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This paper deals with the production-dependent failure rates for a hybrid manufacturing/remanufacturing system subject to random failures and repairs. The failure rate of the manufacturing machine depends on its production rate, while the failure rate of the remanufacturing machine is constant. In the proposed model, the manufacturing machine is characterized by a higher production rate. The machines produce one type of final product and unmet demand is backlogged. At the expected end of their usage, products are collected from the market and kept in recoverable inventory for future remanufacturing, or disposed of. The objective of the system is to find the production rates of the manufacturing and the remanufacturing machines that would minimize a discounted overall cost consisting of serviceable inventory cost, backlog cost and holding cost for returns. A computational algorithm, based on numerical methods, is used for solving the optimality conditions obtained from the application of the stochastic dynamic programming approach. Finally, a numerical example and sensitivity analyses are presented to illustrate the usefulness of the proposed approach. Our results clearly show that the optimal control policy of the system is obtained when the failure rates of the machine depend on its production rate.

With market globalization and technological advancement, manufacturing systems are faced with optimization problems in their global supply chain of production. Production planning problems become more complex when the environmental constraints require optimization of production and reuse of parts returned by customers after use (reverse logistics). Compared to a situation where customer demand is only satisfied by the direct line of production (production from raw materials), the simultaneous control of production and product recovery is very complex [

The remainder of the paper is organized as follows. A literature review is presented in Section 2. Section 3 consists of assumptions of the model and the problem statement. Section 4 provides numerical results and sensitivity analyses to illustrate the usefulness of the proposed approach. The paper ends with the conclusion in Section 5.

Several authors have worked on the study of a combined manufacturing and remanufacturing system. However, few studies today include aspects related to the stochastic dynamics of machines. Stochastic dynamics models allow us to approach more real cases characterized by the presence of random phenomena. Until now, no paper has studied a non-homogeneous Markov process (dependence of failure rates on the production rate) for a hybrid manufacturing/remanufacturing system. The literature on the combined manufacturing/remanufacturing and stochastic aspects, as well as a manufacturing system with production-dependent failure rates is discussed below.

In [

In the preceding paragraph, we note that Reference [

One of the most important results obtained in [

The results of [

This section presents the assumptions used throughout this article, as well as the problem statement.

2) The failure rate of the manufacturing machine depends on its production rate. This assumption is the major motivation of our paper. Other works consider one machine with a production-dependent failure rate or two machines (manufacturing and remanufacturing machines) without deterioration with production speed.

3) The shortage cost depends on parts produced for backlog ($/unit).

4) The inventory cost depends on parts produced for positive inventory ($/unit).

5) The production rate of the manufacturing machine is higher than that of the remanufacturing machine.

6) The remanufacturing machine cannot satisfy the customer demand alone.

7) The manufacturing machine is unable to satisfy the customer demand with its economic productivity, which is why the remanufacturing machine is called upon to fill the demand rate.

8) Manufacturing processes convert the raw materials to finished items.

9) Remanufacturing processes convert used products to as good as new parts

10)New parts (manufactured and remanufactured) satisfy the serviceable inventory.

11)Backorders of unsatisfied demands are permitted.

The system under study as depicted in

Hybrid manufacturing/remanufacturing system

The mode of the machine

The operational mode of the system is described by the random vector

- Mode 1:

- Mode 2:

- Mode 3:

- Mode 4:

With

where

The transition diagram, which describes the dynamics of the considered manufacturing system, is presented in

The dynamics of the system is described by a discrete element, namely

We assume that the failure rate of

Hence,

States transition diagram. (a) Each machine; (b) The system

Let

where

The continuous part of the system dynamics is described by the following differential equations:

where

Let

where constants

The production planning problem considered in this paper involves the determination of the optimal control policies (

where

Based on the value function presented in (8), the optimality conditions and the numerical methods used to solve them (in order to determine the optimal manufacturing and remanufacturing rates) are presented in Appendix A.

The next section provides a numerical example to illustrate the structure of the control policies.

Here, we illustrate the resolution of the model above with a numerical example. Sensitivity analyses with respect to the system parameters are also presented to illustrate the importance and effectiveness of the proposed metho- dology.

This section gives a numerical example for a hybrid manufacturing/remanufacturing system presented in Section 3.

