A numerical method to approximate optimal production and maintenance plan in a flexible manufacturing system

doi:10.1016/j.amc.2004.12.030

Applied Mathematics and Computation

Volume 170, Issue 2, 15 November 2005, Pages 924-940

https://doi.org/10.1016/j.amc.2004.12.030 Get rights and content

Abstract

The simultaneous planning of the production and the maintenance in a flexible manufacturing system is considered in this paper. The manufacturing system is composed of one machine that produces a single product. There is a preventive maintenance plan to reduce the failure rate of the machine. This paper is different from the previous researches in this area in two separate ways. First, the failure rate of the machine is supposed to be a function of its age. Second, we assume that the demand of the manufacturing product is time dependent and its rate depends on the level of advertisement on that product. The objective is to maximize the expected discounted total profit of the firm over an infinite time horizon. In the process of finding a solution to the problem, we first characterize an optimal control by introducing a set of Hamilton–Jacobi–Bellman partial differential equations. Then we realize that under practical assumptions, this set of equations can not be solved analytically. Thus to find a suboptimal control, we approximate the original stochastic optimal control model by a discrete-time deterministic optimal control problem. Then proposing a numerical method to solve the steady state Riccati equation, we approximate a suboptimal solution to the problem.

Introduction

Our model of the production and maintenance planning in a flexible manufacturing system consists of a single workstation producing one part-type through a single operation. The machine is due to failure and there is a preventive maintenance plan to reduce its failure rate. The failure intensity of the machine is supposed to be dependent on its age. The failure rate of the machine is assumed to be a function of its age as well as its maintenance rate.

The research of the production and maintenance planning using stochastic optimal control techniques has drawn much attention lately. Due to the complexity of the manufacturing systems, traditionally, marketing decision making and other decision related areas such as production and maintenance are often treated separately. Clearly, a marketing model with addition of production and maintenance is more realistic and useful from a practical point of view. Because of this reason in this paper, we assume that the demand of the manufacturing product is time dependent and its rate depends on the level of advertisement on that product.

Our objective is to find a control policy which maximizes the expected discounted total profit of the firm, for each set of initial conditions over an infinite time horizon, or equivalently to minimize the negative profit which is defined as the total cost minus the revenue of the firm. While the total cost consists of the cost of the product surplus defined as the discrepancy between total cumulative production and the total cumulative demand, the cost of the repair activity after failure, the cost of the maintenance activity, and the cost of the advertisement. We also assume the repair is more costly than the preventive maintenance.

To solve the optimization problem of this paper, we first write down the necessary conditions for optimality as a set of partial differential equations which can be solved only under restricted assumptions. But under reasonable assumptions, they can not be solved analytically. So we approximate this stochastic optimization problem by a discrete-time deterministic optimal control model which is characterized by a steady state Riccati equation. Then proposing an iterative numerical procedure to solve this Riccati equation, a suboptimal control for the original problem is obtained.

The rest of this paper contains the following sections. Section 2 consists of the literature review in the general area of this research. The problem statement containing the assumptions and the mathematical model of the problem is discussed in Section 3. The optimal control is characterized by the appropriate partial differential equations in Section 4. In Section 5, the approximate discrete-time deterministic optimal control problem is introduced and the corresponding steady state Riccati equation is defined. The iterative numerical procedure to solve this Riccati equation is proposed through a numerical example in Section 6. Finally, the conclusions of the paper are mentioned in Section 7.

Section snippets

Literature review

An interesting feature of many automated production systems is that they can be considered as deterministic systems as long as no machine breakdowns or stoppages occur. Therefore, these systems fall in the category of “piecewise deterministic processes”, according to the terminology of Davis [1]. A class of systems closely related to those considered previously by Sworder [2] and Rishel [3], [4] and called “systems with jump Markov disturbances”. Olsder and Suri [5] have been the first ones to

Problem statement

As mentioned earlier, our model for the production and maintenance planning in a flexible manufacturing system consists of a single workstation producing one part-type through a single operation. The system considered has a state comprising both a continuous and a discrete component. This production system has continuous state variables x, a, and z corresponding to the cumulative production surplus of parts, the machine age, and the demand rate of the part-type respectively. Let u(t) be the

