A matheuristic for aggregate production–distribution planning with mould sharing
Introduction
The increasing pressure to reduce total logistics costs is forcing supply chain managers to rethink production–distribution policies and make best use of their assets across multi-facility networks. This means that they have to embrace new logistics concepts and are confronted with more complex planning problems. This paper develops mathematical models for such an integrated production–distribution aggregate planning problem, based on the case of a large manufacturer of plastic products. These models are then used in a mathematical programming-based heuristic solution approach (or ‘matheuristic’ Maniezzo et al., 2010).
Given its practical importance and academic relevance, researchers have been investigating aggregate production planning in a multi-site environment, inspired by real-life cases from various industries. There is an extensive literature on multi-facility, multi-product, multi-period aggregate production–distribution planning, describing many different problem aspects and complications, and using various solution methodologies ranging from linear programming solvers to metaheuristics.
As in this paper, mixed integer programming formulations are often used in the existing literature. Dhaenens-Flipo and Finke (2001) propose a network flow model with relatively few binary variables for a manufacturer of metal items. A model with varying time scales is presented by Lin and Chen (2007) for a TFT-LCD manufacturer. Kanyalkar and Adil (2007) consider a consumer goods company and solve mixed integer linear goal programming models with different time grids heuristically. Gnoni et al. (2003) augment a mixed-integer linear programming approach with simulation to deal with demand uncertainty for a braking equipment manufacturer in the automotive industry. Along the same lines, Safaei et al. (2010) propose a hybrid mathematical-simulation model in which the simulation is used to reflect dynamics of real-world systems. Leung et al. (2007) have adopted robust optimization to deal with uncertainty for an application in the apparel industry. Similarly, Mirzapour Al-e-hashem et al. (2011) propose a robust multi-objective model in a case study from the paper industry.
Since large-scale problems cannot be tackled with mathematical programming solvers, metaheuristics are increasingly used in the literature to solve such large-scale problem instances. For more details, we refer to recent examples such as the artificial bee colony metaheuristic of Pal et al. (2011) and the genetic algorithm of Fahimnia et al. (2012).
In the existing literature, the possibilities of what products can be produced at which plants is usually given, and the decision is to allocate production volumes to the different plants. Whenever production has to be done in any plant, this often involves a setup or changeover cost, such that binary decision variables are needed next to the (continuous) production volume variables. The complication in the problem we are studying is in the fact that these binary variables are not independent for the different plants. This is due to the fact that the products are produced using injection moulding, such that a certain product can only be produced when the mould is available at the plant. Therefore, the binary variables at the different plants are linked through constraints that ‘track’ where the mould is. This complicates both the modeling and solution approaches a lot. To the best of our knowledge, the only article that contains this complication is that of Aghezzaf (2007). That paper also considers aggregate planning for injection moulding production in multiple facilities. A mixed integer linear programming model is presented that allocates moulds to plants across the planning horizon. Lower and upper bounds for the model are generated using Lagrangian relaxation and linear programming duality. Our paper contributes to the literature by presenting a more generic model that allows more flexibility in exchanging the moulds. Aghezzaf (2007) only allows moulds to be transferred from one plant to another at the end of a period, such that a mould is only available in a single plant during a period. Because time buckets in aggregate planning are reasonably large (typically one month), this is too restrictive and therefore we allow multiple mould moves within a period. Further, we explicitly take the loss of productive time for moulds being transferred into account. Finally, we offer a novel solution approach and show that it is capable of solving large real-life instances.
The remainder of the paper is structured as follows. After a more detailed problem description, the mathematical models are presented in Section 2. These models are used in the matheuristic described in Section 3. Section 4 illustrates the proposed solution methodology with a small-scale example, whereas Section 5 contains the results of the large real-life instance. Section 6 concludes this paper and gives avenues for further research.
