Distribution network design under uncertain demand: robust versus stable configuration Laura Mazzoldi, Ivan Ferretti, Simone Zanoni, Lucio Zavanella Department of Mechanical and Industrial Engineering, University of Brescia, Via Branze 38, I-25123, Brescia, Italy
[email protected],
[email protected],
[email protected],
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Distribution network design under uncertain demand: robust versus stable configuration This paper is focused on the design of a supply chain taking into account the variability of the retailers demand. In particular we study the design of a one-to-many distribution system, considering the products flows from the producers to the different distribution centres and to retailers, as well as the reverse flows from the retailers to the producers. The goal is the definition of the network configuration that can efficiently operate under the uncertainty of the retailers demand. Adapting from the literature of layout two flexibility measures (Braglia et al., 2005), we have identified two possible strategies for the network configuration so as to face such uncertainty: a robust and a stable, where the former produces the minimum expected total cost while the latter determines the minimum variability of transport and handling cost components in the considered time horizon. A numerical analysis has been performed over a realistic case study data, and results have been reported. In particular the main result of this study is the relationship between robustness and stability indices of the distribution network configuration: the proposed model is offered as a management tool, that can support decision making processes. Keywords: facility location, flexibility, robustness, demand uncertainty.
1.
Introduction In general, Supply chain management involves decisions concerning facilities
location (where to locate plants and distribution centres), mix (what is to be produced), and volumes (e.g., quantities to be produced at each site, levels of goods to hold in inventory at each stage of the process). Additional decisions concern information sharing across the different actors of the chain and IT (e.g., how to share information among parties in the process). The location decisions are often the most critical and most difficult within the set of decisions required by an efficient supply chain Daskin et al. (2005).
2
Over the different location decision problems that can be generalized with the facility location decisions, the present paper deals with the Distribution Network Design problem. Distribution Network Design looks at the strategies finalised to the efficient distribution of finished products to customers at the lowest cost. In particular, in this paper we look at the design of the initial configuration of a distribution network as a complex task which may exert a relevant impact on the medium-long time horizon and which determines a decision difficult to be reversed after the implementation. The decision questions investigated are:
How many Producers should be selected?
How many DCs (Distribution Centres) should be opened?
Where should they be located?
Which retailers should be served by which DC? The main factors influencing distribution network design can be divided into the
following two main groups (Chopra and Meindl, 2001). The first one concerns the costs incurred and the second one deals with the service level offered to the customer: o Costs
affected
by
the
network
structure:
inventories,
transportation,facilities and handling, information o Elements of the customer service influenced by the network structure: response time, product variety, product availability, customer experience, order visibility, returnability.
3
During the time period over which the strategic design decisions are effective (i.e. plant locations), each one of the problem parameters (costs, products demand) may fluctuate widely. Parameter estimations may also be inaccurate due to poor measurements or because of the tasks inherent in the modelling process. If the distribution network designed by the decision maker is not efficient with respect to the uncertain environment, the impact of performance inefficiency could be shocking for the organization. Thus in this paper we propose a framework for Distribution Network Design in uncertain environment that may help managers to take decisions explicitly taking into account this variability, in particular we will take into account demand variability at the last stage of the distribution network. Because of the significance of uncertainty, many researchers have addressed stochastic parameters when designing the distribution network. Research that considers uncertainty can be categorized according to two primary approaches (Pan and Nagi, 2010): the probabilistic approach and the scenario approach. For the probabilistic approach, some parameters are regarded as random variables with known probability distributions. In comparison, the scenario approach represents uncertainty through setting up different scenarios, which represent realizations of uncertain parameters. In this work we will follow the scenario approach. The paper is organized as follows, in the next section a literature review on the network design under demand uncertainty problem is presented; then, starting from the basis of previous works on layout design, an explicit definition of the key performance measures used (i.e. robustness and stability) is offered. Afterwards, the model developed, the case study and related results will be given. Finally, the conclusion will
4
summarize the proposed approach to face the uncertainty and the applicability in a realistic context of the model outlined.
2.
Literature review Since supply chain design involves several decisions at the strategic level, it is
desirable to keep the supply chain configuration unchanged over a relatively long period of time once it is determined. Unfortunately the best way to tackle uncertainty in the supply chain design context remains an important and yet unresolved problem (Mo and Harrison, 2005).This is a key reason why the literature on distribution network design under uncertainty consistently grew up in the last decade. However uncertainty remains one of the most challenging and important aspects of supply chain management. According to a recent literature review (Snyder, 2006), more than half of the 150 papers cited were published in the past 10 years and roughly a third were published in the past five years. The growing attention paid to these problems is due to the increased awareness of the uncertainties faced by the greatest part of the firms, as well as to the improvements in both optimization technology and computing power. It has to be observed that the distribution network design problem has some similarity to the facility location problem and to the facility layout problem. Hence, attention will be paid to previous studies already published on this topic for uncertain demand environments. One of the first reviews on the stochastic facility location problems is provided in Louveaux (1993), while a more recent logistic network design models review is offered in Li and Schulze (2011).
