A Two-dimensional Fix and Optimize Algorithm to Solve the Flexible Manufacturing System Lot-sizing with Co-production Problem

In this paper, the flexible manufacturing system lot-sizing with the co-production problem is modeled. It is also simplified using the relationships between variables. One of the common methods for solving such problems is the fix and optimize algorithm, which is generally used in a one-dimensional approach. Also, a two-dimensional fix and optimize algorithm is applied to solve the problem. This algorithm is compared with two common algorithms using simulated data series.

In this paper, Flexible Manufacturing System Lot-sizing with Co-production problem is modeled using mixed-integer programming. The production of products in this flexible system varies with the change of production mode, and a different mixture of products is produced for each production mode. Also, the planning interval includes T periods, and the demand for each product in each given period is constant. In each period, a fixed setup cost is added to the production and maintenance variable costs, if production occurs. The objective function of the model minimizes the sum of fixed setup costs and production and maintenance variable costs of inventory in each period and each production mode. Problem constraints include setup forcing constraints, inventory balance constraints, initial inventory constraints, co-production constraints, production mode constraints, non-negative variables constraints, and binary variable constraints. Among the methods proposed to solve this group of problems, the fix and optimize method is one of the most effective and general methods. The basic idea of ​​this approach is that due to the difficulty of solving the main problem with a longtime interval, a problem with a shorter time interval called the time window is solved instead. Except for the variables in the time window, other integer variables are considered continuous variables, so the resulting problem is easier to solve. In the following steps, the time window variables in the current step are assumed constant, and this repetition will continue until the end of the desired periods. Time windows can be considered with or without overlap. In this paper, two innovative one-dimensional fix and optimize algorithms based on time and production mode variables and a new two-dimensional algorithm based on time and production mode variables are applied to solve the model using simulated data at three levels of small, medium, and large scales. MATLAB 2016 software is used to code the algorithms of this study, and numerical calculations are performed by a personal computer with Intel®Core™i3-7100@3.90GHz processor and 8 GB RAM. 

The research results indicated the significant superiority of the proposed two-dimensional algorithm in terms of response time over the two one-dimensional algorithms. It is important to note that in terms of the quality of the answer in the studied problems, no significant difference was observed. 

In many real cases, due to the fact that the cost parameters in different production situations (e.g., the oil, gas, and petrochemical downstream industries) are close to each other and in practice, determining production conditions in accordance with other parameters such as demand is independent of production costs, the efficiency of the proposed algorithm will be more visible in this article. The most important limitation in this study was the lack of real data for a flexible production system with correlated products, which is why simulated data were used to validate the model and test the proposed algorithms.

In the future, researchers can use real-time case studies based on the proposed model and algorithms in this paper. They can also add other features to the model, such as limited production capacity and allowable shortages. Manufacturing plants that have features similar to this study can benefit from the findings to optimize production costs.

Applying the results of this research can increase the productivity of production units and the use of non-renewable energy resources.

In this paper, a mathematical model (MILP) was proposed for the Flexible Manufacturing System Lot-sizing with Co-production problem. In addition, an innovative two-dimensional fix and optimize algorithm was developed.