The supply chain under consideration in this paper consists of three stages, namely production plants, distribution warehouses and customers. The design of the network, i.e. the number and location of plants and warehouses, is given, such that the problem at hand is an allocation problem at the tactical level, i.e. deciding which products to produce where and through which warehouse products should be shipped to the customers. Products are being produced using injection moulding. Each stock-keeping unit (SKU) is made from its unique mould. The moulds can be exchanged between the different plants such that each SKU can be manufactured in any plant as soon as the corresponding mould is present. However, since the moulds are very expensive compared to the value of the products being produced on them, only a single mould is available per SKU. Therefore, only a single plant can produce a certain SKU at any given time, depending on whether or not the mould is present in the plant.
The European branch of the global manufacturer in case, having plants, warehouses and customers spread across Europe, has always produced each SKU at a single location so far. This means that moulds always stay in the same plant and are not exchanged between multiple plants. Because of changes in the product mix being demanded across their European market, and because of the increasing pressure to reduce costs, the company is reconsidering this strategy and wants to quantify the savings potential of mould sharing. If a mould always stays in the same plant, large volumes of the (low-value) product have to be transported from that plant to customers across the whole of Europe. By sharing the mould across plants, production can occur closer to the final market and transportation costs decrease significantly. On the other hand, mould sharing increases costs because (i) the variable manufacturing cost can be higher than in the mould's ‘home’ plant, and (ii) handling costs may be higher in the warehouses. Also, the transportation of the moulds themselves and the overhead for coordinating the mould exchanges incur extra costs.
The modeling and solution approach presented in this paper is capable of making this cost trade-off, while also taking into account capacity restrictions in both plants and warehouses. These capacity restrictions may force certain volumes of products to be allocated to a production–distribution combination that is more expensive (if capacity at the cheapest option is depleted). Further, the capacity restrictions could also necessitate producing certain volumes of (seasonal) products beforehand and keeping them in inventory to cover peak demand periods. The resulting inventory holding costs are also taken into account in the overall cost trade-off. The proposed model and solution approaches will help providing the answer to the company's question about the savings potential of increasing flexibility by allowing the possibility of sharing moulds across different plants.
Section snippets
Mathematical model
In this section, we present mathematical models for the multi-product multi-period aggregate production–distribution planning problem. As explained above, these models will be able to make the trade-off between the distribution cost savings that mould sharing enables with the additional costs they incur, all within the limited production and warehousing capacities of a given production–distribution network.
We propose two mathematical models for exploiting the possibility of sharing moulds
Solution approaches
To solve the aggregate production planning problem discussed in this paper, the mathematical models presented above can be implemented in an existing MILP solver and fed with the necessary data from the instance at hand. However, since the models are NP-hard, it is impossible to solve large-scale instances to optimality in this manner. Further, the LP-relaxations are not tight at all and produce a very poor lower bound. As a result, the calculation times to reach proven optimality quickly
Illustrative example
The mathematical models and the matheuristic presented above have been implemented in C++ using Gurobi 4.6. This section introduces an illustrative example that considers five SKU's, three plants, two warehouses, and four customers. The production times and costs for each of the SKU's at each of the plants are given in Table 1, the transportation costs are shown in Table 2 and the demand for the next three periods is given in Table 3. Each period consists of 4000 time units. Moving a mould from
Computational results
For further validation and evaluation of the proposed model and matheuristic solution approach, a realistic dataset was derived from real-life data obtained from the European branch of a global plastics manufacturing company. This dataset contains four plants, seven warehouses and the demand for 293 SKU's across 37 aggregate customer regions for 6 months.
Conclusion
This paper considers the multi-period, multi-product aggregate production–distribution planning problem for a producer of injection-moulding plastic products that has multiple plants and distribution warehouses across Europe. The goal is to quantify the savings potential that is achieved by exchanging moulds between multiple plants and to generate aggregate plans that can realize this savings potential. To this end, a mixed integer linear program is presented that can be solved to optimality
References (12)
Production planning and warehouse management in supply networks with inter-facility mold transfers
European Journal of Operational Research
(2007)- et al.