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In Logendran and Terrell (1988) demands are stochastic and price-sensitive, and plants are incapacitated. The Authors used a heuristic to select facilities to be opened and the allocation of clients to facilities. The quantities to be transported from plants to customers are optimized separately for each plant-customer combination in view of the random demand of each customer to each plant. Chen et al. (2007) considers the planning of a multi-product, multi-period, and multi-echelon supply chain network that consists of several existing plants at fixed places, some warehouses and distribution centres at undetermined locations, and a number of given customer zones. The supply chain planning model is constructed as a multi-objective mixed-integer linear program (MILP) to satisfy several conflict objectives, such as minimizing the total cost, raising the decision robustness in various product demand scenarios, lifting the local incentives, and reducing the total transport time. In Louveaux and Peeters (1992) it is possible to find a stochastic incapacitated facility location problem in which demands, variable production and transportation costs as well as selling prices can be random, uncertainty of the random variables is captured and appreciated by the discussion of different scenarios. Patriksson (2008) considers a class of stochastic MPEC (Mathematical Programs with Equilibrium Constraints) traffic models with explicit possible uncertainties in travel costs and demands. Results are presented on the stability of the optimal solutions to perturbations in the probability distributions of travel time and demand, establishing the robustness of the model, while the main objective of Zhang et al. (2007) is to study the remanufacturing logistics network design problem from third party logistics supplier (3PLS) perspectives under the uncertain environment, and Pishvaee et al. (2011) proposes a robust optimization model for handling the inherent uncertainty of input data
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in a closed-loop supply chain network design problem. Recently Hasani et al. (2011) proposed a model considering multiple periods, multiple products, and multiple supply chain echelons as well as uncertain demand and purchasing cost. Uncertainty of parameters in the proposed model is handled via an interval robust optimisation technique. The model assumptions are well matched with decision making environments of food and high-tech electronics manufacturing industries. In the network design problem under uncertainty considerations, Shimizu et al. (2011) formulates a global logistics network design problem under uncertainties and present a solution method in terms of the stochastic programming known as recourse model; moreover, a method to support decision-making on robust design under noisy environment in terms of the knowledge from multi-objective optimization is presented. Piplani and Saraswat (2012) discusses a typical service network of a company providing repair and refurbishment services for its products: a mixed-integer linear programming (MILP) model that minimises the total cost subject to flow balance and logical constraints is presented, taking into consideration that the optimal network configuration depends on several factors such as the number of products returned, percent of faulty products and warranty fraction of modules, which are uncertain in nature: a min-max robust optimisation model is then developed and solved to address these uncertainties. A class of production-distribution planning problems with non-stochastic uncertain demand is presented in Blanchini et al. (1997). The study models the system as a dynamic game between two players who control the flows on a network with node and arc capacity constraints.A survey on strategic aspects of facility location, including problems under uncertainty, is offered in Owen and Daskin (1998).
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Van Landeghem and Vanmaele (2002) proposed a method, which uses simulation to develop risk profiles based on uncertain values of various parameters, is proposed. One of the approaches proposed aims at finding the decision policy that yields the most stable outcome, i.e. with low variability of the key performance measures (such as service level or total supply chain inventory). Another approach tries to find a policy able of reducing the total number of changes to the plan over the time horizon, while keeping the key performance measures fixed at their target level. The outcome is a planning process for tactical demand chain planning. The study of Romero Morales et al. (1999) proposed a multi-period singlesourcing problem (MPSSP) for the logistics network design evaluation in a dynamic environment. The Authors consider a dynamic demand pattern as well as inventory policies. So as to solve the problem, a class of heuristic solution approaches are presented. Lalwani et al. (2006) proposed a method to identify the factors of a distribution network structure that change over time and mainly affect its performances. For investigating the sensitivity of a scenario the approach proposed combines the use of simulation, Taguchi technique and ANOVA with distribution design modelling. The simulation model developed, based on real world data of a European after-sales business in the automotive industry, indicates that when developing the network a careful consideration has to be given to reliably estimating the inventory holding costs and the mechanism for determining the capital holding charge. The model designed shown that optimum design is most at risk due to the uncertainties associated with stock holding costs and delivery frequencies rather than customer volume changes and transport tariffs.