Hamilton–Jacobi–Bellman equation

This optimal control problem belongs to the class of problems considered by Rishel. Under appropriate assumptions of smoothness for the control, the following set of Hamilton–Jacobi–Bellman partial differential equations characterizes an optimal control: $ρ V (β, a, x) = \min \{ϕ^{β} (a, x, u, v) + \frac{\partial}{\partial a} V (β, a, x) f (u, v) + \frac{\partial}{\partial x} V (β, a, x) (u (t) - z (t)) + \sum_{α \in E} q_{β α} (a, v) [V (α, ϕ (a, α), x) - V (β, a, x)]\}, \forall β \in E, over θ \in Θ (β) .$ We have defined the cost rate function ϕ^β(a, x, u, v) as a function of the state variables only. Therefore, we can deduce from

A discrete-time approximation model and the steady state Riccati equation

The state Eqs. (1), (2), (3) can be quantized to get a discrete-time approximation of the original model as follows: $x (k + 1) = x (k) + u (k) - z (k), x (0) = x_{0},$ $a (k + 1) = a (k) + f (u (k), v (k)), a (T) = 0, k > T,$ $z (k + 1) = (c_{0} + 1) z (k) + c_{1} w (k), z (0) = z_{0} .$ To quantize the performance criterion (8), the following changes should be made: $J^{β} (x, a, z, u, v, w) = E \{\sum_{k = 0}^{\infty} \frac{1}{(1 + r)^{k}} [ϕ^{ξ (k)} (a (k), x (k), u (k), v (k)) + w (k) - π z (k)] / x (0) = x_{0}, a (0) = a_{0}, z (0) = z_{0}, ξ (0) = β\},$ where r is a positive discount rate in each period. First about the discounting term, notice that $\sum_{k = 1}^{\infty} \frac{1}{(1}$

A numerical example illustrating the iterative numerical procedure to solve the Riccati equation

As a numerical example, assume the following values for the parameters of the problem: $d = 2, s = 0.1, p = 1, e = 2, f = 1, q = 0.5, b_{1} = 1, b_{2} = 1, c_{0} = 0.2, and c_{1} = 1 .$ So the numerical values of the matrices will be $A = \begin{matrix} 1 & 0 & - 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1.2 \end{matrix}, B = \begin{matrix} 1 & 0 & 0 \\ 1 & - 1 & 0 \\ 0 & 0 & 1 \end{matrix}, Q = \begin{matrix} 2 & 0 & 0 \\ 0 & 0.1 & 0 \\ 0 & 0 & 1 \end{matrix}, R = \begin{matrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0.5 \end{matrix} .$ The following values for the initial conditions are also assumed: $x (0) = x_{0} = 3, a (0) = a_{0} = 1, and z (0) = z_{0} = 2 .$ The numerical procedure for the solution of the steady state Riccati equation is proposed in the following manner. Consider a case of a finite number of stages

Conclusions

The simultaneous planning of the production and the maintenance in a flexible manufacturing system is considered in this paper. It is different from previous researches in this area in two separate ways. First, the failure rate of the machine is supposed to be a function of its age. Second, we assume that the demand of the manufacturing product is time dependent and its rate depends on the level of advertisement on that product. These assumptions are more realistic and make the results of this

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A numerical method to approximate optimal production and maintenance plan in a flexible manufacturing system

Abstract

Introduction

Section snippets

Literature review

Problem statement

Hamilton–Jacobi–Bellman equation

A discrete-time approximation model and the steady state Riccati equation

A numerical example illustrating the iterative numerical procedure to solve the Riccati equation

Conclusions

Markov Models and Optimization

Feedback control of a class of linear systems with jump parameters

IEEE Transactions on Automatic Control

Control of systems with jump Markov disturbances

IEEE Transactions on Automatic Control

Dynamic programming and minimum principles for systems with jump Markov disturbances

SIAM Journal on Control

Optimal control of production rate in a failure-prone manufacturing system

IEEE Transactions on Automatic Control

Optimality of zero-inventory policies for unreliable manufacturing systems

Operations Research

An algorithm for the computer control production in flexible manufacturing systems

IIE Transactions

Near-optimal manufacturing flow controller design

International Journal of Flexible Manufacturing Systems

Dynamics and design of a class of parameterized manufacturing flow controllers

IEEE Transactions on Automatic Control

Perturbation analysis for the design of flexible manufacturing system flow controllers

Operations Research

Hedging policies for failure-prone manufacturing systems: Optimality of JIT and bounds on buffer levels

IEEE Transactions on Automatic Control

On the ordering of hedging points in a class of manufacturing flow control models

IEEE Transactions on Automatic Control

Optimal control of manufacturing flow and preventive maintenance

IEEE Transactions on Automatic Control

Production control of a manufacturing system with multiple machine states

IEEE Transactions on Automatic Control