ИЗВЕСТИЯ РАН. ТЕОРИЯ И СИСТЕМЫ УПРАВЛЕНИЯ, 2014, № 2, с. 57-67
УДК 62.40
MULTIPERIOD AND STOCHASTIC FORMULATIONS FOR A CLOSED LOOP SUPPLY CHAIN WITH INCENTIVES
© 2014 г. I. Litvinchev, Y. A. Rios, D. Özdemir, L. G. Hernandez-Landa
Moscow, Russia, Computing Center Russian Academy of Sciences
E-mail: litvin@ccas.ru Mexico, Faculty of Mechanical and Electrical Engineering, UANL Turkey, Faculty of Engineering, Yasar University, Universite Caddesi Received March 05, 2013
Abstract — Reverse logistics network design problem we focus on is about locating distribution centers, inspection centers and remanufacturing facilities, and determining the acquisition price as well as the amount of returned goods to be collected depending on the unit cost savings and competitor's acquisition price. We introduce the multiple periods setting and stochastic demand formulated by scenarios. We develop two mathematical programming models to determine the pricing strategy of the recovered products together with the optimal network that must be designed to be the most profitable closed cycle. Our methodology is based on a Golden Section Search with some flexibility that enables us to fix the used product acquisition price and then solve the model as an integer linear programming. Moreover, we establish dependent size fixed costs of opening a distribution, an inspection, and a remanufacturing centers, and show that they have a strong impact on the Golden Section search behavior.
DOI: 10.7868/S0002338814020127
Introduction. "Reverse Logistics" or "Closed Loop Supply Chains" are the networks designed with the goal of addressing a fUll process of product delivery and at the same time the product recovery at the end of its usable life for remanufacturing or disposal. A reverse logistics network was defined in [1] as the relationship between the used product market and the market for new products: when these two markets coincide then we have a closed-loop network. In [2] the members of the closed loop supply chain were divided in two groups: the members of the traditional logistics chain, including raw material suppliers, manufacturers, retailers and demand markets, and members of the reverse logistics chain, including market demand, recovery centers and manufacturers.
The closed loop supply chain management has had an increasing interest due to the actual focus on ecologically sustainable development, e.g., in [3] the authors consider that good management of a logistics plan demonstrates the commitment of the company with the environment. Moreover, the economic benefits of reusing serviceable waste are substantial compared to the exclusive use of new raw materials.
Currently, the European Union has a legislation to deal with the pollution caused by electronic wastes: Waste of Electric and Electronic Equipment (WEEE) Directive [4]. The general purpose of these directives is to reduce the electrical and electronic waste and to promote its reuse, recycling and the other forms of recovery in order to reduce the final disposal. The WEEE directive covers a wide range of products: small or large household appliances, telecommunications equipment, lighting equipment, electric tools, toys, sport and leisure equipment, medical devices, monitoring and controlling devices, and automated devices.
Producers must now recover and recycle a predetermined fraction of sold products. These activities involve collection of used products, inspection and separation to determine whether the product is recoverable or not, reuse, recycling, remanufacturing or repairing the product, disposal of the unrecoverable products, and redistribution of recovered remanufactured products [5].
We focus on a reverse logistics network design problem where distribution centers, inspection centers and remanufacturing facilities must be located, the acquisition price as well as the amount of returned goods to be collected depending on the unit cost savings and competitors acquisition price must be determined. The objective of this study is to determine the used product collection strategies in two different
Factories I
Distribution/ Inspection Centers J
Clients K
■й
-Û Q.
Fig. 1. Closed loop schema
reverse logistics networks extended from the model proposed in [6]. The first network considers multiple periods along the time while the second one takes into account a stochastic demand formulated by scenarios. We develop two mathematical programming models to determine the pricing strategy of the recovered products together with the optimal network that must be designed to be the most profitable closed cycle.
