of variables k+ and k at 0; and Xmax the variables k+ and k at maximum. For every element in Xs, corresponds 3 non-zero variables: k+, k and the corresponding slack variables in the two first equations of (8). To each element of X0 corresponds 2 non-zero variables: the corresponding slack variables in the two first equations of (8). To each element of Xmax corresponds 2 non-zero variables: the corresponding k+ and k variables in the two first equations of (8). So the number of non-zero variables is at least 2k+| Xs |. Therefore, for an extremal optimal solution, we have | Xs | ≤ 2. In addition, if the two variables have the same sign, the solution would not be optimal so we can transfer values without changing the cost. The two exceptions, named respectively the positive and negative fractional customer, consists of one customer raising his demand by a fraction of its past demand and one customer withdrawing a fraction of his past demand. Following this observation, we define the binary variables wk+ that indicates whether customer k
C, s
(14)
i=0
gmax
W, k
(10)
S
fk = 1
gmax
0
W, s
f k+ = 1
f k+
,
j
i=0
fk
C
cjks ,
i
wk+dk +
µj Qj
M, j
i=0
k C
k C
i
gmax
k C
(8)
k
z k )+
subject to
k C
C
gmax
(z k+
+
k C,s S
subject to + k
ks dks
k C
+ i i 2g
i=0
k C
z k+
ks dks ,
f k dk
(18)
f k+ dk k
(19)
C
(20)
s S
z k+
Mk dk wk+,
zk
k
ks dks
(21)
C
Mk dk (1
wk ),
k
C
(22)
s S
z i+
z i+
+ i gM ,
2ig (
ks
+ (1
ks
(1
s S
2ig (
zi
(23)
i = 0...gmax
f k+ ) M ), i
)M
s S
k (1
C , i = 0...gmax f k ) M ),
k
(24)
C , i = 0...gmax (25)
k
,
is ,
+ k,
µj ,
wk+, ks
wk ,
0
f k+ , f k
{0,1}
(26) (27)
Constraints (10)–(11) are the initial constraints of problem (8), Constraints (12)–(13) impose that the demand lost by a customer is gained by another one, and the sum of all lost demands do not exceed Γ. Constraints (14)–(17) ensure that there is exactly one positive and one 100
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negative fractional customer and those fractional customers are not of the type that double their demand or have a null demand. Constraints (18)–(19) require that the fractional customer does not lose or gain more than their original demand. Finally, Constraints (20)–(25) impose that the variables z take, at optimality, their expected value as previously defined. The interpretation of the dual variables σks is the marginal cost of increasing by one the demand of product s by customer k; the maximal value of this marginal cost is Mk, the maximum unit transition cost from a plant to the customer k; the value of M is max(Mk ) .
5. Numerical results To demonstrate the proposed approach for the design of robust distribution network under uncertainty, we tested the method on an instance derived from the real distribution network of a Canadian pulp and paper company. The problem parameters have been dictated by the real case from our industrial partner. The distribution network has 2 mills, 10 demand zones (the customers) and 10 potential warehouses. The total demand of all the customers is 495000 units. Every demand is a multiple of g = 1000. The pulp and paper company owns some warehouse out of 10 while the rest of warehouses belongs to third parties’ logistics companies (3 PLs) that operates them. The pulp and paper company signs leasing contracts with these 3 PLs companies to secure a certain amount of space over a certain time of period (1 or 2 years) for its distribution operations. Our models aim to support the decision maker at the pulp and paper company that aims to minimize its total distribution cost and need to make a trade-off between investing in its own warehouses or leasing warehouses owned by the 3 PLs companies. Using our methodology, we solve this instance with a value of Γ ranging from a minimum value of 0 (this value is equivalent to deterministic case with fixed demand) to a maximum value of 80000 with a step of 10000 units. Tables 1 and 2 present the numerical results of the optimal solution (Γ = 0). Table 1 shows the quantity of product transiting from warehouse (wi) to customer (Ci) for periods 1 and 2 while Table 2 gives the percentage occupancy of open warehouses. Indeed, in this case, 4 warehouses (w1, w2, w7, w10) are open and are used up to their maximum capacity except warehouse w7 used at 86% in period 2. On the other hand, Table 1 shows that all customers are served by a single warehouse (single sourcing policy) except customer 6 supplied by three (3) warehouses in period 1 and two (2) warehouses in period 1. We remark that our solution proposes almost a single sourcing except for Customers 5 and 6. Note that this single sourcing is only applied from the warehouse point view and not from the company level. In other word, the customer is receiving all its products from one warehouse to keep the cost as low as possible. In addition, this practice is quite used in North America (NA), for distribution to maintain a steady and stable relation between the warehouse and the customer and to keep the cost low (closest warehouse as distances to travel are quite long in NA). To overcome this characteristic of the optimal solution having almost single sourcing concept, we could have added extra
xk C
4.5. The global design problem Finally, the last problem is to determine the design variables that could not be changed once fixed. The design problem is:
min
(Q ) +
(
jks Qj i j
+ oj r j )
j W k K s S
subject to jks Qj oj
dks ,
s S j W
oj ,
jks
oj
k
C
(28)
s S
j
{0,1},
W,
j
k
C,
s
S
W
0
jks
In the design problem (28), determining the design variables that could not be changed once fixed requires setting the value of Qj related to the storage availability at warehouse j, to minimize the fixed inventory cost rj, the variable inventory cost ij and the transportation cost Φ(Q), knowing that we will have to face the worst demand for that network. A binary variable oj determines whether warehouse j is open or not. To make sure that there is always enough space in the wared. houses to fulfil any demand it is required to have j W Qj s S s Since Problem (28) is convex with respect to variables γjks we can use a Benders’ decomposition approach to solve the following network design problem: Min Z + j W k K s S ( jks Qj i j + oj r j ) subject to jks Qj oj
dks ,
s S j W
Z+
k
C
Table 1 Quantity of product transiting from warehouse (wi) to customer (Ci): Optimal solution.
