Green supply chain network design to reduce carbon emissions

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Abstract

We consider a supply chain network design problem that takes CO2 emissions into account. Emission costs are considered alongside fixed and variable location and production costs. The relationship between CO2 emissions and vehicle weight is modeled using a concave function leading to a concave minimization problem. As the direct solution of the resulting model is not possible, Lagrangian relaxation is used to decompose the problem into a capacitated facility location problem with single sourcing and a concave knapsack problem that can be solved easily. A Lagrangian heuristic based on the solution of the subproblem is proposed. When evaluated on a number of problems with varying capacity and cost characteristics, the proposed algorithm achieves solutions within 1% of the optimal. The test results indicate that considering emission costs can change the optimal configuration of the supply chain, confirming that emission costs should be considered when designing supply chains in jurisdictions with carbon costs.

Introduction

With the globalization of supply chains, the distance between nodes in the distribution network has grown considerably. Longer travel distances lead to increased vehicle emissions on the transportation routes, resulting in an inflated carbon footprint. Hence, there is a need to effectively and efficiently design eco-friendly supply chains, to both improve environmental conditions and the bottom line of the organization. Network design is a logical place to start when looking to green a supply chain design. Wu and Dunn (1995) cite transportation as the largest source of environmental hazards in the logistics system. This claim is supported by the fact that transportation via combustion engine vehicles accounted for 27% of the Canadian greenhouse gas (GHG) inventory in 2007 (Environment Canada, 2009). And while heavy duty diesel vehicles, such as diesel tractors commonly used in logistics, account for only 4.2% of vehicles on the road, they accounted for 29.2% of Canadian GHG emissions from transportation in 2007. Thus, reducing the number of vehicle kilometers travelled through the strategic placement of nodes could play a significant role in reducing the carbon footprint of the nation.

Supply chain design models have traditionally focused on minimizing fixed and operating costs without taking carbon emissions into account. Recent studies, however, started to take emissions into account. This includes Cruz and Matsypura, 2009, Nagurney et al., 2007, Benjaafar et al., 2010, Merrick and Bookbinder, 2010, Ando and Taniguchi, 2006.

This paper develops a green supply chain design model that incorporates the cost of carbon emissions into the objective function. The goal of the model is to simultaneously minimize logistics costs and the environmental cost of CO2 emissions by strategically locating warehouses within the distribution network. A three echelon, supply chain design model is proposed that uses published experimental data to derive nonlinear concave expressions relating vehicle weight to CO2 emissions. The resulting concave mixed integer programming model is tackled using Lagrangian relaxation to decompose it by echelon and by warehouse site. The nonlinearity in one of the subproblems is eliminated by exploiting its special structure. This decomposition results in subproblems that require less computational effort than the initial problem. By keeping most of the features of the original problem in the subproblems, a strong Lagrangian bound is achieved. A primal heuristic is proposed to generate a feasible solution in each iteration using information from the subproblems. The quality of the heuristic is measured against the Lagrangian bound. Test results indicate that the proposed method is effective in finding good solutions.

The remainder of the paper is organized as follows. In the next section we look at the emission data, followed by the problem formulation in Section 3. We then delve into the Lagrangian relaxation procedure and proposed heuristic in Sections 4 Lagrangian relaxation, 5 A primal heuristic for generating feasible solutions, respectively. Finally, we test the algorithm and heuristic in Section 6, and conclude in Section 7.

Section snippets

Emissions data

Few comprehensive data sets exist that show the relationship between vehicle weights and exhaust emissions. While the exact emission levels will depend on the engine type, terrain driven and the driver tendencies, the general relationship between vehicle weight and emissions will not change (i.e. linear, concave or convex relationship). This section reviews the available emissions data and draws conclusions about the relationship between emissions and the vehicle operating weight.

The most

Problem formulation

Let us define the indices i = 1,  , m, j = 1,  , n and k = 1,  , p corresponding to plant locations, potential distribution centers (DCs) and customers, respectively. A distribution center at location j has a maximum capacity Vj and a fixed cost gj. Each customer has a demand of dk. The variable cost of handling and shipping a production unit from a plant at location i to distribution center j is cij. Similarly, hjk denotes the average handling and shipping cost to move a production unit from distribution

Lagrangian relaxation

Given the difficulty in solving [FLM] directly, we use Lagrangian relaxation to exploit the echelon structure of the problem. It is important to select the constraints for relaxation, as relaxing more constraints may deteriorate the quality of the bound and heuristics. We relax constraints (2) using Lagrangian multipliers, μj, since they link the echelons of the supply chain. This leads to the following subproblem:[LR-FLM]:mini=1mj=1nf(xij)+j=1nk=1pf(dkyjk)+i=1mj=1n(cij-μj)xij+j=1nk=1p(h

A primal heuristic for generating feasible solutions

While the Lagrangian algorithm provides the Lagrangian bound, it does not reveal the combination of product flows, customer assignments and open facilities that will produce this result. Hence, heuristics are commonly used in conjunction with Lagrangian relaxation algorithms to generate feasible solutions.

To generate feasible solutions, we devise a primal heuristic based on the solution of the subproblems. Subproblem [SP1] generates the assignments of customers to distribution centers and

Numerical testing

The solution algorithm is implemented in Matlab 7 and uses Cplex 11 to solve the subproblems, the heuristic and the master problems. The test problems were generated similar to the capacitated facility location instances suggested by Cornuejols et al. (1991). The procedure calls for problems to be generated randomly while keeping the parameters realistic. The coordinates of the plants, distribution centers and customers were generated uniformly over [10, 200]. From the coordinates, the

Conclusions

This paper integrates the cost of carbon emissions into supply chain network design. The new problem formulation minimizes the combined expenses associated with the fixed costs to set up a facility, the transportation cost to move goods and the cost of emissions generated on the shipping lanes. The resulting network design model has practical applications for supply chain design, particularly in regions that have a carbon tax or cap-and-trade system.

This work offers two primary contributions to

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