Stochastic demand is an important factor that heavily affects production planning. It influences activities such as purchasing, manufacturing, and selling, and quick adaption is required. In production planning, for reasons such as reducing costs and obtaining supplier discounts, many decisions must be made in the initial stage when demand has not been realized. The effects of non-optimal decisions will propagate to later stages, which can lead to losses due to overstocks or out-of-stocks. To find the optimal solutions for the initial and later stage regarding demand realization, this study proposes a stochastic two-stage linear programming model for a multi-supplier, multi-material, and multi-product purchasing and production planning process. The objective function is the expected total cost after two stages, and the results include detailed plans for purchasing and production in each demand scenario. Small-scale problems are solved through a deterministic equivalent transformation technique. To solve the problems in the large scale, an algorithm combining metaheuristic and sample average approximation is suggested. This algorithm can be implemented in parallel to utilize the power of the solver. The algorithm based on the observation that if the remaining quantity of materials and number of units of products at the end of the initial stage are given, then the problems of the first and second stages can be decomposed.
To satisfy customers and improve supply chain performance are among the most important objectives of firms. Supply chain performance is influenced by activities such as purchasing, manufacturing, transport, and sales. We focus on problems encountered in purchasing and production planning, such as utilization of discount policies and making decisions before demand has been realized. It has been pointed out that discounts may have an impact on purchase prices [
Stochastic programming is an optimization framework that considers uncertain parameters to define optimal solutions. It has been applied in fields such as financial and planning control and manufacturing and capacity planning. With this model, uncertainty can be overcome, and firms can reduce supply chain expense to gain a competitive advantage.
The main contribution of this study is to construct a two-stage stochastic linear programming model for multiple materials and products, while considering discount policies from different suppliers and their effects on production planning. The model strives to answer the question of the quantity of material to be ordered from a specific supplier. Production planning at all stages provides basic information such as required labor, operational hours, and production lines. At a small scale, the problem can be solved directly through a deterministic equivalent. In a larger scenario, random search combined with sample average approximation is applied.
The remainder of this paper is organized as follows. The next section reviews the related literature. Section 3 presents the proposed two-stage stochastic mixed-integer programming mathematical models, and Section 4 describes the framework for applying random search and sample average approximation to solve the large-scale problem. Conclusions are drawn in Section 5.
Material procurement is the initial activity in production planning. In this phase, orders are allocated based on supplier price quotations to minimize total expenditures. Suppliers use quantity discounts to increase order sizes. Various methods for order allocation based on quantity discounts have been proposed, including mixed integer programming and stochastic programming optimization models. Stochastic programming can effectively account for uncertainty in modeling. Moheb-Alizadeh et al. [
Production planning is another process whose role is important in order to minimize the total cost and control the leverage of the manufacturing phase. Many scholars have used stochastic programming to construct aggregate planning for production from many aspects, especially when the demand is unstable and cannot be predicted accurately. Mahdavi et al. [
Several approaches have been developed to solve stochastic programming problems. Some of the most common are random search methods, stochastic approximation, stochastic quasi-gradient methods, and sample average approximation. Random search (stochastic) algorithms use randomness or probabilities to modify the solution process, and can be useful for ill-structured global optimization problems, whose objective function may be nonconvex or in a mixed continuous-discrete domain. Such algorithms include simulated annealing, tabu search, genetic algorithms, evolutionary programming, particle swarm optimization, and ant colony optimization [
Inspired by stochastic programming in order allocation and aggregate planning in a real production environment, this paper aims to optimize purchasing at two stages and form two aggregate plans for the production stages after receiving material from the procurement phase. The proposed model simultaneously considers the effect of decisions on the expected total cost at both stages and solves them together. The result is globally optimal for all scenarios, since the first stage considers impacts on all scenarios in the second stage. The results include essential information such as the number of hires, total production lines needed, and total overtime hours in each scenario of customer demand under some resource limitations and constraints.
Our two-stage stochastic linear programming model is constructed.
Our objective is to minimize the expected total cost of stages 1 and 2. The main cost components of stage 1 are for purchasing and manufacturing. Costs in stage 2 are more complex and include purchasing and manufacturing costs, balanced by income from selling products, salvage products, and salvage materials.
The expected total cost for both stages can be presented as
where
and
Subject to:
where
With
Several instances of the problem were created to validate the model, from which it can be seen that materials with a high difference in cost between the first and second stages take more priority in purchasing activity in the first stage. As a result, they are often purchased to cover usage over the whole planning horizon. These decisions are in contrast to remaining materials with a low difference in cost between the two stages. For these materials, the decision-maker will apply a delay strategy, purchasing only a fraction of the materials in the first stage, and deciding on subsequent purchases after demand has been realized.
If the cost is high to manufacture one product or to open one line, a manufacturer often accepts a lost sale. However, products with high selling prices are often manufactured in large quantities, which can create overage products in some scenarios.
The above example is a small-scale problem solved through a deterministic equivalent transformation to validate the model. If the number of scenarios is large, then a random search combined with sample average approximation is suggested, which can be solved by parallel computing.
Let
Stochastic linear programming of optimization problems is popular due to its effectiveness in describing real phenomena. In this study, a two-stage stochastic linear programming approach is proposed for the procurement and manufacturing process, considering material all-unit discounts to order optimal quantities for production. In the first stage, managers must make decisions without demand realization, and available information consists only of suppliers’ quotations and the manufacturer’s capacities and facilities. The decisions in the first stage propagate their effects to the second stage. However, in the second stage, based on first-stage outcomes and stochastic market demand realization, correct actions can be taken to determine the optimal order allocation and production planning. To solve the problem in the large scale, an algorithm combining metaheuristic and sample average approximation is suggested. This can be implemented in parallel to utilize the power of the solver.
The authors thank Vietnam National University Ho Chi Minh City (VNU-HCM), who kindly sponsored this research.