Pravin Kumar Johri

Optimal partition for shop floor control in semiconductor wafer fabrication

Abstract Wafer fabrication is a very complex manufacturing process. It is imperative that the mean and the variance of the cycle time be kept small. For the purpose of scheduling, the set of wafer fabrication process steps need to be partitioned into zones of roughly equal sizes, where size refers to the sum of the processing times of the steps in a zone. By formulating the problem as a dynamic program, we find the optimal partition so that the sum of the squares of the deviations from a target size is minimized.

Implied constraints and an alternate unified development of nonlinear programming theory

Abstract This paper offers an alternate unified view of nonlinear programming theory from the perspective of implied constraints. Optimality is identically characterized for both constrained and unconstrained problems in terms of implied constraints. It is shown that there is a weaker condition than the Guignard constraint qualification for the existence of finite multipliers in the Karush-Kuhn-Tucker conditions. Surprisingly, this condition does not directly qualify the constraints but instead qualifies the objective in terms of implied constraints. More surprisingly, the existence of the finite multipliers follows directly from this objective qualification — it is not necessary for the point to be a local optimum. Methods for generating implied constraints are used to obtain a more general sufficient condition for local and global optimality. A single unified formulation of duality shows that duality is nothing more than an effort to generate the tightest implied constraint. Duality theorems hold in general for this formulation — convexity is not required — and the existence of the duality gap in prior formulations is easily explained. The algorithmic potential of this approach is highlighted by showing that the Simplex method systematically tries to imply the objective from the constraints of the problem.

Optimal Repair of a 2-Component Series-System with Partially Repairable Components

A 2-component series system is maintained by one repairman. The up-times of the components are s-independent r.v.s with exponential distributions. The time required to repair a failed component is the sum of a number of s-independent, exponential r.v.s. Components can be partially repaired, and a working component can fail even while the system as a whole is not functioning. The analysis finds repairman allocation policies which maximize the system availability. Under the assumption that it is permissible to reassign the repairman instantaneously among failed components, the explicit form of optimal policies is obtained. And, the optimal policies are characterized when the time for such a reassignment is allowed to be an exponential r.v.

Implied constraints and LP duals of general nonlinear programming problems

A very surprising result is derived in this paper, that there exists a family of LP duals for general NLP problems. A general dual problem is first derived from implied constraints via a simple bounding technique. It is shown that the Lagrangian dual is a special case of this general dual and that other special cases turn out to be LP problems. The LP duals provide a very powerful computational device but are derived using fairly strict conditions. Hence, they can often be infeasible even if the primal NLP problem is feasible and bounded. Many directions for relaxing these conditions are outlined for future research. A concept of local duality is also introduced for the first time akin to the concept of local optimality.

FURTHER INSIGHT INTO THE STRUCTURE OF BOLD AND TIMID POLICIES

A gambler repeatedly plays a game until either he becomes broke or his fortune becomes equal to or exceeds a target amount. The gambler is allowed to make multiple bets, i.e. stake integral amounts on different alternatives of the game, and more than one bet may win simultaneously. The objective is to determine a strategy that maximizes the probability of attaining the target amount. When all bets have the same gain and alternatives of betting exist such that the relevant plays are feasible, the following results are obtained. For unfavorable games, a bold type of policy is shown to be optimal. A timed type of policy is shown to be best within a restricted class of policies for favorable games. In general, optimal policies contain multiple bets. Based on a numerical example, this is established for roulette also. DYNAMIC PROGRAMMING