This paper deals with a periodic replacement policy after the expiry of a warranty for a product sold with a two-dimensional non-renewing failure replacement warranty. We model product failures using the one-dimensional approach and use a more appropriate formulation to model the effect of age and usage to the product’s degradation. Under the periodic replacement policy, for a given usage rate y, the product is repaired with minimal repair when it fails at age x, x<τy and it is replaced with a new one if its age reaches τy. We obtain the optimal value of τy which minimizes the long-run average cost to the buyer and give a numerical example to illustrate the optimal solution. Keywords: Periodic replacement policy, Two-dimensional warranty, Long-run average cost, Accelerated failure time (AFT) model.

The ever-growing awareness of environmental protection has significantly influenced the method of manufacturing products. Due to the introduction of new processes, the management of sustainable manufacturing shows different characteristics to those of traditional systems. Sustainable manufacturing systems have attracted a great deal of attention in the past 20 years as an emerging manufacturing approach. Particularly in the last 10 years, the number of papers focusing on the topic of sustainable manufacturing systems' management has increased rapidly. More and more practical factors have been considered and integrated into this area which makes it more complex, but closer to reality. This paper aims to classify the mathematical problems dealing with the management of sustainable manufacturing systems. More than 100 related papers mainly from 1994 to 2015 have been selected and reviewed and divided into three categories according to the main elements in a manufacturing system: production planning and control, inventory management and control and manufacturing network design. The development of each category is summarised and the corresponding mathematical problems are discussed to provide a general overview of the relevant research fields and identify future research directions.

"This paper deals with a mathematical model for periodic replacement policy based on the number of failures. The product is sold with a two-dimensional non-renewing failure replacement warranty. We model product failures using the one-dimensional approach which allows to modeling the effect of age and usage to the product's degradation. Under the periodic replacement policy, for a given usage rate y, the product is repaired with minimal repair when it fails at the first n-1 failures and it is replaced with a new one at n-th failure. We obtain the global optimal values of failures which minimizes the expected cost per unit time to the buyer and give a numerical example to illustrate the optimal solution. Keywords: periodic replacement, two-dimensional warranty, global solution"