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Pan and Nagi (2011) considered the design of a supply chain for a new market opportunity with uncertain customer demands. A robust optimization model was proposed where expected total costs, cost variability due to demand uncertainty, and expected penalty for demand unmet were the three components in the objective function. In this model uncertainty was captured via scenario approach while the objective was to choose one partner for each echelon and simultaneously decide for each partner the production plan, inventory level, and backorder amount. Finally Georgiadis et al. (2011) proposed a mathematical model for designing supply chain networks under time varying demand uncertainty. Uncertainty is captured in terms of a number of likely scenarios possible to materialize during the lifetime of the network. The problem is formulated as a mixed-integer linear programming problem.
3.
Robustness and stability definition There are a variety of performance measures that can be used for the distribution
network design under uncertain: some of them are discussed in Mo and Harrison (2005) and in Snyder (2006). Ideally, we would like to design a distribution network that has the lowest total cost (or highest total profit) under the whole set of the possible demand scenarios. However, this is usually unachievable. A distribution network configuration that has the lowest total cost (or highest total profit) for some demand scenarios may not perform well for other profiles. Firstly, the concept of robust design was introduced by Genuchi Taguchi in the 1960s, and it was subsequently accepted and used in the field of experimental design and quality control. The basic idea of robust design is to make a manufacturing process insensitive to noise factors. Starting from this concept, a set of studies proposed the application to the design of supply chains, namely robust supply chain design. 9
The literature review of Snyder (2006) illustrates a rich variety of approaches to the optimization under uncertainty that appeared in the literature and their application to facility location problems. In particular, the paper discusses the different "robustness" performance addressed by different authors and it compares the methods adopted to achieve the solution. Looking at the layout problem, as proposed in the literature review section, from such a point of view some similarities exist with the distribution network design, where the strategic decision of location uncertainty plays an important role. Under this kind of problems, the study of Rosenblatt and Lee (1987) and the following of Braglia et al. (2003) proposed a robustness-oriented approach to the facility layout problem under stochastic demand scenarios: the property of layout robustness can be regarded as the ability to minimize the total expected costs. The subsequent contribution of Braglia et al. (2005) proposed an alternative approach, named stability. The concept of layout stability significantly differs from the concept of robustness: stability aims at reducing the variability in the performance of a given layout, which is a consequence of the exogenous instability of the system. Hence, coming back to the similarity with the layout problem, we explicitly define the following performance measures for the distribution network design problem(which differs from “robust optimization” definitions, but which we consider more convenient for the problem under study) :
robustness: property of a distribution network configuration which is able to guarantee the lowest expected cost in the long-run, over the potential scenarios determined by the changes in the external context;
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stability: property of a distribution network configuration to show a limited sensitivity to demand variability i.e. a configuration which, regardless of the changes in the external context, will always perform similarly to the levels given in the planning phase.
Given the two previous definitions, it is also possible to define:
the robust distribution network as the configuration that minimizes the total expected costs over the different scenarios;
the stable distribution network as the configuration that minimizes the variability of the transport and handling cost components during the considered time horizon. Considering the robustness performance measure, the definition here adopted it
could be observed that is the same of the first measure proposed by Mo and Harrison (2005) for solving the supply chain robust problem. However it should be emphasised that the approach to supply chain robustness here adopted is significantly different from the robust optimization theory (Ben-Tal and Nemirovski, 2002) therefore it should not be confused with that.Viceversa, the stability measure here adopted is directly inspired from the layout performances introduced in Braglia et al. (2005). When considering the stability performance measure, it is possible to observe how variance is usually considered as a standard measure of risk. For companies that are risk adverse, the optimal supply chain design which incorporates attitudes prone to risk may be different than the design approach adopted by those companies which adopt expected costs as the main decision criterion (Mo and Harrison, 2005). Moreover, the
11
minimisation of the sensitivity to demand variability may lead to a set of important benefits:
reduced need for rearrangement of central distribution capacities;
reduced need for transport fleet rearrangement. However, it is also opportune to assess a comparison between the stable
configuration and the robust solutions. This is a reasonable approach, because the choice of the stable solution may imply a poor performance of the whole network, in terms of expected log-run costs, making the choice made affected by a low profitability. On the other hand, the choice of the robust configuration may lead to a significant variability of the performance itself. In the case of a variability level close to the one offered by the stable configuration, a significant saving in the average total costs may be detected. Therefore, when a stable configuration is preferred to the choice of the robust one, it is also opportune to appreciate some additional performance indicators, such as the following parameters:
the deterioration of the solution performance in terms of increase in the expected long run costs;
the improvement of the performance of the solution, which may be appreciated by the decrease in variability of the considered cost components.
