یک مدل بهینه سازی قوی برای برنامه ریزی تولید تحت عدم قطعیت دو مرحله ای چند محصولی

Production planning (PP) is one of the most important issues carried out in manufacturing environments which seeks efficient planning, scheduling and coordination of all production activities that optimizes the company’s objectives. In this paper, we studied a two-stage real world capacitated production system with lead time and setup decisions in which some parameters such as production costs and customer demand are uncertain. A robust optimization model is developed to formulate the problem in which minimization of the total costs including the setup costs, production costs, labor costs, inventory costs, and workforce changing costs is considered as performance measure. The robust approach is used to reduce the effects of fluctuations of the uncertain parameters with regards to all the possible future scenarios. A mixed-integer programming (MIP) model is developed to formulate the related robust production planning problem. In fact the robust proposed model is presented to generate an initial robust schedule. The performance of this schedule could be improved against of any possible occurrences of uncertain parameters. A case from an Iran refrigerator factory is studied and the characteristics of factory and its products are discussed. The computational results display the robustness and effectiveness of the model and highlight the importance of using robust optimization approach in generating more robust production plans in the uncertain environments. The tradeoff between solution robustness and model robustness is also analyzed.

Production planning is one of the most significant issues carried out in manufacturing systems which seeks efficient planning, scheduling and coordination of all production activities that optimizes the company’s objective. The goals of production planning are determining the optimal quantity of production, inventory and other manufacturing parameters to satisfy fluctuating demands over a planning horizon [1]. Production planning has taken substantial attention in operations research literature. Some researchers have used a hierarchical approach for production planning that called hierarchical production planning (HPP). Hierarchical production planning is a renowned approach to handle the complexity of multi-level production planning and scheduling problems in real-world systems [2] and [3]. Other researchers have used multi criteria decision making (MCDM) approaches for production planning [4] and [5]. Comprehensive surveys on production planning in the literature were provided to cover different assumptions of manufacturing environment [6], [7], [8] and [9]. Mula et al. [10] provided a profound review on mathematical models for production planning under uncertainty. Stadtler [11] considered dynamic multi-item multi-level lot-sizing problem in general product structures with single and multiple constrained resources as well as setup times. He proposed a relax-and-fix heuristic, called internally rolling schedules with time windows to solve the problem. Akartunali and Miller [12] studied multi-item multi-level production planning problems with backlogging. They developed a time window MIP based heuristic that can generate good quality feasible solutions in reasonable computational time for various kinds of lot-sizing problems. Wu et al. [13] proposed two new MIP models to formulate capacitated multi-level lot-sizing problems with backlogging. They developed an effective optimization framework that achieves good quality solutions in reasonable computational time. Wu et al. [14] studied the capacitated multi-level lot-sizing problem with setup times. They presented a developed relax-and-fix based heuristic that uses domains derived from several strategies of relax-and-fix and a LP relaxation technique. Ramezanian et al. [15] studied a multi-product multi-period capacitated multi-stage production planning with sequence-dependent setups. They proposed a more efficient mathematical model for the problem and used two MIP-based heuristics for solving related problem. Many real-world planning problems involve noisy, incomplete or inaccurate data. In the literature, different approaches have been used to deal with different forms of uncertainty. Mathematical programming and stochastic approaches were used to formulate uncertainty in manufacturing systems [16], [17] and [18]. Another approach to incorporate uncertainty in production planning models is fuzzy approach [19], [20] and [21]. Recently, robust optimization model, strong technique to contrast uncertainty, was used to deal with uncertainty in the systems. Robust optimization can be very efficient and useful because of generation of the good and stable solutions for any possible occurrences of uncertain parameters [22]. The concept of robust optimization in operation research is introduced by Mulvey et al. [22]. They developed a robust counterpart approach with a nonlinear regularization function that penalizes the constraint violations and uncertainties are addressed via a set of discrete scenarios. Robust optimization has yielded a series of solutions that are progressively less sensitive to realizations of the data in a set of scenarios. The optimal solution of robust optimization model will be robust with respect to optimality if it remains ‘close’ to the optimal if input data change: this is called solution robustness. The solution is also robust with respect to feasibility if it remains ‘almost’ feasible for small changes in the input data: this is termed model robustness. Bai et al. [23] showed that the traditional stochastic linear program fails to determine a robust solution despite the presence of a cheap robust point. They examined properties of risk-averse utility functions in robust optimization. They argued that a concave utility function should be incorporated in a model whenever the decision maker is risk averse. Ben-Tal and Nemirovski [24] proposed a robust optimization approach to formulate continuous uncertain parameters. Ben-Tal and Nemirovski [24], Ben-Tal and Nemirovski [25] and Ben-Tal et al. [26] developed robust theory of linear, quadratic and conic