s S
µ r , , j , Qj
,
r,
r = 1…m
j W
oj ,
jks
oj
j
{0,1},
W,
j
k
C,
s
S
W
0
jks
where r,
=
r ks dks k C,s S
k C r
i M,s S
(zr+k
+
is Qr
is
gmax
zr k )+
(z r+i
zr i)
i=0
(29)
In the above problem (29), Δ represents the finite set of possible values of the binary variables k , k+, wk+, wk , f k+ , f k , and hr(δ) = (σr,δ,zr,δ,αr,δ,μr,δ), with r = 1, …, mδ is the set of extreme points of Problem (9) with the value of the binary variable fixed. 101
Customer
Period
w1
C1 C1 C2 C2 C3 C3 C4 C4 C5 C5 C6 C6 C7 C7 C8 C8 C9 C9 C10 C10
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
33000 27000 28000 22000 14000 26000
w2
w7
w10
13000 12000 22000 18000
5000
12000 18000 24000 18000 18000 6000 24000
28000 24000 32000 18000 15000 38000
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M. Ouhimmou, et al.
two periods defined in the fourth column. The value of the configuration goes from the first warehouse (w1) to the last warehouse (w10); a value of 1 means that the warehouse (wi) is opened. The next column shows the number of open warehouses. Depending on the values of Γ, sometimes it is advantageous to use a Third-Party Logistics provider (3 PL) (Selviaridis and Spring 2007) as shown in the seventh column of Table 5. In the eighth column, it is the total cost in the case of single warehouse (Hill and Galbreth, 2008). The last column gives the CPU time using Cplex solver version 12.6 on a standard PC with an Intel Xeon 2.67 GHz 64-bit Dual core processor with 72 GB RAM. The model is implemented in the modelling with OPL studio. The running time ranges from 5 s (Γ = 0) to more than 4366 s (Γ = 80000). Table 5 shows clearly that the running time increases quickly as the value T increases. It is possible that some sectors may be very sensitive to invest to hedge against such uncertainty and are not willing to apply such approaches. However, the power of robust optimization is that you can choose the level of robustness of your decision; the higher the budget, the better is your protection against uncertainty and higher probability of satisfying your customer's demand. Note that about 7% is the maximum price of robustness and will provide a 100% of satisfying your customer demand. One could choose lower budget (2 or 3%) but the protection against uncertainty will be lower as well (probability of not satisfying customer is higher). For sure, sectors were customers' service is very important and penalties of not or delayed delivering products are very high will be more open to apply such techniques to hedge against uncertainty. This characteristic is illustrated in the next subsection.