The considered computation domain

The production system will be able to meet the demand rate over an infinite horizon and reach a steady state if

the following condition is satisfied:

and the data presented in

The production policies, illustrated in

Production rate of at mode 1

. Numerical data of the considered system

2 | 50 | 1 | 0.5 | 0.5 | 1.2 | 1.3 | 1.15 |

1.25 | 1/80 | 1/100 | 1/60 | 1/15 | 1/15 | 0.09 |

Production rate of at mode 2

Boundary of at mode 1

creases. The traces of

According to the classical results as in [

Examining Figures 3 to 6, we see that the optimal stock levels depend directly on the level of returned products. Consequently, the optimal production control policy consists of one of the following rules:

1) Set the productivity of

Boundary of at mode 2

Production rate of at mode 1

2) Reduce the productivity of

3) Set the productivity of

4) In Figures 7-10, the optimal policies of

5) The relation between the inventory, the stock of returned products and the production rate of

The computational domain of

1) Produce at the maximal rate (or at

2) Set the production rate to zero when the current stock level is larger than a threshold value.

Production rate of at mode 3

Boundary of at mode 1

Boundary of at mode 3

3) The results of

The results of

Based on the results from Figures 3-10, the production rates of

where

With numerical methods, the results show that

In the hybrid manufacturing/remanufacturing system consisting of two machines and one type of product, with a constant failure rate such as the one described in [

The optimal policy of the proposed joint optimization of production and machine reliability is given by (10)- (12). To validate and illustrate the usefulness of the developed model, let us confirm the obtained results through a sensitivity analysis. Several experiments were conducted to ensure that the structure of the policies obtained is maintained under the variation of the model parameters, and can therefore be used in practice.

A set of numerical examples were considered to measure the sensitivity of the control policies obtained in mode 1 (both machines are producing) and to illustrate the contribution of this paper. We analyze the sensitivity of the control policies according to the backlog costs in the first section. In the subsequent section, we examine the sensitivity of the optimal policies according to different values of the return rate. The sensitivity analysis enables the tracking of variations to the policy boundaries.

In this section, we will perform sensitivity analysis on the backlog cost.

This section analyzes the sensitivity of the optimal threshold values with respect to the return rates.

When the return rate takes four values:

When

Variation of at mode 1: Effect on

Variation of at mode 1: Effect on

Variation of at mode 1: Effect on

of the remanufacturing machine is set directly to its maximal value instead of

Through the observations drawn from the sensitivity analysis, the results demonstrate conclusively that the resulting policy is optimal and enhances machine reliability. Control policies for our systems consider an extension of the multi-hedging point structure. Without in any way limiting the generality of this proposal, this model is based on certain assumptions relating to a pair of machines (manufacturing and remanufacturing machines) which are not identical and which operate in parallel. Given certain conditions, extended versions of this model might be adopted across a number of industrial sectors.

Variation of at mode 1: Effect on

Although some of the concepts of reverse logistics, such as the facility location models including return flows, inventory management models, production and transportation planning models, have been put into practice for years, it is only fairly recently that the integration of aspects related to the stochastic dynamics of machines has been a real concern for the management of reverse logistics systems. This paper confirms that it is possible to integrate production-dependent failure rates in a hybrid manufacturing/remanufacturing system subject to random failures and repairs, in order to minimize the overall incurred cost. The machines produce one type of final product.

The failure rates of the manufacturing machine depend on its production rate. To take into account its availability, the company will then maximize the recovery of its products used from the market, allowing it to eventually minimize the use of raw materials which become increasingly rare. We developed the stochastic optimization model of the considered problem with two decision variables (production rates of manufacturing and remanufacturing machines). The stock levels of new and returned products were the state variables. From the numerical study, it was found that for two parallel machines, when the failure rates of the machines depend on the production rate, the hedging point policies are optimal within a five-threshold feedback policy, and the reliability of the machines is enhanced. A numerical example is given to illustrate the utility of the proposed approach. The sensitivity analyses show that the structure of the results obtained is maintained. This approach takes into account both the multi-objective aspect and the dynamics of machines. However, the model is far from perfect, and leaves much to be desired, especially in the case involving multiple machines, multiple products, random return rate, the quality of remanufactured products (non-conforming products) and the returns control policy, such as the pricing policy.