These calculations should be performed to allow the analyst appreciate the level of risk associated with the deteriorated performance of the distribution network configuration. Moreover, some comments are needed to underline how the expected result is that the stable configuration should be more centralised than the robust one, due to the
12
risk pooling effect. Anyway, it has to be claimed that due to the fact that risk pooling effect decreases as the correlation between demands from the markets becomes more positively correlated, the benefits, deriving from risk pooling, to have a more centralised stable configuration (compared to the robust one) decrease while the market demands become more positively correlated.
4.
Problem definition We formulate a two-stage distribution network problem considering three
different kinds of location: the Producers’ facilities (in brief, Ps) the Distribution Centres (in brief, DCs), and Retailers (in brief, Rs). At each P, there is the opportunity of shipping products both to the DCs or directly to the Rs. At the DCs, products are shipped to the Rs and, therefore, there is the responsibility for managing the product flows from the Ps to the Rs. Finally, the Rs are the places where products are sold to the customers. We consider a network with limited capacity for both the P’s and the DC’s. In addition, some distinctive features of the problem are considered:
products distributed are perishable;
products may return from the customers.
As far as the limited shelf life of the products distributed is concerned, an industrial case observed (as explained in the next sections, too) inspired the model development. In particular, the reference case belongs to the sector of fresh food distribution and, therefore, this specific feature have been accurately incorporated in the problem definition, giving adequate relevance to both return flows and shelf life ends.
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As far as the return option is concerned, due to some regulation restrictions in the transport operation, it is considered that only the Ps may receive the returned products, directly from the Rs, while the DCs cannot receive the returned products. Basically, returned products are defined as those products that cannot be sold at the Rs, because they are defective or because the their shelf life is already expired. In some cases, these products are also affected by an end-of-life cost and they may be subjected to specific regulations concerning their disposal. The structure of the distribution network considered is shown in Figure 1, where the dashed lines represent the returned product flows. R’s
DC’s P’s
Figure 1. The distribution network structure. It is also opportune to highlight some of the hypotheses introduced, as the reference case implied their assumption. 1. The problem definition refers to a “green field” situation, so there isn’t any solution to consider as starting point for the optimisation.
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2. Demand uncertainty is captured and modelled using different scenarios over time: demand may vary, from a period to another, up to the end of the time horizon set at the preliminary stage. Thus the optimisation can be carried out by a multi period Mixed Integer Programming, comparing the performance over different scenarios. 3. A preliminary hypothesis is that no modification could be introduced in the network configuration during the evolution of the market demand. In other terms, the optimisation is carried out in long-run terms appreciated over a mix of possible demand scenarios. Consequently, it is assumed that, once adopted the configuration of the network, it will not be modified up to the end of the time horizon assumed. This is necessary as the network performance is evaluated in long-run terms. 4. The model is developed as a bi-objective Mixed Integer Programming, where the two conflicting functions that have to be minimized are the total cost and the transport and handling cost components total variance: two weight coefficient have been introduced in order to implement a robust or a stable solution: when the value of the first coefficient (the weight of total cost minimization function) is set to 1 and the second one (the weight of total variance minimization function) is set to 0, a robust solution is expected, while in case the first coefficient is set to 0 and the second one is set to 1 a stable solution is reached. It is also possible to set both coefficient to the same value (e.g. 0.5) in order to obtain a “balanced” solution, or to choose different values of the coefficients in order to give more importance to one or the other of the two considered functions.
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5.