quadratic problems. Their approaches are used vastly in many engineering and design problems. Bertsimas and Sim [27] and Bertsimas and Thiele [28] proposed robust optimization methods for discrete optimization in continuous spaces. Leung and Wu [29] considered aggregate production planning (APP) problem in which some parameters are uncertain. They proposed a robust optimization model to minimize total production cost in manufacturing systems. They analyzed their proposed model solutions with single-period and multi-period data and considered the tradeoff between solution and model robustness. Leung et al. [30] studied robust model for multi-site production planning problem. They applied their priory analysis for evaluating the robustness of considered problem. Also, Leung et al. [31] presented a developed robust optimization model for achieving production plan for production planning of perishable products in an uncertain environment. Zanjani et al. [32] focused on a multi-period, multi-product sawmill production planning problem with considering the uncertainty in quality of raw materials. They proposed two robust optimization models with different variability measures, namely the upper partial moment of order 2 (UPM-2) and the upper partial variance (UPV). They analyzed the tradeoff between the plan’s robustness and raw material consumption and expected backorder/inventory cost. Mirzapour Al-E-Hashem et al. [33] and [34] studied multi-site aggregate production planning problems under uncertainty. They presented a multi-objective robust optimization model for the studied problem. The aim of this paper is to address a multi-period multi-product multi-machine two-stage production system under uncertainty. A robust optimization model is proposed to determine the robust production plans that are less sensitive to realizations of the uncertain data. The uncertainties of customer demands and production, inventory and subcontracting costs are addressed via a set of discrete scenarios. A mixed-integer programming model is presented to minimize the total costs (consisting of the production costs, subcontracting costs, inventory costs, backorder costs, setup costs, labor costs, hiring costs and layoff costs). A set of data from an Iranian refrigerator factory is used to test the effectiveness and efficiency of the proposed model. Table 1 shows the main features of the mentioned robust production planning models and our developed model. Table 1. Main features of robust optimization models for production planning problem. Features Leung and Wu [29] Leung et al. [30] Leung et al. [31] Kazemi Zanjani et al. [32] The proposed model Main topic Robust aggregate production planning Robust multi-site production planning Robust production planning for perishable products Robust sawmill production planning Robust production planning for two-stage production system No. of stages Single-stage Single-stage Single-stage Single-stage Two-stage No. of products Multiple Multiple Multiple Multiple Multiple No. of periods Multiple Multiple Multiple Multiple Multiple No. of machines Single Single Single Multiple Multiple Subcontracting Considered Considered Not considered Not considered Considered Workforce Considered Considered (k-types) Considered (single type) Not considered Considered (2-types) Labor hiring Allowed Allowed Allowed Not allowed Allowed Labor lay off Allowed Allowed Allowed Not allowed Allowed Backorders Not included Not included Not included Allowed Allowed Variation term Absolute form Absolute form Absolute form Quadratic form Absolute form Setup Not considered Not considered Considered Not considered Considered Lead time Not considered Not considered Not considered Not considered Considered Uncertain parameters Demand, lay off cost, hiring cost, subcontracting cost Demand, production cost, labor cost, inventory cost Demand, regular time and overtime production cost, inventory cost Units of products produced by processes (yield of processes) Demand, regular time and overtime production cost, subcontracting cost, labor cost Table options The remainder of this paper is structured as follows: In Section 2, the proposed robust optimization model for two-stage production planning system is demonstrated. Moreover, the deterministic MIP model for two-stage production system and the background to robust optimization model are described in this section. Computational results, multiple aspects of robustness and general implications of the framework are highlighted in Section 3 using a case-oriented example. Finally, some concluding remarks are given in Section 4.

The purpose of this paper is to develop a robust optimization model for two-stage production systems in which the objective function is to minimize the total costs of production over the planning horizon. This paper is concentrated on multi-period multi-product multi-machine two-stage production systems with setup decisions, lead times and uncertain parameters. In order to deal with the uncertainty of parameters and generate the robust production plan that is less sensitive to the change in the uncertain data assuming a scenario set with associated probabilities, robust optimization is used. A mixed-integer programming model that can be used to compute optimal robust production plan is proposed. The computational experiments obtained from a set of real-world data for an Iranian refrigerator factory show that the proposed robust model is more practical for handling uncertain parameters in the production environments. The tradeoff between optimality and infeasibility is used for obtaining robust solution based on the opinion of decision-makers. The results show the robustness and effectiveness of the model in real-word practical production planning problem. Also, the results obtained by the robust MIP model indicate the advantages of robust optimization in generating more robust production plans over the considering expected value of uncertain parameters in deterministic programming model. It may be interesting to extend the MIP model to formulate a multi-stage production system in order to obtain more realistic production plans. Also, real data from other companies can be used to validate the model.