Table 2 Percentage occupancy of warehouses: Optimal solution. Period
w1
w2
w7
w10
1 2
100% 100%
100% 100%
100% 86%
100% 100%
Table 3 Quantity of product transiting from warehouse to customer: Robust solution (Γ = 20000). Customer
Period
w1
C1 C1 C2 C2 C3 C3 C4 C4 C5 C5 C6 C6 C7 C7 C8 C8 C9 C9 C10 C10
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
43000 17000
w2
w3
12000 18000 28000 12000
10000
24000
18000
32000 26000
7000 12000 13000
22000
w10
15000 26000
3000 2000 28000 24000 34000 16000 15000 38000
5.1. Feasibility and price of robustness
constraints to forbid single sourcing by imposing two or more warehouses to supply each customer. Table 3 shows the configuration of a robust solution when Γ increases to 20000. Open warehouses to serve customers are those with the greatest capacity. So the inventory and the transportation cost will increase. Table 4 gives the change in the percentage occupancy of warehouses when Γ increases to 20000. This table shows that warehouse 3 is opened in both periods but has been underutilized (68% in period 1 and 46% in period 2) compared to the other three (3) warehouses fully utilized. Compared to the deterministic instance (Tables 1 and 2), the method has closed warehouse 7 and opened warehouse 3 to serve the customers while w1, w2, and w10 are open in both instances. Note that flows between warehouses and customers have changed between the two solution (Γ = 0 and Γ = 20000). Table 5 presents the results of the network design problem with different values of Γ. This table shows that robustness has a price. The later is called price of robustness in (Bertimas et al., 2004). We observe that for a demand variation that accounts for 16.16% of the total demand, the rise in total cost is 7.46% while the rise in storage cost is 8.76% (the storage cost rises from 7060100$ to 7678500$). Depending of the context and the industry, this cost would be a small price to pay for more robustness, but too high in other contexts like competitive markets. The first column of Table 5 presents the value of Γ for the test. The second column gives the optimal storage and transportation cost, while the third shows the cost variation compared the optimal solution. The fifth column gives the configuration of the open warehouses in the
In this section, we present a set of experiments to calculate the price of robustness and the probability of feasibility of the model (probability of meeting customer's demand). The probabilities of satisfying the demand defined by PГ and optimal storage and transportation cost as a function of Γ are given in Figs. 2 and 3 for periods 1 and 2, respectively. The results show clearly that the more the budget of the robust solution increases, the greater the probability of satisfaction of the demand. For example in Fig. 2, the two point (Γ = 20000, Cost = 10173450) and (Γ = 50000, Cost = 10463415) have a probability of meeting customer's demand of 26% and 56% respectively. This high increase of probability (from 26% to 56%) has generated a small increase of 2.77% of total cost (price of robustness is low compared to the increase of probability of meeting customer's demand). The probabilities are calculated by executing the Algorithm below. Algorithm. (Satisfaction probability of demand) For each Гj, the values of the first column of Table 5 (For a given configuration (column 5), we have Qj, the maximum quantity of product that could pass through the warehouse j per period) Generate 100 randomly value of Гi (Гi = 0, .., 80000) For each Budget Гi d + Гi then C = C+1 (C, counts the number of If j W Qj s S s times the condition is true) Satisfaction probability of demand (PГ) = C/100 The pulp and paper company is facing a big challenge of uncertain demand and needed to decide what is the least cost distribution network that hedges against uncertainty. Our approach allows them to choose the best locations of their own warehouses (owned by the company) and what is the best 3 PL company to partner with to minimize distribution costs. This approach helps the company to negotiate and come into agreement (leasing contract over time) with the appropriate 3 PL with the convenient and well-located warehouses to deliver the customers of the pulp and paper company.
Table 4 Percentage occupancy of warehouses: Robust solution (Γ = 20000). Period
w1
w2
w3
w10
1 2
100% 100%
100% 100%
68% 46%
100% 100%
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Table 5 Results of the network design problem with different values of Γ Γ(units)
Total cost($)
Cost
Period
Variation (%) 0
10 060 000
10 000
10 1463 42
0.86
20 000
10 173 450
1.13
30 000
10 175 200
1.15
40 000
10 322 967
2.61
50 000
10 463 415
4.01
60 000
10 529 983
4.67
70 000
10 561 483
4.98
80 000
10 810 550
7.46
100%
10,900,000 $
90%
10,800,000 $
80%
Open
|w1|, …, |w10|
warehouses
|1|1|0|0|0|0|1|0|0|1| |1|1|0|0|0|0|1|0|0|1| |1|1|0|0|0|0|0|1|0|1| |1|1|0|0|0|0|0|1|1|1| |1|1|1|0|0|0|0|0|0|1| |1|1|1|0|0|0|0|0|0|1| |1|1|1|0|0|0|0|0|0|1| |1|1|1|0|0|0|0|0|0|1| |1|1|0|0|0|0|0|1|1|1| |1|1|0|0|0|0|0|1|0|1| |1|1|0|0|0|1|0|1|0|1| |1|1|0|0|0|0|0|0|0|1| |1|1|1|0|0|0|0|1|0|1| |1|1|0|0|0|0|0|0|0|1| |1|1|1|0|0|0|0|1|0|1| |1|1|1|0|0|0|0|0|0|1| |1|1|1|0|1|0|0|0|0|1| |1|1|0|0|0|0|0|0|0|1|
4
60%
10,600,000 $
50%
10,500,000 $
40%
10,400,000 $
30%
P
10,300,000 $
20%
10,200,000 $
10%
10,100,000 $ 0% 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000
Fig. 2. Satisfaction probability of demand (PΓ) and optimal storage and transportation cost as a function of Γ for period 1. 100%
10,900,000 $
90%
10,800,000 $
80%
10,700,000 $
P
70% 60%
10,600,000 $
50%
10,500,000 $
40%
10,400,000 $
30%
P
10,300,000 $
20%
10,200,000 $
10%
5 4 4 5 5 5 5 5
Cost($) (Single warehouse)
Time (s)
10 108 908
5.64
10 190 455
26.14
10 203 650
432.6
10 225 200
1 476.6
10 399 877
2 870.8