Model design
For the formulation of the model, the following notation has been introduced: Indices: i 1,2...I: index of the potential location sites for Producers (Ps)
j 1,2...J: index of the potential location sites for Distribution Centres (DCs)
k 1,2...K: index of the Retailers (Rs)
t 1,2...T: number of periods considered Parameters: csij: transportation cost for shipments between the i-th P and the j-th DC [€/(ton·km)]; csik: transportation cost for shipments between the i-th P and the k-th R [€/(ton·km)]; csjk: transportation cost for shipments between the j-th DC and the k-th R [€/(ton·km)]; crki: transportation cost for reverse shipments between the k-th R and the i-th P [€/(ton·km)]; dsij: travel distance between the potential i-th P and the potential j-th DC [km] dsik: travel distance between the potential i-th P and the k-th R [km] dsjk: travel distance between the potential j-th DC and the k-th R [km] tsij: travel time between the potential i-th P and the potential j-th DC [hours]
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tsik: travel time between the potential i-th P and the k-th R [hours] tsjk: travel time between the potential j-th DC and the k-th R [hours] g: fixed cost of setting up a P [€] f: fixed cost of setting up a DC [€] o: operation activity costs (includes: receiving, handling, picking and shipping activities) at the DCs [€/ton] e: operation cost for shipping products from P to R (includes additional picking and shipping activities) [€/ton] dkt: products demand in k-th R in the t-th period [ton] p: maximum capacity of the P [ton/period] r: maximum capacity of the DC [ton/period] sl: maximum time of total transportation activities (from P to R) so as to guarantee the shelf life of the product [hours] top: time spent in operation activities (including: receiving, handling, picking and shipping activities) at the P or at the DC to prepare the shipment for the R [hours] α: percentage of the returned products [%] A1: relative weight of “total cost minimization” objective function A2: relative weight of “total variance minimization” objective function
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Decision variables: qijt: quantity shipped from the i-th P to the j-th DC during the t-th period [ton] qikt: quantity shipped from the i-th P to the k-th R during the t-th period [ton] qjkt: quantity shipped from the j-th DC to the k-th R during the t-th period [ton] xi = 1, if a P is located and set up at potential site i, 0, otherwise yj = 1, if a DC is located and set up at potential site j, 0, otherwise wij = 1, if the j-th DC is served by the i-th P, 0, otherwise zjk = 1, if the k-th R is served by the j-th DC, 0, otherwise uik = 1, if the k-th R is served by the i-th P, 0 otherwise bki = 1, if the returned products of the k-th R is shipped to the i-th P, 0 otherwise 5.1 Robust solution The model for obtaining the robust configuration (i.e. the configuration with the lowest expected total cost) can be formulate as a multi period MIP: min TC ,
(1)
where I
J
T
J
K
T
TC qijt csij dsij o q jkt cs jk ds jk i 1 j 1 t 1
I
K
T
j 1 k 1 t 1
K
I
T
qikt csik dsik e d kt crki dsik bki i 1 k 1 t 1
k 1 i 1 t 1
18
I
J
g xi f y j , i 1
(2)
j 1
s.t.
J I qikt q jkt d kt , j 1 i 1
k, t
(3)
qikt uik d kt
i, k, t
(4)
q jkt z jk d kt
j, k, t
(5)
j
(6)
k
(7)
wij xi
i, j
(8)
wij y j
i, j
(9)
z jk y j
j, k
(10)
u ik xi
i, k
(11)
j, t
(12)
i, j, k
(13)
i, k
(14)
I
w
ij
yj ,
i 1
J I uik z jk 1, j 1 i 1
I
K
qijt q jkt , i 1
ts
ij
k 1
wij ts jk z jk top sl
tsik uik top sl
19
K J qijt qikt xi p , k 1 j 1
i, t
(15)
j, t
(16)
i, k
(17)
k
(18)
K
q
yj r ,
jkt
k 1
bki xi
I
b
ki
1,
i 1
The solution of the model proposed will allow the user to decide the locations of the two types of facilities (i.e., the Ps and the DCs), while considering simultaneously the forward and the reverse flows, together with their correlation, the limited capacity of the facilities and the constraints on the maximum time for transportation (so as to guarantee the shelf life of the products). The objective of the model is to minimize the total costs of the system, which include both the fixed costs and the variable ones. In particular: constraint (3) ensures that the demand of k-th R in the t-th period is always satisfied (from at least one P or DC); constraint (4) permits to deliver quantity qikt from the i-h P to the k-th R in the t-th period only if the i-th P is enabled to deliver goods to the k-th R; similarly constraint (5) permits to deliver quantity qjkt from the j-th DC to the k-th R in the t-th period only if the j-th DC is enabled to deliver goods to the k-th R. Constraint (6) declares that each activated DC can be served only by one P, while constraint (7) declares that each R can be served only by one P or by one DC. Constraints (8), (9), (10) and (11) ensures that a delivery from a point (P or DC) to another (DC or R) can exists only if both locations exists and are selected to be a part of the network. Constraint (12) ensures that all goods delivered to the j-th DC from any P, are then delivered to Rs. Constraints (13) and (14) ensure that time spent in transport
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and operation activities (at Ps or at DCs) does not exceed shelf life of products sl. Furthermore in t-th period, the i-th P (constraint (15)) nor the j-th DC (constraint (16)) cannot deliver quantities greater than its production/storage capacity. Constraint (17) declares that the k-th R can return goods to the i-th P, only if i-th P is selected to be a part of the network (independently on the forward choice of delivering goods from i-th P to the considered R; in other words a certain P1 can deliver goods to the considered jth R, which can return goods to another P2). Finally constraint (18) ensures that k-th R returns goods to only one P. This model has been coded using FICO Xpress® and, assessing several experimental tests, it has been possible to observe that the optimal solution may be efficiently solved up to 40 locations (5 Ps, 8 DCs, 27 Rs) and 10 time-periods to capture the different scenarios describing the uncertainty of the demand. 5.2 Stable solution Demand at each R varies from period to period, this variability induce the fluctuations of the transport and handling cost components within the distribution network. The variance of the considered cost components between each facility has been taken into account as main source of variability. Four different variance components been taken into account separately, named: fijt: transport and handling costs related to a delivery from the i-th P to the j-th DC during the t-th period [$]; fikt: transport and handling costs related to a delivery from thei-th P to the k-th R during the t-th period [$];
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fjkt: transport and handling costs related to a delivery from the j-th DC to the k-th R during the t-th period [$]; fkit: transport and handling costs related to a reverse shipment from the k-th R to the i-th P [$]. The different variance components are computed as follows: I
J
12 2fij .
