stochastic dynamic programming

Productivity of production systems depends on proper planning in various fields such as production, maintenance and repairs, inventory control, etc. Considering the major and direct impact of the production plan on other fields, it is necessary to prepare this plan with a suitable approach so that the production systems can be properly managed and costs can be reduced as much as possible. Limitations such as the need to supply a certain amount of demand in a certain period of time, the possibility of producing defective products, and the imposition of costs due to frequent system setup, make managers a challenge in providing an accurate plan. In this research, an attempt is made to plan the production process in a multi-period single product system by taking advantage of the Markov decision process and taking into account the above-mentioned constraints. In the following problem, using the stochastic dynamic programming technique, the best possible action will be selected and in other words, the best production volume will be determined for each period. The goal is to determine the volume of production in each period and for different states, in such a way that at the end of the permitted periods of production, the entire demand can be covered with the lowest cost. The effectiveness of the model is examined by solving a numerical example and analysis of the effects of changing parameters on the results of the problem has been presented. The results show that there is a direct relationship between the average cost rate and the costs related to setting up the system and producing each product unit, and the relationship between the said rate and the probability of producing healthy products and the production capacity is inverse. Increasing the production capacity after a certain threshold will not affect the average cost rate.

In many industries, some of production capacity will be out of reach when a machine deteriorates. Preventive Maintenance (PM) will enhance the machine condition, but it occupies some of production capacity. The PM will reduce the production capacity and will cause the customer's order delay if it is unnecessary, in contrast the probability of unexpected failure will be increased and will have the same or worst consequences if the PM is too late. One of the ways to deal with this problem is Condition-Based Maintenance with Discrete Monitoring (CBMWDM) whose main challenge is to find an optimal inspection scheme. If the time interval between inspections is too short, the inspection cost will be increased although it will diminish unnecessary PMs and unexpected failures. On the other hand, if the interval between inspections is too long, the sum of unexpected failure and backorder costs will be increased although the cost of inspection will be reduced. Hence, simultaneous planning of the inspections and PM actions should be emphasized. In this paper, a single product single machine system with Markovian deteriorating conditions and demand uncertainty is considered. The main objective is to integrate the inspections and maintenance planning in a tactical level and finite horizon that minimizes the average cost of inspections, maintenances, and backorders. For this purpose, a stochastic dynamic programming model is presented whose structure is dependent to appointed inspection scheme. The state variable of the model is an ordered pair whose components represent the demand and machine condition. The demand is a discrete random variable with arbitrary distribution and the machine condition will be determined after each inspection. Corresponding to each inspection scheme and for each outcome of the state variable, optimal PM decision is obtained and consequently the optimal inspection scheme is determined. The optimal inspection scheme and corresponding PM decisions determine simultaneous inspections and preventive maintenance planning. The strategy of the new model is analyzed for a six-month horizon. Numerical results of the model show that the total average cost is non-decreasing in machine state and demand. Secondly, at the time of inspection, the preventive maintenance should be executed at the same time or earlier if machine state is worse or demand is more.