Production planning of a multi-site manufacturing system by hybrid modellinga case study from the automotive industry
International Journal of Production Economics
(2003) - et al.
A robust optimization model for multi-site production planning problem in an uncertain environment
European Journal of Operational Research
(2007) - et al.
A multi-objective robust optimization model for multi-product multi-site aggregate production planning in a supply chain under uncertainty
International Journal of Production Economics
(2011) - et al.
Computational complexity of the capacitated lot size problem
Management Science
(1982) - et al.
An integrated model for an industrial production–distribution problem
IIE Transactions
(2001)
Cited by (47)
The capacitated multi-level lot-sizing problem with distributed agents
2021, International Journal of Production EconomicsCitation Excerpt :Increasing customer demands for individualized products push manufacturing companies towards agile and modularized production processes. The onset of digitalization and Industry 4.0 enables them to efficiently and safely connect with other companies in order to build up collaboration networks and to overcome inefficiencies by sharing resources among each other (Raa et al., 2013). This can be done by coordination or collaborative planning, which are key elements of supply chain management (Stadtler, 2008).
Heuristic solution methods for the selective disassembly sequencing problem under sequence-dependent costs
2021, Computers and Operations ResearchA production and distribution planning of perishable products with a fixed lifetime under vertical competition in the seller-buyer systems: A real-world application
2021, Journal of Manufacturing SystemsCitation Excerpt :Varthanan et al. [82] proposed a novel analytic hierarchy process based heuristic discrete PSO to solve a difficult PDP. Raa et al. [27] added an integrated PDP for the plastic products considering the possibility of mold sharing between the factories. Nasiri et al. [83] developed an integrated PDP for a three-level SC with probabilistic demand.
Tactical sales and operations planning: A holistic framework and a literature review of decision-making models
2020, International Journal of Production EconomicsEnhancing supply chain production-marketing planning with geometric multivariate demand function (a case study of textile industry)
2020, Computers and Industrial EngineeringCitation Excerpt :Sarkar (2013) also addressed the production-inventory problem for a deteriorating item in a two-echelon supply chain with transportation consideration. There are also lots of papers to deal with the production-distribution problem (see Torabi & Moghaddam, 2012; Fahimnia et al., 2013; Raa, Dullaert, & Aghezzaf, 2013; Su, Huang, Fan, & Mak, 2015; Zheng, Zhang, Hana, & Lu, 2016 and references therein). Quality improvement is taken into consideration by Kim and Sarkar (2017) in a joint replenishment problem to clean a complex multi-stage production system from defective items.
Project schedule performance under general mode implementation disruptions
2020, European Journal of Operational ResearchCitation Excerpt :A matheuristic solution approach combines the benefits of metaheuristics and exact approaches (Maniezzo et al., 2010). Recently, matheuristic approaches have been used to solve numerous combinatorial optimisation problems, such as the nurse rostering problem (Della Croce & Salassa, 2014), routing problems (Archetti & Speranza, 2014; Doerner & Schmid, 2010; Hemmati, Hvattum, Christiansen, & Laporte, 2016), the multidimensional knapsack problem (Wilbaut, Salhi, & Hanafi, 2009), production planning (Raa, Dullaert, & Aghezzaf, 2013), liner shipping network design (Brouer, Desaulniers, Karsten, & Pisinger, 2015; Brouer, Desaulniers, & Pisinger, 2014; Karsten, Brouer, Desaulniers, & Pisinger, 2016) timetabling (Fonseca, Santos, & Carrano, 2016) and personnel scheduling (Smet, Wauters, Mihaylov, & Vanden Berghe, 2014). Toffolo, Santos, Carvalho, and Soares (2016) recently proposed a matheuristic solution procedure to solve the multi-mode multi-project scheduling problem.