(19)
i 1 j 1
J
K
22 2f jk .
(20)
j 1 k 1
I
K
32 2fik .
(21)
i 1 k 1
K
I
42 2fki .
(22)
k 1 i 1
So as to calculate the stable distribution network, according to the definition given (i.e. the network configuration which is able to originate the lowest variation of the different cost components, over the horizon of demand uncertainty set) among different variability dimensions we chose the total variance of transport and handling cost components as the key performance indicator. Therefore the stable network configuration is the configuration that minimizes the following relationship:
4 2 min VoC tot 2f . f 1
(23)
22
5.3 Combining robust and stable solution As stated in the previous section the model has been developed as a bi-objective Mixed Integer Programming, where the two conflicting functions in (1) and in (23) have to be minimized. The most common approach to multi-objective optimization is the weighted sum method (Arora, 2004): M
U wmvm x .
(24)
m 1
Here w is a vector of weights typically set by the decision maker such that M
w
m
1 and w > 0. If objectives are not normalized, wm’s need not add to 1. As with
m 1
most methods that involve multi-objective function weights, setting one or more of the weights to zero can result in weakly Pareto optimal points. The relative value of the weights generally reflects the relative importance of the objectives. This is another common feature of weighted sum methods. If all the weights are omitted or are set to 1, then all objectives are treated equally. The weights can be used in two ways. The user may either set w to reflect preferences before the problem is solved or systematically alter w to yield different Pareto optimal points (generate the Pareto optimal set). In fact, most methods that involve weights can be used in both of these capacities, i.e. to generate a single solution or multiple solutions. This method is easy to use, and if all the weights are positive, the minimum of Eq. (24) is always Pareto optimal. However, there are a few recognized difficulties with
23
the weighted sum method (Das and Dennis, 1997 and Koski, 1985). First, even with the use of some of the methods discussed in the literature for determining weights, a satisfactory a priori selection of weights does not necessarily guarantee that the final solution will be acceptable; one may have to re-solve the problem with new weights. In fact, this is true of most weighted methods. The second problem is that it is impossible to obtain points on nonconvex portions of the Pareto optimal set in the criterion space. Although nonconvex Pareto optimal sets are relatively uncommon, some examples are noted in the literature (KOski, 1985; Stadler and Dauer, 1992 and Stadler, 1995). The final difficulty with the weighted sum method is that varying the weights consistently and continuously may not necessarily result in an even distribution of Pareto optimal points and an accurate, complete representation of the Pareto optimal set. Adopting the weighted sum method to the present work, the resulting model is defined as follows: min TC A1 VoC A2 ,
(25)
s.t. (3) – (18). When the value of coefficient A1 is set to 1 and A2 is set to 0, a robust solution is expected, while in case A1 is set to 0 and A2 is set to 1 a stable solution is reached. It is also possible to set both coefficient to the same value (e.g. 0.5) in order to obtain a “balanced” solution, or to choose different values of the coefficients in order to give more importance to one or the other of the two considered functions. It should be noted how the objective function in Eq. (25) can be interpreted as a variation of the objective
24
function of the mixed-integer linear program of Pan and Nagi (2010), where infeasibility penalty, i.e.. customer demand not met, is not considered.
6.
