In this paper, we will show that the economic lot-sizing problem with rework of defectives is NP-hard. Therefore, we reduce it to the well-known PARTITION problem. This is in line with the findings for similar models that in-vestigate lot-sizing with remanufacturing

Buscher, Udo

Rudert, Steffen

English

Technische Universität Dresden: Qucosa

        Thomas W. Günther,  Xenia Börner,  Melanie Mischer On the complexity of the economic lot- sizing problem with rework of defectives Dresdner Beiträge zur Betriebswirtschaftslehre – Nr. 182/22  Steffen Rudert  Udo Buscher       On the complexity of the economic lot-sizing problem with rework of defectives   Herausgeber: Technische Universität Dresden Fakultät Wirtschaftswissenschaften Die Professoren der Fachgruppe Betriebswirtschaftslehre ISSN: 0945-4810 https://doi.org/10.25368/2022.322 Autoren: Kontakt: Dipl.-Wirtsch.-Ing. Steffen Rudert Prof. Dr. Udo Buscher Lehrstuhl für Betriebswirtschaftslehre, insbesondere  Industrielles Management 01062 Dresden Telefon: (0351) 463-33470 E-Mail: bwlim@tu-dresden.dehttps://tu-dresden.de/bu/wirtschaft/bwl/limEmpfohlene Zitation: Rudert, S., Buscher, U. (2022). On the complexity of the economic lot-sizing problem with rework of defectives. Dresdner Beiträge zur Betriebswirtschaftslehre, Nr. 182/22, Dresden, https://doi.org/10.25368/2022.322 On the complexity of the economic lot-sizing problem with rework of defectives 1 On the complexity of the economic lot-sizing problem with rework of defectives Steffen Rudert1 and Udo Buscher1 1 Faculty of Business and Economics, TU Dresden, 01069 Dresden, Germany Abstract. In this paper, we will show that the economic lot-sizing problem with rework of defectives is NP-hard. Therefore, we reduce it to the well-known PARTITION problem. This is in line with the findings for similar models that investigate lot-sizing with remanufacturing. Keywords: lot-sizing, rework, complexity. 1 Introduction We consider a planning problem based on the dynamic lot-sizing problem that was first published by [1]. This seminal work has been extend by many researchers and led to various models to capture different practical situations [2]. The addressed model in this paper faces an imperfect production process that generates a fraction of defective items that can be reworked to serve the same demand as initially perfect quality items. Different from remanufacturing with external product returns, models that consider rework of defectives have only rarely been considered. [3] introduced a remanufacturing model including a joint and a separate setup case for production and remanufacturing. [4] showed that both models are NP-hard. [5] investigated internal returns in a multi-product model with limited capacity and rework of defectives. This paper continues with the problem definition, including the MIP formulation in section 2. Afterward, sec-tion 3 demonstrates that the model presented here is NP-hard, and section 4 provides concluding remarks. 2 Problem definition and formulation The imperfect production process that is the basis of our planning problem produces perfect quality items, called serviceables, but also a fraction of 𝛽 defectives items. These units can be reworked to serve the same demand as initially perfect quality items. Consequently, all goods satisfy the same demand 𝑑𝑡 in every period. Backordering ofdemand and disposal of defectives are not allowed. There are inventories of serviceables 𝐼𝑠,𝑡, and of defectives 𝐼𝑑,𝑡,at the end of each period. ℎ𝑠 denotes the costs associated with holding a serviceable and ℎ𝑑 with holding a defectiveitem for one