Transportation Research Part D 22 (2013) 45–48
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A multi-objective optimization model for sustainable logistics facility location
a,⇑, b, a
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a School of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China
b Department of Civil, Architectural and Environmental Engineering, Illinois Institute of Technology, Chicago, IL 60616, USA
article info
Sustainable facility location Service reliability
Carbon dioxide emissions
1. Introduction
This paper offers an exploratory study of sustainable facility location. The methodology, based on the classical uncapacitated facility location problem, provides decision makers with a multi-objective optimization model to determine the trade-off among economic, service and environmental considerations. Our results indicate that it may be desirable to open more facilities than optimal from a narrow economic perspective to reduce the car- bon dioxide emissions of transport and to improve service reliability.
Ó 2013 Elsevier Ltd. All rights reserved.
Road freight transport has been a rapidly growing contributor to carbon dioxide (CO2) emissions (European Commission, 2010). Logistics system design models have traditionally focused on minimizing economic cost or maximizing customer ser- vice level without taking CO2 emissions into account. Recent studies, however, started to address the challenging require- ments of sustainable facility location.
This paper develops a multi-objective uncapacitated facility location problem (UFLP) with an environmental objective in the context of sustainable development. The model is to simultaneously minimize economic cost, CO2 emissions, and max- imize service reliability by strategically locating facilities within a logistics network.1
2. Model formulation and solution algorithms
2.1. Formulation of the model
The UFLP is a classical facility location problem and forms the basis of many location models that have been used in logis- tics network design. Its finds the best location of facilities and the allocation of customers that minimizes transportation and fixed costs (Daskin et al., 2003).
To be competitive, logistics service providers need to offer suitable levels of service to customers, and this is often related to facility location layouts, distances from customers to facilities, time requirements for delivering goods, the capability of facilities to supporting customer’ demands, and the connectivity of transportation routes.
Here, customer service reliability is defined as the probability of one facility providing the required goods to customers within a given time given a set of constraints. One of the constraints considered is the need to limit CO2 emissions. While there are a number of factors affecting emissions levels, we only consider the amount of goods being transported and the
⇑ Corresponding author. Tel.: +86 13851962244. E-mail address: (T. Xifeng).
1 For recent work in this field see, Chaabane et al. (2011) and Elhedhli and Merrick (2012).
1361-9209/$ – see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.trd.2013.03.003
46 T. Xifeng et al. / Transportation Research Part D 22 (2013) 45–48
distance travelled assuming, road vehicles are identical, that average speed is known, and the gradient of a road is not a factor.
Based on the UFLP, the problem is formulated as a mixture of three mathematical programming formulations: minimum economic cost, maximum customer service reliability and minimum CO2 emissions. We define the indices i = 1, 2,. . ., I and j = 1, 2,. . ., J as corresponding to customers and candidate facilities; each customer having a demand hi. A logistics facility at location j has a fixed cost fj. The unit cost of shipping from a facility at location j to customer i is cij. The distance between a facility at j and customer i is dij. The speed of trucks shipping from facilities to surrounding customers is modeled as a ran- dom variable v with a distribution function Fv(). Ti is the time deadline of customer i; evf is the CO2 emissions of a fully loaded vehicle; eve is the CO2 emissions of an empty vehicle, and w is the weight limit for a vehicle. Two binary location variables are introduced; Xj taking the value one if a facility is open at candidate location j and zero otherwise; Yij takes a value of one if customer i is assigned to facility j and zero if not. The model can be formulated as:
fjXj þ cijhidijYij ð1Þ
j1⁄41 i1⁄41
dij1⁄2ðevf eveÞhi=wþevedhi=weYij ð3Þ
j1⁄41 i1⁄41 XJ
Yij1⁄41 8i ð4Þ
Yij 6Xj 8i; j ð5Þ
Xj2f0;1g 8j ð6Þ
Yij 2f0;1g 8i; j ð7Þ
The objective function (1) minimizes the sum of fixed location and transportation costs. The objective function (2) maxi- mizes minimum service reliability and function (3) minimizes CO2 emissions from transportation. Constraints (4) guarantee that each customer is assigned to exactly one logistics facility, and constraints (5) state that a customer cannot be assigned to a facility unless it is open. Constraints (6) and (7) are standard integrality constraints.
2.2. The hybrid algorithm
A classical technique is utilized to solve the multi-objective optimization problem, which applies the e-constraint meth- od. From the perspective of sustainable development, the environmental impact is considered as the priority, with the eco- nomic and the service objectives formulated as constraints. After transforming the multiple objectives into one, the greedy heuristic is used to construct a feasible solution by greedily dropping facilities from the solution until no further improve- ment can be obtained. In detail, the procedure is:
Step 1: let the current number of facility locations k = J, that is, exists facilities at all candidate sites.
Step 2: allocate each customer to the nearest facility among k facility locations, and compute the CO2 emissions and the cost and the service reliability to each customer.
Step 3: if the economic cost is lower than the decision maker’s expectation, and the service reliability is higher than the specified limit, stop and return the last results; otherwise, go to Step 4.
Step 4: select one facility and ensure that the increase in CO2 emissions is smaller than if its customers are reallocated to other nearby facilities, while guaranteeing that the economic cost and service reliability satisfy the decision makers. Step 5: drop the facility from the solution and let k = k 1, then go to Step 2.
The demand of each customer location.
Customer location C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22 C23 C24 C25
Demand(ton) 166 156 88 59 163 191 79 141 99 170 159 50 176 199 113 180 126 48 56 93 155 169 162 77 116
Minfð1 Fvðdij=TiÞÞYijg ð2Þ
The fixed location cost of each candidate facility.