10 500 434
3 864.7
10 598 883
3945.8
10 601 483
4 235.5
10 898 760
4 366.8
3 PL
1 1 1 1 1 1 1 1 1
a high level of robustness can be achieved at a relatively low cost. This kind of observation has been reported by many authors, such as Daskin et al. (2005). All our results are coherent with the existing literature. Our approach allows considering of customer's demand uncertainty in the network design problem and giving a trade-off between robustness and optimality of the solution. Our approach is valid as long as the global demand is constant while it varies by customer, region or by family product, etc. This is the case of many sectors such as retailers, manufacturers, etc. In addition, Humanitarian logistics where they need to deal with emergencies that are uncertain can be addressed using our proposed approach. This method is able to solve real, but small, problems with quite large admissible variations. However, larger problems could take several hours to solve. Future works should aim to formulate a mathematical model that allows for mixed integer programming instances that are easier to solve, and to implement acceleration strategies to reduce computation running time. Another research direction could take into account uncertainties in transportation costs since fuel prices can be volatile. Furthermore, supply could be uncertain due many reasons and can impact the whole supply chain and deserves to be studied. In this paper, we only focus on the demand side as we are dealing with the distribution network of the pulp and paper company that was facing the issue of the demand uncertainty. In our case, we suppose the pulp mills are capable to deliver the right amount of final products to the warehouses and customers. This research could be extended to consider supply uncertainty and see to what extent this will affect the configuration of the distribution network. Finally, it would be interesting to compare our proposed approach to fuzzy programming (FP) and stochastic programming (SP). While FP requires (among others) the extraction of appropriate fuzzy rules, SP could necessitate the development of an efficient solution Algorithm to deal with the huge combinatorial MIP optimization problem resulting from the probabilistic scenarios.
10,700,000 $
70%
P
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
Configuration
10,100,000 $ 0% 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 Gamma ( Gamma (
Fig. 3. Satisfaction probability of demand (PΓ) and optimal storage and transportation cost as a function of Γ for period 2.
6. Conclusion
Acknowledgement
This paper has addressed the design problem of robust distribution network under demand uncertainty based on a real industrial case study in the pulp and paper sector. We based our robust optimization approach on the work of Thiele et al. (2009) and broadened its use to include situations where the uncertainty does not essentially lay in the value of the coefficients, but rather in their distribution across a given set, in our case the set of customers. Our experimental results show that
The authors would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC), CRSNG RGPIN 2014-05705 for financial supports of this project. They are also grateful to the partner pulp and paper industrial company who provided the real case study as well as the data.
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Appendix 1. Benders decomposition method Benders decomposition offers a solution approach for combinatorial optimization problems like mixed-integer programs based on the concept of delayed constraint generation (we generate the constraints dynamically one by one). Thus, it can solve problems with complicating variables (integer variables). When we fix these complicating variables, we obtain an easy-to-solve problem (generally a transportation problem or a linear program). The method consists in alternating the sequence of a linear program and a pure integer program. The following figure illustrates this method by considering a mixed integer program:
The Benders master problem equates the original problem. This relaxed master problem provides a lower bound on the objective value of the original problem. The Benders Algorithm consists in sequentially solving the sub problem using its dual problem and the Benders master problem. At each iteration, we compute a lower and upper bound of the objective value of the original problem by solving the primal and master problem. Indeed, we insert the dual solution of the dual problem into the master problem and solve it to get new values of the integer variables which we then insert into the dual sub-problem to compute a new value for dual variable. The selection of the dual variables (when dual alternative optima exist) affects the quality of Benders' cut (the best cut, also called Pareto cut) and consequent running time. No other cut can dominate a Parteo cut. The algorithm terminates when the bounds (lower and upper) draw close enough or equate each other, creating an optimal solution, a combination of both the master problem and the dual problem solutions. A finite number of iterations between the primal sub-problem and the master problem exist because of the finite number of extreme points and extreme rays of the dual problem. The main advantage of Benders decomposition involves the ability to find an optimal solution without generating all extreme points and extreme rays of the dual sub-problem. Note that the Benders master problem, a pure integer program, remains very time consuming. The necessity to solve this pure integer program at each iteration constitutes a major drawback to this method. For more details about the Benders Algorithm, see Benders (1962) and Geoffrion (1974).
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