period. At the beginning and at the end of the finite planning horizon 𝑇, all inventories must be empty. The setup cost 𝑅𝑝 incur when there is production 𝑝𝑡  at one period and 𝑅𝑟 when there is rework 𝑟𝑡; each indicated bya binary variable 𝑦𝑡  and 𝑧𝑡 respectively. Similar to other models, we assume that the unit production costs are neg-ligible for both processes, see [6]. Finally, the objective of the planning problem is to minimise the sum of the total cost, which comprises of all holding and all setup costs. The MIP formulation for our model reads as follows: min ∑ ℎ𝑠𝐼𝑠,𝑡 + ℎ𝑑𝐼𝑑,𝑡 + 𝑦𝑡𝑅𝑝 + 𝑧𝑡𝑅𝑟𝑇𝑡=1(1) subject to 𝐼𝑠,𝑡 = 𝐼𝑠,𝑡−1 + (1 − 𝛽)𝑝𝑡 + 𝑟𝑡 − 𝑑𝑡   for all 1, … , 𝑇 (2) 𝐼𝑑,𝑡 = 𝐼𝑑,𝑡−1 + 𝛽𝑝𝑡 − 𝑟𝑡   for all 1, … , 𝑇 (3) 𝑝𝑡 ≤ 𝑦𝑡 ∙ 𝑑𝑡,𝑇   for all 1, … , 𝑇 (4) 𝑟𝑡 ≤ 𝑧𝑡 ∙ 𝑑𝑡,𝑇   for all 1, … , 𝑇 (5) 𝐼𝑠,0 = 𝐼𝑑,0 = 𝐼𝑠,𝑇 = 𝐼𝑑,𝑇 = 0 (6) 𝑝𝑡 , 𝑟𝑡 , 𝐼𝑠,𝑡 , 𝐼𝑑,𝑡 ≥ 0   for all 1, … , 𝑇 (7) 𝑦𝑡 , 𝑧𝑡 ∈ {0; 1}   for all 1, … , 𝑇 (8)S.Rudert, U.Buscher  2 The objective function (1) minimizes the sum of the inventory and the setup costs for production and rework. The inventory balance equations for serviceables and defectives are listed in (2) and (3). The formulation for the binaries for production and rework can be found in (4) and (5). As stated above, all inventories are zero at the beginning and the end of the planning horizon (6). The variable domains are specified by (7) and (8).  Indices  𝑡 Actual period 𝑇 End of the planning horizon 𝑘 Production period 𝑙 End of the planning horizon Parameters  𝑑𝑡 Demand at period t 𝑑𝑘,𝑙 Demand from period 𝑘 to 𝑙 𝛽 Defective rate at production ℎ𝑠 Holding cost per serviceable ℎ𝑑 Holding cost per defective 𝑅𝑝 Setup cost for production 𝑅𝑟 Setup cost for rework Variables  𝑝𝑡 Production amount in period 𝑡 𝑟𝑡 Rework amount in period 𝑡 𝐼𝑠,𝑡 Inventory of serviceables at the end of period 𝑡 𝐼𝑑,𝑡 Inventory of defectives at the end of period 𝑡 𝑦𝑡 Binary variable for production setup 𝑧𝑡 Binary variable for rework setup 𝐶 Total cost Table 1. Notation 3 The  proof of NP-hardness We will show now that our problem of the single-item dynamic lot-sizing with rework of defectives is NP-hard. It can be reduced to the well-known PARTITION problem, which can be found in [7]:  Problem PARTITION: For the given positive integers 𝑎1, … , 𝑎𝑛, does  there exist a set 𝑆 ⊂ 𝑁 = {1, … , 𝑛} such that ∑ 𝑎𝑖𝑖∈𝑆 = ∑ 𝑎𝑖𝑖∈𝑁∖𝑆 .   Moreover, our proof is based on [4], and we will use the following parameters:   𝛽 = 0.5          𝑅𝑝 = 1          𝑅𝑟 = 1          ℎ𝑠 = 3          ℎ𝑑 = 0  Due to 𝛽 = 0.5, half of the demand is fulfilled by production directly and the other half by rework: 𝛽 ∑ 𝑝 = ∑ 𝑟. As the setup costs are equal (𝑅𝑝 = 𝑅𝑟 = 1) and defectives are stored at no cost (ℎ𝑑 = 0), it is not reasonable to store one serviceable for just one period (ℎ𝑠 = 3). Even the non-optimal approach of producing and reworking simulta-neously in one period would lead to a better result than storing one serviceable for one period. A few parameters are set due to technical purposes:  𝑇 = 𝑛 On the complexity of the economic lot-sizing problem with rework of defectives   3 𝑑𝑡 = 𝑎𝑡    for all 1, … , 𝑇 𝑎𝑡 > 0   for all 1, … , 𝑇 𝑎𝑡 ≤    𝛽 ∑ 𝑎𝑖𝑡−1𝑖=1   for all 2, … , 𝑇  Hereby, the condition 𝑎𝑡 ≤ 𝛽 ∑ 𝑎𝑖𝑡−1𝑖=1  is required so that there are enough defective items available for rework at any time. While the production option is always possible, rework needs a sufficient stock of defective items to satisfy the demand. Without this condition, there must probably be production to satisfy 𝑎𝑡 as there may not be enough defective items available for the rework option. For the remanufacturing model, a