More precisely, we have a closed loop supply chain of three levels (Fig. 1), where the manufactured product is shipped to distribution centers and from there it is distributed to the clients. From those clients, the used product is recovered and sent to the inspection centers where it is decided if it would be discarded or sent to the manufacturing plants (dashed lines of Fig. 1). Moreover, we must choose the best distribution centers, the best inspection centers (distribution center with a mark in the figure), and the remanu-facturing factories (factory marked with an R in the figure) in order to meet the levels of demand in a mul-tiperiod or stochastic demand environment, such as minimizing transportation costs between the three levels of the chain. Furthermore, the solution should offer the best acquisition price of the used product as we consider that there is competition with other organizations as in [3].
Our methodology is as follows. First, we propose two new nonlinear integer programming models, one for the multiperiod case and the other for the stochastic demand. Then, we adapt a Golden Section Search that enables us to fix the used product acquisition price and then solve the model as an integer linear programming. In this stage we explore the trade-off between quality and computational time. Moreover, we establish a new manner of determining the fixed costs of opening a distribution, an inspection, and reman-ufacturing centers. We show that these new cost have a strong impact on the Golden Section search behavior.
The rest of this work is structured as follows. Below we give a brief literature review to underscore some work done on closed loop supply chain and customer incentive models. In Section 1 the multiperiod nonlinear model is presented together with the stochastic nonlinear model. Section 2 exhibits our Golden Search Method we tune the accuracy of the branch-and-bound method for the solution of the subproblem. In this same section we introduce a new manner of computing the fixed costs of the problem. Section 3 is related to the experimental results on random generated instances to show that our methodology is efficient. A final section concludes this work.
Literature review. The reverse logistic network problem has been broadly studied (see [1, 2, 7—10] and the references therein). Moreover, there are some excellent reviews by Pokharel and Mutha [3] and Fleischmann et al. [11].
With respect to reverse logistics that consider a multiperiod setting we can mention the work [12] where the authors present a genetic algorithm. In [13], a multiperiod reverse logistics network is designed under risk by a stochastic mixed integer linear programming. In [14], a multicommodity formulation is presented to use a reverse bill of materials by integer linear programming.
The stochastic demand point of view has been studied in [13, 15, 16] by integer linear stochastic programmings. In [17] the authors integrate a sampling strategy with an accelerated Benders decomposition, while [18] proposes a robust optimization model for handling the inherent uncertainty of input data.
There are few pricing models for acquiring used products [3]. We can cite [19] where the remaining value in the used products that can recovery is the company's main motivation for the collection operation. They use nested heuristics based on a tabu and Fibonacci searches. In [20] a simulation model is presented to calculate the collection costs.
Our methodology is as follows. First, we propose two new nonlinear integer programming models, one for the multiperiod case and the other for the stochastic demand. Then, we adapt a Golden Section Search that enables us to fix the used product acquisition price and then solve the model as an integer linear programming problem. In this stage we explore the trade-off between quality and computational time. Moreover, we establish a new manner of determining the fixed costs of opening a distribution, an inspection, and remanufacturing centers. We show that these new cost have a strong impact on the Golden Search behavior.
Our work is based on the concept of [6], where the aim is locating distribution centers, inspection centers and remanufacturing facilities, determining the acquisition price as well as the amount of returned goods to be collected depending on the unit cost savings and competitor's acquisition price to minimize the transportation costs, fixed costs and used product acquisition costs. A mixed-integer nonlinear programming problem studied in [6] becomes a mixed integer linear one when the acquisition price is set to a given value. The best value of the acquisition price is determined by the Golden Section search. Our work extends [6] by introducing the multiperiod framework and stochastic demand. Moreover, we improve the Golden Section search by introducing some flexibility that improves significantly the execution times without losing the quality of the solutions.
1. Nonlinear mixed integer programming models. We consider that a single type of product is produced, sent to the distribution centers, distributed to the clients, returned to the inspection centers, and returned to a remanufacturing plant. Our models must be an effective tool to decide which distribution and inspection centers must be established and which factories
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