Facility location F1 F2 F3 F4 F5
Location cost ($) 78,500 46,900 92,200 58,700 89,800
46,300 85,400
76,000 85,300
C22 C23 C24 C25
11 176 292 237 266 201 10 240 75 195 251 15 32 131 251 85 245 42 15 197 43 226 164 147 264 73 283 292 29 196 97 225 106 258 242 171 179 26 181 90
CO2 emissions (Kg)
5.3137 103
T. Xifeng et al. / Transportation Research Part D 22 (2013) 45–48
The distance between each customer location and each candidate facility location.
Distance (km) C1
C4 C5 C6 C7 C8
111 269 252 114 123 222 155 112 294 213 158 164 249 120 44 242 182 53 133 262 246 229 39 98 139
57 257 264 48 25
38 115 14 95 10 247 26 207 269 226 192 221 212 75 211
C9 C10 C11
204 141 66 168 77 157 256 130 131 168 211 223 271 121 60 126 55 255 108 257 204 147 176 41
77 113 258
183 163 49 27 232 230 127 18 176 53
219 161 76 276 228 267 210 56 222 21
234 151 128 184 257 202 158
90 212 115
C15 C16 C17 C18
171 42 153 135 267 128 257 177 253 103 116 264 270 163 209 141 282 278 189 132 245 90 136 224
61 102 143 111 101 258 285 259 27 103 26 140 79 37 84 150
147 69 26 21 267 70 259 214 262 282
42 119 295 194 269 145 55 187 70 159
218 183 177 131
74 129 48 183 288 29
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10
85 241 297
47 276 147 100 42 209 86 152 124 166 122 11 262 53 88 13 173 241 272 182 104 40 65 25 251 156 154
75 100 132 94
Best solution to the sustainable facility location problem.
Facility location
F1 F2 F3 F4 F5 F6 F7 F8
Allocation of customers
C20, C22 C24
C2, C7, C10
C1, C4, C6, C8, C18
C5, C12, C14
C3, C9, C15, C17
C11, C13, C16, C21, C23
Service reliability
Fig. 1. Impact of the number of facilities on economic cost and CO2 emissions.
3. Computational experiments
We assume there are 25 customers, Cj (j = 1, 2, . . ., 25), the demand of each customer hi as shown in Table 1. There are 10 candidate facility locations designated as Fj (j = 1, 2, . . ., 10) with the fixed location cost of each shown in Table 2. Table 3 lists the distances between each customer and candidate facility location, dij. The shipment cost per ton-kilometer is cij= $1. The truck considered is fully loaded with 25 tons, the CO2 emissions of a fully loaded vehicle is evf = 1.096 kg/km, the CO2 emis- sions of an empty vehicle is eve = 0.772 kg/km; and the truck’s speed is normally distributed with a mean of 80 km/h and
48 T. Xifeng et al. / Transportation Research Part D 22 (2013) 45–48
Fig. 2. Impact of the number of facilities on economic cost and service reliability.
variance 102. Customers have the same time deadline of 2 h. The limits of service reliability and costs are Rij =0.9, and TC = $700,000.
The computations are solved by coding in MATLAB Version 7.13 (R2011b), and executing the computational program, the results are shown in Table 4.
4. Sensitivity analysis
The solution provides a reasonable compromise solution. When relevant parameters are kept unchanged, and the model only includes the economic objective, the optimum locations of facilities are at F2, F6 and F10, and the minimum cost is $380,725. The poorest service reliability is 0.691, and the CO2 emissions are 9.377 103 kg. When the model includes both the economic and the service objectives, but assuming the latter has priority, the optimum locations of facilities are F2, F4, F6, F7, F8, F9 and F10, the maximum service reliability is 0.993, while the cost is $584,503, and CO2 emissions are 5.506 103 kg.
Focusing on CO2 emissions, service reliability and the number of open facilities, Fig. 1 shows how emissions fall with more facilities, and Fig. 2 how reliability increases with more facilities. As seen in the figures, costs fall with decreasing numbers of facilities in the earlier phases, and is minimized when there are three facilities, they then begin to rise because of the saving in locating facilities is fully absorbed into transport costs. For longer distances, the number of facilities decreases, and CO2 emissions rise, while service reliability decreases. As the environmental impact of transport increases disproportionately with the cost of operating facilities, the environmentally friendly solution requires more open facilities than does economic cost effectiveness.
5. Conclusions
The paper has offered an extension of the classical UFLP, in which the economic objective, service objective and environ- mental objective are considered simultaneously as part of the sustainable facility location design. The computational tests demonstrate that the optimum solution obtained is a reasonable trade-off of the three component objectives, and suggest that more logistics facilities be opened to decrease CO2 emissions and improve service level.
Acknowledgements
The authors gratefully acknowledge the editor-in-chief and anonymous reviewers for their valuable comments and sug- gestions that will significantly improve the paper. This research is partially supported by the Fundamental Research Fund for the Central Universities (No. B12020025), PRC.
References
Chaabane, A., Ramudhin, A., Paquet, M., 2011. Designing supply chains with sustainability considerations. Production Planning & Control 22, 727–741. Daskin, M.S., Snyder, L.V., Berger, R.T., 2003. Facility Location in Supply Chain Design. Working Paper. Northwestern University.
Elhedhli, S., Merrick, R., 2012. Green supply chain network design to reduce carbon emissions. Transportation Research Part D 17, 370–379.
European Commission, 2010. Energy and Transport in Figures. European Commission, Brussels.
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