similar assumption exists, and all returns are already available in 𝑡 = 1 to guarantee that the remanufacturing option is always available. Based on the parameters, the optimal solution for the planning problem generates the total costs of 𝐶 = 𝑛. Consequently, there is either production or rework to satisfy demand but not both. Likewise, there are no serviceables stored at all. Let us assume that there is production and rework for one period while there is only one or the other for the rest. As all demands are positive for every period (𝑎𝑡 > 0), the total cost would exceed 𝑛 = 𝑇. Let us further assume that one serviceable is stored for one period. If this item is transferred to the next period, the total costs are reduced by at least 2. Hence, there should not be storage of serviceables at all. Finally, we end up in a situation in which there is either production or rework at one period, but not both. Also, there is no storage of serviceables at any time. Our planning problem presented in this paper can be reduced to the PARTITION problem. The answer to PARTITION is positive, which means there is a solution to the PARTITION problem if and only if there are 𝑛 positive integers 𝑎𝑛 with 𝐴 = 𝑎1, 𝑎2, … , 𝑎𝑛 that form two subsets 𝑆 ⊂ 𝑁 and 𝑁 ∖ 𝑆 for which ∑ 𝑎𝑖𝑖∈𝑆 = ∑ 𝑎𝑖𝑖∈𝑁∖𝑆 . One of the subsets contains all periods where production is used to meet demand, while the other subset contains all periods where only rework is used and there is no storage of serviceables at all. Contrary, if the subsets do not match, there must be at least one period in which production and rework are necessary, and the total cost would exceed 𝑛 = 𝑇. Conversely, this instance will have no solution to the PARTITION problem. 4 Concluding remarks We investigated on the complexity of our model for an imperfect production process where there is a rework of defectives. Various models for remanufacturing have already proven to be NP-hard. To the best of our knowledge, this paper is the first to address this for a model with a rework of defectives. We could show that our problem can be reduced to the well-known PARTITION problem, which is NP-hard. References 1. Wagner, H.M., Whitin, T.M.: Dynamic version of the economic lot size model. Management Science, 5, 89-96 (1958). 2. Brahimi, N., Absi, N., Dauzèré-Pérès, S., Nordli, A.: Single-item dynamic lot-sizing problems: An updated survey. European Journal of Operational Research, 263(3), 838-863 (2017). 3. Teunter, R.H., Bayindir, Z.P., van den Heuvel, W.: Dynamic lot sizing with product returns and remanufacturing. Interna-tional Journal of Production Research, 44(20), 4377-4400 (2006). 4. Helmrich, M.J.R., Jans, R., van den Heuvel, W., Wagelmans, A.P.: Economic lot-sizing with remanufacturing: complexity and efficient formulations. IIE Transactions, 46, 67-86 (2014). 5. Goerler, A., Voß, S.: Dynamic lot-sizing with rework of defective items and minimum lot-size constraints, International Journal of Production Research, 54(8), 1-14 (2016). 6. Kilic, O.A., van den Heuvel, W.: Economic lot sizing with remanufacturing: Structural properties and polynomial-time heuristics. IISE Transactions, 51(12), 1318-1331 (2019). 7. Garey, M.R., Johnson, D.S., 1979. Computers and Intractability: A guide to the Theory of NP-completeness. W.H. Freeman and Company, New York, NY.   S.Rudert, U.Buscher  4  

On the complexity of the economic lot-sizing problem with rework of defectives

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On the complexity of the economic lot-sizing problem with rework of defectives

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