Multi-Product Production System with the Reduced Failure Rate and the Optimum Energy Consumption under Variable Demand

The advertising of any smart product is crucial in generating customer demand, along with reducing sale prices. Naturally, a decrease in price always increases the demand for any smart product. This study introduces a multi-product production process, taking into consideration the advertising- and price-dependent demands of products, where the failure rate of the production system is reduced under the optimum energy consumption. For long-run production systems, unusual energy consumption and machine failures occur frequently, which are reduced in this study. All costs related with the production system are included in the optimum energy costs. The unit production cost is dependent on the production rate of the machine and its failure rate. The aim of this study is to obtain the optimum profit with a reduced failure rate, under the optimum advertising costs and the optimum sale price. The total profit of the model becomes a complex, non-linear function, with respect to the decision variables. For this reason, the model is solved numerically by an iterative method. However, the global optimality is proved numerically, by using the Hessian matrix. The numerical results obtained show that for smart production, the maximum profit always occurs at the optimum values of the decision variables.