Optimal control of production rate in a failure prone manufacturing system

Abstract

We address the problem of controlling the production rate of a failure prone manufacturing system so as to minimize the discounted inventory cost, where certain cost rates are specified for both positive and negative inventories, and there is a constant demand rate for the commodity produced.The underlying theoretical problem is the optimal control of a continuous time system with jump Markov disturbances, with an infinite horizon discounted cost criterion.We use two complementary approaches. First, proceeding informally, and using a combination of stochastic coupling, linear system. arguments, stable and unstable eigenspaces, renewal theory, parametric optimization etc., we arrive at a conjecture for the optimal policy. Then we address the previously ignored mathematical difficulties associated with differential equations with discontinuous right hand sides, singularity of the optimal control problem, smoothness and validity of the dynamic programming equation etc., to give a rigorous proof of optimality of the conjectured policy.It is hoped that both approaches will find uses in other such problems also.We obtain the complete solution and show that the optimal solution is simply characterized by a certain critical number, which we call the optimal inventory levelo If the current inventory level exceeds the optimal, one should not produce at all, if less, one should produce at the maximum rate, while if exactly equal one should produce exactly enough to meet demand. We also give a simple explicit formula for the optimal inventory level.

Markovian approximation for manufacturing systems of unreliable machines in tandem

“…By comparing all possible pairs of (s, S), the optimal pair of (s * , S * ) which minimizes the average running cost (12) can be found. We remark that when c P = 0, the average running cost (12) is convex in s when S is kept constant (see [1,2,8,19,24]). In general the average running cost (12) is not necessarily convex.…”

“…For one-machine one-part-type system, Akella and Kumar [1] have shown analytically that the Hedging Point Production (HPP) policy is optimal in the sense that the average running cost of the system is minimized. Later Boukas and Yang [3] extended Akella and Kumar's idea to allow simultaneous planning of production and maintenance.…”

Optimization and Analysis of a Manufacturing–Remanufacturing–Transport–Warehousing System within a Closed-Loop Supply Chain

“…The number of products being transported at time t is described as follows: For machines productions rates, we use the hedging point policy [28], which ensures that the amount of products does not exceed the manufacturing stock capacity.…”

Near optimal manufacturing flow controller design

“…Gershwin, Akella, and Choong (1985) proposed a heuristic approximation of the value function and investigated further the structure of the optimal controller when the production surplus/backlog levels enter certain subsets of the production surplus space that render the optimal control indeterminable unless additional optimality conditions are added. Akella and Kumar (1986) solved analytically the Hamilton-Jacobi-Bellman equation to obtain the optimal value function for a small flow controller of a one-part-type/one-machine manufacturing system. Bielecki and Kumar (1988) studied the steady-state probability distribution of production surplus/backlog under a hedging-point flow controller, and used it to determine the optimal hedging points.…”