Model application and results The proposed model has been applied to a producer, distributors and retailers of
fresh food. The network design problem in this context is particularly complex, mainly because of the constraint introduced by the constraint on the maximum transport lead time, from the P to the R (both direct or via DC), which should be less than 24 hours, so as to guarantee the appropriate shelf life of the product at the R. Such a constraint increases the total costs of the network because, in general, it is necessary to have a higher number of DCs with respect to other types of networks. Moreover, it is necessary to adopt particular vehicles and warehouses, so as to maintain the products at the right temperature. We consider the situation in which the given R number is 27 and their location is defined. For each R, the demand is known and it is defined as a uniform distribution, which mean is related to the population density of the city where each R is located; moreover the distribution is defined in the range [0.5dj 1.5dj], where dj is the mean value of demand for retailer j. It has to be underlined that the mean demand values are very different among Rs: in fact, the maximum dj value considered is about 30 times the minimum one. Based on these distributions, yearly demands were sampled for each R for an horizon of 10 periods (corresponding to 10 years), so as to apply the multi period MIP model. In the following table, the most relevant parameters of the case study are offered, as implemented in the model. Parameter g f e o p
Value 250,000 [€] 180,000 [€] 33 [€/ton] 37 [€/ton] 50,000 [ton/year]
25
r top sl
25,000 [ton/year] 5 [hours] 24 [hours] 3%
Table 1. Parameters of the industrial case. Applying the MIP model for the optimisation of the robust solution (A1 = 1, A2 = 0) within the set of parameters given, a solution with 2 Producers and no Distribution Centres is obtained. The application of the heuristic, developed for the stable configuration (A1 = 0, A2 = 1), lead to a solution with 5 Producers and 8 Distribution Centres (i.e. all potential Ps and DCs are involved in the network). As expected at a preliminary evaluation, while comparing the two solutions, the stable exhibited a total expected cost greater than the robust one (with a significant difference, equal to 36.77%), while the decrease in the variability of the flows was equal to 90.84%.
Figure 2. Distribution network: robust configuration.
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Figure 3. Distribution network: robust configuration, forward and reverse flows. Assigning different values to A1 and A2 coefficients, the following results has been obtained. A1
A2
Activated DCs 0
ΔTC (%)
-
Activated Ps 2
-
ΔFV (%) -
1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 -
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
5 5 5 5 5 5 3 5 5 5
1 8 8 8 8 8 3 8 8 8
29.76% 31.82% 33.26% 32.17% 31.35% 34.72% 14.54% 32.72% 33.51% 36.77%
-72.13% -81.17% -78.77% -73.06% -82.13% -83.63% -50.20% -72.99% -82.80% -90.84%
Robust solution
Balanced solution
Stable solution
Table 2. Distribution network configurations for different levels of A1 and A2 coefficients.
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Figure 4. (0-1) normalized values of TC and FV of the different distribution network configurations for different levels of A1 and A2 coefficients.
7.
Conclusions The distribution network design is a critical step within the assessment of
strategic decisions for a supply chain management. In particular, this step which exerts a significant influence on the system performance, especially when demand is uncertain and variable over time. In this work we have adapted from the layout flexibility taxonomy (Braglia et al., 2003) two flexibility criteria that can be integrated within a strategic decision making process for the distribution network design. In order to take this fact into account, two alternative strategies were identified and discussed, i.e. the identification of a robust or a stable distribution network configuration. Moreover also intermediate solutions have been analysed, giving to the two conflicting objective functions different relevance coefficients. The present paper formally defined and discussed the properties of the different strategies, which may be founded on the adoption of a more stable configuration or a more robust one. In particular, the discussion made also reference to previous studies by the authors themselves, which were published on the features of layout flexibility.
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The study proposed was motivated by the real problem observed and faced in a fresh food company in Italy. In particular, the company produces a number of product types, starting from different factories located in Italy, and distributing them all over the country through a multi level distribution network. So as to find the solution, a model has been presented and the results obtained helped the company managers to proceed to take the decision about the best network configuration. A further development of the present study is to consider as source of risk not only the retailers demand changes but also risk due to the uncertainties associated with stock holding costs and delivery frequencies changes as proposed by Lalwani et al. (2006). An additional further development of interest is to look for a performance indicator taking into account the reliability of the network itself, as firstly introduced in Snyder and Daskin (2005), the aim of which is to minimize costs taking into account, together with the expected transportation costs, the possible failures at the facilities.
8.
References
Arora, J.S., 2004. Introduction to Optimum Design, Second edition. Elsevier. Ben-Tal, A. and Nemirovski, A., 2002. A robust optimization methodology and applications. Mathematical Programming, Series B, 92, 453–480. Blanchini, F., Rinaldi, F. and Ukovich, W., 1997. A network design problem for a distribution system with uncertain demands. SIAM Journal on Optimization, 7 (2), 560– 578.
