A study of two warehouse inventory systems with a production unit is developed in this article with some constraints which are of practical applicability to optimize the total production cycle and its cost. A production unit evolves in three different states to retain its quality and prolong its lifetime: the state of producing items, the state of reworking the identified defective items, and the state of being idle. It processes the items up to a certain time point. The screening process starts immediately after a product comes out of the production unit. The classified non-defective items are first stored in own warehouse ($OW$), after filling to its maximum capacity, and the remaining items fill in the first block $\left({RW}_1\right)$ of the rental warehouse $\left(RW\right)$. All identified defective items are stored in the second block $\left({RW}_2\right)$ of $RW$. The holding cost of an item is higher in $RW$ than $OW$. All defective items are sent to the production unit for re-do processes as a single lot immediately after the stop of the production and re-do items are stored in ${RW}_1$ to satisfy the demand. The items in the ${RW}_1$ are of higher priority in satisfying the demands after the stop of the production unit in producing new items as to deduce the total cost. Demand is assumed as both time and advertisement dependent and is encouraged once production starts. The deterioration rate differs in both warehouses. No backlog is entertained. The study is directed to achieve optimum total cycle cost towards the attainment of the optimum production time slot and the entire cycle of the system. We have arrived at explicit expressions for the total cost function of the entire production cycle. An analytic optimization process of the discriminant method is employed in the form of an algorithm to arrive at the optimum total cost. It provides a numerical illustration of a specific environment. The implications of the current research work are as follows. The optimum utility of production units in three different states in arriving at the optimum total cost is extensively studied with respect to deterioration, demand, and production rates. It also examined the influence of fluctuating deterioration, demand, and production parameters in arriving at optimum deterioration cost, holding cost, and total cycle cost, as they have important managerial insights. The effect of rental charges on the optimum total cost is examined as the system is used for multi-purpose storage.

In this paper, we present a two-stage stochastic model of carcinogenesis in a two level random environment. The random environment switches between two levels, say, 1 and 2 alternately. When the environment is in level 1, a normal cell either divides into two normal cells or dies; and an intermediate cell divides into two intermediate cells or dies. When the environment is in level 2, a normal cell either divides into two intermediate cells or divides into one normal cell and one intermediate cell or divides into two normal cells or dies; and an intermediate cell either divides into two malignant cells or divides into one intermediate cell and one malignant cell or divides into two intermediate cells or dies. It is assumed that, once a malignant cell is produced, it generates a malignant tumor with probability 1. We obtain the mean numbers of normal, intermediate and malignant cells.

A two-mutation model of carcinogenesis which evolves under the influence of three level random environment on the production process is formulated and analyzed. A random environment occupies one of the levels 1, 2 and 3 at any time t according to a Markov process. When the environment is in level 1, a normal cell either divides into two normal cells or dies; and an intermediate cell divides into two intermediate cells or dies. When the environment is in level 2, a normal cell either divides into one normal cell and one intermediate cell or dies and an intermediate cell either divides into one intermediate cell and one malignant cell or dies. When the environment is in level 3, a normal cell either divides into two intermediate cells or dies and an intermediate cell either divides into two malignant cells or dies. It is assumed that, once a malignant cell is produced, it generates a malignant tumor with probability 1. We obtain the mean numbers of normal, intermediate and malignant cells at any time t.