Production, Manufacturing and Logistics
Optimal production run time for a deteriorating production system under an extended inspection policy

https://doi.org/10.1016/j.ejor.2008.05.008Get rights and content

Abstract

A deteriorating production system is subjected to random deterioration from an in-control state to an out-of-control state with a general shift distribution. In order to reduce the defective items, part inspection policy, under which production inspections are performed only at the end of the production run, and full inspection policy are both considered in the literature. Moreover, the former dominates the latter. Since the product produced towards the end of a production cycle are more likely to be defective, it can further economize the inspection costs that they are directly reworked without inspection. In this paper, we propose an extended product inspection policy for a deteriorating production system. Product inspections are performed in the middle of a production cycle, and after the inspection, all products produced until the end of the production run are fully reworked. Based on the model, we show that there exists a production run time and a corresponding unique inspection policy such that the expected total cost per item per cycle is minimized. Finally, numerical examples are provided to illustrate our extended inspection policy, and indicate that such product inspection model will reduce the quality-related cost than part inspection does.

Introduction

The classical economic manufacturing quantity (EMQ) model assumes that a production system always remains in the in-control state, and that all the items produced are of perfect quality without any post-sale cost. This assumption may not be true in general. In many practical situations, a production system continuously deteriorates due to usage or age such as corrosion, fatigue, and cumulative wear. As a result, without taking any maintenance action to the system, the production process will eventually shift to the out-of-control state in which more defective items are produced than in the in-control state.

Many researchers have investigated the impact of process deterioration and machine breakdown on the production policy. Since this paper focuses on process deterioration, we confine ourselves in reviewing the following related works. Rosenblatt and Lee (1986) initially studied the effects of process deterioration on the traditional EMQ model, where the random degeneration from the in-control state to the out-of-control state is exponentially distributed. Hariga and Ben-Daya (1998) extended Rosenblatt and Lee (1986) to incorporate a more generalized assumption that an elapsed time until process shift is arbitrarily distributed, and provide distribution-based and distribution-free bounds on the optimal cost. Lee and Rosenblatt (1987) considered an inspection mechanism to monitor the imperfect production system and simultaneously determined the production cycle and the inspection schedule. Djamaludin et al. (1995) propose an inspection policy under which the last certain number of products in a lot are tested by burn-in method and those failed products during the test are scrapped. Tseng (1996) replaced a preventive maintenance policy with an inspection policy for an imperfect EMQ model. On the basis of Tseng (1996) model, Wang and Sheu (2000) provided several useful properties for obtaining an optimal production/maintenance policy. Djamaludin et al. (1994) utilized lot size to control the warranty cost per item for products under free repair warranty (FRW), where the production system can go into an out-of-control state with a given probability each time that an item is produced. Yeh et al. (2000) reformulated the Djamaludin et al. (1994) model to consider that the production process is subject to a random deterioration from an in-control state to an out-of-control state, taking both the restoration cost and the inventory holding cost into account and obtaining the bounds of the optimal production run length. Chen and Lo (2006) extended Yeh et al. (2000) to consider allowable shortages for the imperfect production processes. By taking the difference between warranty cost after sale and rework cost before sale into account, Lee and Park (1991) simultaneously determined the optimal production cycle and the inspection schedule. Yeh and Chen (2006) employed the idea given in Djamaludin et al. (1995) and propose a product inspection scheme called last-K inspection policy for the deteriorating production system investigated in (1994). Kim and Hong (1999) extended Rosenblatt and Lee (1986) model to consider a deteriorating production system with a general shift distribution, and studied the optimal production run length under the assumption that all defective items are detected and reworked at some costs after the production run is over. Although taking the full inspection schemes would eliminate the defective items, it incurs higher inspection costs. Wang (2005) extended Kim and Hong (1999) to take a moderate inspection policy, where product inspections are only performed at the end of the production run. In real production, the product produced towards the end of a production cycle are more likely to be defective, so it can further economize the inspection cost that all products produced at the end of the production run are directly reworked without inspection. Therefore, an extended inspection policy should be considered to balance the above quality-related costs. In this paper, we reformulate the work of Wang (2005) to consider an extended product inspection policy, where product inspections are performed in the middle of the production run, and after the inspection, all products until the end of the production run are fully reworked.

The remainder of this paper is organized as follows. The mathematical model is developed in Section 2. In Section 3, we first investigate the properties of the optimal extended inspection policy when production lot size is predetermined. Further, the uniqueness property and upper bound of the optimal production lot size is investigated and an efficient searching procedure is provided. In Section 4, numerical examples are provided to illustrate the application of our model. Finally, some conclusions are drawn in Section 5.

Section snippets

Notations and assumptions

We use the following notations throughout the paper:

    K

    setup cost per cycle

    d

    demand rate

    p

    production rate (>d)

    h

    holding cost per item per unit time

    r

    restoration cost per cycle

    cm

    manufacturing cost per unit product

    cI

    inspection cost per item

    cR

    rework cost per item (>cI)

    cW

    post-sale(warranty) cost for a defective item (>cI+cR)

    X

    the elapsed time until the production process shifts to the out-of-control state

    f(x),F(x)

    probability density function, distribution function of X, respectively

    F¯(x)

    survival function

Optimal production run time and corresponding inspection policy

In this section, some useful properties for determining the optimal run time t and the corresponding inspection policy (u1,u2) are proposed. Finally, we provide an efficient solution procedure for searching t, u1 and u2.

Numerical example

Consider a deteriorating production system with Weibull shift distribution F(t)=1-e-(t/α)β, where scale parameter α>0 and shape parameter β1. Without loss of generality, we assume that α=1. The following nominal values are selected for the model parameters: cm=0, K=2000, r=1500, p=1200, d=350, h=0.5, θ1=0.2, θ2=0.85. Table 1 summarizes the optimal policy and resulting expected total cost per item for various combinations of (cI,cR,cW) and β.

For comparison, Table 1 also lists the results of the

Conclusion

In this paper, we propose the optimal run time t under an extended inspection policy (u1,u2)for a deteriorating production system, where a general shift distribution is considered. That is, in a production cycle, product inspections are performed from the (pu1t) th until the (pu2t) th, and all products from (pu2t) th until the end of the production run are reworked without inspections. We show that there exists an optimal production run time t and a unique corresponding inspection

Acknowledgements

We are thankful to the editor and anonymous referees reviews for their valuable comments and suggestions to an earlier version of this paper. This research was partially supported by the National Natural Science Foundation of China(Grant no. 60574055).

References (13)

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