29
Braglia, M., Zanoni, S. and Zavanella, L., 2003. Layout design in dynamic environments: strategies and quantitative indices. International Journal of Production Research, 41, 995–1016 Braglia, M., Zanoni, S. and Zavanella, L., 2005. Robust versus Stable Layout design in stochastic environments. Production Planning and Control, 16, 71–80. Chen, C.L., Yuan, T.W. and Lee, W.C., 2007. Multi-criteria fuzzy optimization for locating warehouses and distribution centers in a supply chain network. Journal of the Chinese Institute of Chemical Engineers, 38, 393–407. Chopra, S. and Meindl, P., 2001. Supply Chain Management: Strategy, Planning, and Operations. Upper Saddle River: Prentice-Hall. Das, I., and Dennis, J.E., 1997. A closer look at drawbacks of minimizing weighted sums of objectives for pareto set generation in multicriteria optimization problems. Structural Optimization, 14, 63–69. Daskin, M.S., Snyder, L.V. and Berger, R.T., 2005. Facility location in supply chain design. In Langevin, A., Riopel, D. (Eds.) Logistics Systems: Design and Operation, New York: Springer. Georgiadis, M.C., Tsiakis, P.,Longinidis, P. and Sofioglou, M.K., 2011. Optimal design of supply chain networks under uncertain transient demand variations. Omega, 39, 254– 272. Hasani, A., Zegordi, S.H. and Nikbakhsh, E., 2011. Robust closed-loop supply chain network design for perishable goods in agile manufacturing under uncertainty. International Journal of Production Research, DOI: 10.1080/00207543.2011.625051.
30
Koski, J., 1985. Defectiveness of weighting method in multicriterion optimization of structures. Communications in Applied Numerical Methods, 1, 333–337. Lalwani, C.S., Disney, S.M., Naim, M.M., 2006, On assessing the sensitivity to uncertainty in distribution network design. International Journal of Physical Distribution & Logistics Management, 36, 5 – 21. Li, L. and Schulze, L., 2011. Uncertainty in Logistics Network Design: A Review. Proceedings of the International MultiConference of Engineers and Computer Scientists 2011, Vol II, IMECS 2011, March 16–18, 2011, Hong Kong. Logendran, R. and Terrell, M.P., 1988. Uncapacitated plant location allocation problems with price sensitive stochastic demands, Computer and Operations Research, 15, 189-198. Louveaux, F.V., 1993. Stochastic Location Analysis. Location Science, 1, 127-154. Louveaux, F.V. and Peeters, D., 1992. A dual-based procedure for stochastic facility location. Operations Research, 40, 564-573. Mo, Y. and Harrison, T.P., 2005. A conceptual framework for robust supply chain design under demand uncertainty. In Geunes, J., Pardalos, P.M. (Eds.) Supply chain optimization. New York: Springer. Owen, S.H. and Daskin, M.S., 1998. Strategic facility location: a review. European Journal of Operational Research, 111, 423-447. Pan F. and Nagi R., 2010. Robust supply chain design under uncertain demand in agile manufacturing. Computers and Operations Research, 37, 668–683.
31
Patriksson, M., 2008. Robust bilevel optimization models in transportation science. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Science, 366, 1989–2004. Piplani, R. and Saraswat, A., 2012. Robust optimisation approach to the design of service networks for reverse logistics. International Journal of Production Research, 50, 1424–1437. Pishvaee, M.S., Rabbani, M., Torabi, S.A., 2011, A robust optimization approach to closed-loop supply chain network design under uncertainty. Applied Mathematical Modelling, 35, 637–649. Romero Morales, D., van Nunen, J.A.E.E. and Romeijn, H.E., 1999. Logistics network design evaluation in a dynamic environment. In Speranza, M.G., Stähly, P. (Eds.). New Trends in Distribution Logistics, Berlin: Springer-Verlag. Rosenblatt, M.J. and Lee, H.L., 1987. A robustness approach to facilities design. International Journal of Production Research, 25, 479–486. Shimizu, Y., Fushimi, H. and Wada, T., 2011. Robust Logistics Network Modeling and Design against Uncertainties. Journal of Advanced Mechanical Design, Systems, and Manufacturing, 5, 103–114. Snyder, L.V., 2006. Facility location under uncertainty: a review. IIE Transactions, 38, 537–554. Snyder, L. V. and Daskin, M. S., 2005. Reliability models for facility location: The expected failure cost case. Transportation Science, 39, 400–416.
32
Stadler, W., 1995. Caveats and boons of multicriteria optimization. Microcomputers in Civil Engineering, 10, 291–299. Stadler, W., and Dauer, J.P., 1992. Multicriteria optimization in engineering: A tutorial and Survey. In M. P. Kamat (Ed.), Structural optimization: Status and promise. Washington, D.C.: American Institute of Aeronautics and Astronautics. Van Landeghem, H. and Vanmaele, H., 2002. Robust planning: a new paradigm for demand chain planning. Journal of Operations Management, 20, 769-783. Zhang, Y., Xie, L., Hang, W. and Cui, X., 2007. A Robust Model for 3PLS to Design a Remanufacturing Logistics Network under the Uncertain Environment. Proceedings of the IEEE International Conference on Automation and Logistics, August 18-21, 2007, Jinan, China.
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