Production, Manufacturing and LogisticsCapacitated dynamic lot sizing problems in closed-loop supply chain
Introduction
Introduced by Florian and Klein [6], the capacitated dynamic lot sizing problem (CDLP) and its variations have received quite a lot of attention from both academic researchers and industrial practitioners because of their vast application to a variety of practical problems, e.g. production planning. The traditional CDLP addresses a single item production planning problem in which deterministic but time-varying demands must be satisfied from either stock or production with limited capacity in a discrete-time finite horizon, and the decision maker should decide how much and when to produce in order to minimize the total production and holding costs. Obviously, in the traditional CDLP product flows are from manufacturers to customers. However, nowadays environmental legislation and economical arguments motivate companies to take full responsibility for their products during their entire life cycles [12], and a popular method is to set up a program for the collection and further processing of used products, which incurs product flows from customers to manufacturers. Possible options to process returned products include remanufacturing, repair, recycling and disposal [12]. So far, products that are being returned and processed include single use cameras, machine tools, medical instruments, copiers, etc. ([11], [12], [13]). Recent research on reverse logistics are reviewed by Dekker et al. [5].
This study is motivated by existing literature and our collaboration with an iron factory in Northeastern China. Besides producing steel from ore, the factory also collects scrap steel from two sources: (1) the processes of steel production; (2) collection centers located at the neighborhoods of the factory. The scrap steel is used to produce steel again in electronic furnaces, and the amount of scrap steel received in the planning horizon can be forecasted from the past data. Now, more than 20% of steel produced by this factory is made out of scrap steel. However, due to the limited number of furnaces in the factory, some of the production procedures are capacity constrained.
In this paper, we study the capacitated dynamic lot sizing problem arising in closed-loop supply chains. When attempting to formulate a general model of this problem, the scenario in our mind is as follows. A factory produces a certain kind of product of which the demands during a finite planning horizon are deterministic but time-varying, and should be satisfied without backlogging. From the environmental and economical perspective, the factory collects used products from customers. Two options are available for processing the returned products: disposal and remanufacturing. Demands can be satisfied by newly produced products or remanufactured products with the same quality commitment. Different from former research on similar problems, the capacities of production, remanufacturing and disposal are limited in the factory.
A brief review of related literature is stated below. Florian and Klein firstly proposed CDLP in 1971 [6], and this problem and its variations have been widely studied by researchers since then. The complexity of the CDLP mainly depends on the capacity parameter structure (variable or constant); but it is generally NP-hard even for several special cases [7], [2]. However, CDLP is not NP-hard in the strong sense; pseudo-polynomial algorithms for the general case have been proposed in the literature (see for instance [4]). Interested readers should refer to a recent comprehensive review on single item lot sizing problem by Brahimi et al. [3].
Compared to the huge amount of literature on dynamic lot sizing problem, much fewer studies have been done on dynamic lot sizing problem in closed-loop supply chain and most of them appeared in the recent few years ([8], [9], [10], [11]). Richter and Sombrutzki [9] firstly studied the uncapacitated dynamic production planning and inventory control model and some of its extensions under a Wagner/Whitin cost structures, and proved the zero-inventory-property of the optimal solutions. Richter and Weber [10] considered a similar problem with variable manufacturing and remanufacturing cost, and for the case of time-constant cost and demand data they proved the optimality of a policy starting with remanufacturing before switching to manufacturing and gave an estimation for the optimal switching point. Golany et al. [8] analyzed a production planning problem with remanufacturing, and provided the problems’s general formulation and assessed its computational complexity under various costs structures; they proved that the problem is NP-hard for general concave-cost structures. When cost are linear, they obtained an algorithm based on transforming the problem into the transportation problem in a special way. Beltran and Krass [1] studied the case where the demands can be totally satisfied by ordered items and unprocessed returned items, and the returned items can also be disposed; they proposed several useful properties of the optimal solutions which leads to an efficient dynamic programming algorithm. Teunter et al. [11] studied dynamic lot sizing with product returns and remanufacturing, and they analyzed this problem under two cases according to the set-up cost schemes: there is either a joint set-up cost for manufacturing and remanufacturing (single production line) or separate set-up costs (dedicated production lines). For the joint set-up cost case, they presented an exact, polynomial-time dynamic programming algorithm. For both cases, they suggest modifications of the well-known Silver Meal (SM), Least Unit Cost (LUC) and Part Period Balancing (PPB) heuristics.
All the models mentioned above are under the assumption that production and remanufacturing is uncapacitated. This assumption limits their application in settings where capacity constraints exist. The aim of this paper is to solve the capacitated dynamic lot sizing problem in closed-loop supply chain, and to analyze how capacity constraint affects the operations planning. A general model of this problem is formulated, and several important properties of the problem are identified when the cost functions are concave. Special cases of the general model are analyzed and solved to optimality.
The rest of the paper is organized as follows: Section 2 introduces a general formulation of the capacitated dynamic lot sizing problem with production, remanufacturing and disposal. In Sections 3, we characterize several useful properties of the problem when all the cost functions are concave. Special cases of the general problem are analyzed and solved in Section 4. Numerical experiments are conducted in Section 5, and Section 6concludes this paper.
Section snippets
A mathematical model of capacitated dynamic lot sizing problem with production, disposal and remanufacturing
In this section, we propose a general mathematical model of CDLP with production, disposal and remanufacturing options. The setting where the problem arises can be described as follows. A factory produces a certain kind of product, of which demands are deterministic but time-varying during a finite planning horizon and should be satisfied without backlogging. The factory is also responsible for processing used products returned from customers. The amount of the returned products is regarded
Several useful properties of the problem with concave cost functions
It is easy to observe that the set of feasible policies of the general model is a closed bounded convex set. Let E denote the set of the extreme points of the feasible set. When all the cost functions are concave, according to the optimization theory, an optimal policy can be obtained by searching only in E. Next, we characterize several properties of E. Proposition 1 If a feasible policy belongs to E, then Proof Suppose there exists such a policy in E and has
Several special cases of the general model
Usually the practical settings are much simpler than that described in the above general model. For example, the capacities of some operations may be unlimited, or limited but constant. It is often not necessary to consider production, disposal and remanufacturing simultaneously. In this section, we discuss several simplified cases of the general model.
Numerical experiments
In this section, we examine the efficiencies of the above proposed algorithms. The parameters of the test problems are generated randomly within a reasonable range. The reasonability is based upon our experiences during collaboration with the local iron factory mentioned in Section 1. All the cost functions are fixed charge functions. In practical settings with returned products, a too long planning horizon is meaningless since the data about the amount of returned products are forecasted and
Conclusion
In this paper, we study the capacitated dynamic lot sizing problem with returned products. Different from studies on traditional dynamic lot sizing problems that focus on producing new products, this paper considers the scenarios where returned products are also remanufactured or disposed under capacity constraints. A general model of this problem is formulated, incorporating production, disposal and remanufacturing operations. Several useful properties of the problem are characterized when the
Acknowledgements
The authors owe special thanks to an anonymous referee for insightful comments on this paper. This paper is financially supported by the Natural Science Foundation of China (NSFC 70625001, 70721001). Liaoning Provincial Scientific Project, PhD program funding of Ministry of Education (MOE) and 111 project of MOE in China with number B08015.
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2020, European Journal of Operational ResearchCitation Excerpt :As the capacitated lot-sizing problem, the LSR problem with bounded capacity remains NP-hard. A number of interesting complexity results and useful properties for the capacitated LSR problem with different cost structures have been found by Pan, Tang, and Liu (2009). Very few studies have been dedicated to developing exact methods to solve the difficult variants of the deterministic LSR problem.