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The paper considers a production facility that might deteriorate suddenly at some point during the production run time; after deterioration, nonconforming items are produced in a greater rate compared to the rate before deterioration. Moreover, the production facility may ultimately break down; consequently, the production lot is aborted before completion. If breakdown happens, corrective action is started immediately; otherwise, the production lot is completed and preventive repair is implemented at the end of the production cycle to enhance system reliability. The mathematical model is formulated under general distributions of failure, corrective, and repair times, while the numerical examples are solved under exponential failure and uniform repair times. The formulated model successfully determines the optimal lot size in addition to the optimal process parameters (mean and standard deviation) simultaneously.

In real-life situations, most production systems are unreliable to a significant degree, and process deterioration occurs at some point in the production run time. A process shift from in-control state to out-of-control state will cause the production of more nonconforming items. This process may go worse and ultimately machine breakdown happens causing production interruption, and consequently plans for meeting demand are severely affected. Building on the previous argument, the need for more realistic modeling of EMQ (Economic Manufacturing Quantity) is rising in the manufacturing field; such modeling should take into consideration system variables which affect performance in a random manner; some of those variables include the following.

Time to shift from in-control state to out-of control state.

Time to breakdown.

Corrective maintenance time.

Preventive maintenance time.

Rate of producing nonconforming items before and after process shift; this can be handled in various ways.

Assuming nonconforming items are produced only after deterioration and they are produced at constant rate.

Assuming nonconforming items are produced before and after a sudden deterioration; accordingly, the process mean shifts to a new value and stays unchanged. The rate of producing nonconforming items is obviously greater after deterioration, and in both cases, the rate depends on the value of the process mean.

Or deterioration might be considered to happen gradually following a specific pattern (e.g., linear, quadratic, or exponential).

Usual costs related to maintenance, production, and inventory such as corrective and preventive repair costs, inventory holding cost, shortage penalty cost, cost of producing nonconforming items, and set-up cost.

Obviously, the two factors that have a great influence on EMQ decisions are process deterioration and machine breakdown. During the last few years, researchers from all over the world have done a tremendous amount of work investigating those two issues and their effect on EMQ.

Dagpunar [

Giri and Dohi [

Giri and Yun [

Hsieh and Lee [

Chiu et al. [

Chakraborty et al. [

Pentico et al. [

Up to our knowledge, no one has considered process targeting and production lot sizing simultaneously under process deterioration and machine breakdown conditions. In this paper we are going to study the joint effect of process deterioration, machine breakdown, and random repair times (corrective and preventive) on the optimal lot-sizing decisions in addition to process parameters (mean and standard deviation). We will consider process deterioration to happen suddenly, meaning that the process mean immediately shifts from original value

While determining the EMQ (or production run time) is important for production planning, one can question the economic benefits that can be drawn from a model that aims to find the optimal values of the process parameters: mean and standard deviation. For a quality characteristic, the performance of the process is quantified using the specification limits, lower specification limit (

The model under consideration is describing a production facility which may shift from in-control-state to out-of-control state at any random time during a production run; in both states, nonconforming items are produced in different rates depending on the process mean value. The production facility may also break down at any time during a production cycle. Once a shift to the out-of-control state has occurred, it is assumed that the production process will stay in that state until the whole lot has been produced or machine breakdown occurs. If the machine breakdown happens during the production run, then the interrupted lot is aborted and a new lot is started after corrective maintenance when all available inventory is depleted (no resumption—NR—policy).

Assuming that the production process starts in an “in-control” state at time

Figures

Breakdown/no shortage.

Breakdown/shortage.

No Breakdown/no shortage.

No Breakdown/shortage.

When deterioration happens after time

Now we solve an example by assuming the following.

Time to shift from in-control to out-of-control state is following a uniform distribution;

Also we assume the corrective and preventive repair times to follow uniform distribution functions:

From the results shown in Table

On the other hand, under a fixed value of deterioration factor

And we notice reasonably that the average cost per unit time increases when any of the two parameters (

Dependent of the optimal policy (

1.005 | 2.626 | 254.496 | 1013.53 |

1.01 | 2.612 | 253.715 | 1109 |

1.02 | 2.558 | 252.72 | 1428.28 |

1.03 | 2.501 | 254.141 | 1730.86 |

1.04 | 2.495 | 255.399 | 1759.87 |

1.005 | 2.654 | 254.496 | 1083.91 |

1.01 | 2.638 | 253.715 | 1178.08 |

1.02 | 2.582 | 252.72 | 1492.01 |

1.03 | 2.522 | 254.141 | 1789.71 |

1.04 | 2.515 | 255.399 | 1818.25 |

1.005 | 2.682 | 254.496 | 1152.19 |

1.01 | 2.665 | 253.715 | 1244.87 |

1.02 | 2.606 | 252.72 | 1553.82 |

1.03 | 2.542 | 254.141 | 1846.79 |

1.04 | 2.536 | 255.399 | 1874.88 |

In what follows we consider a model to determine the optimal production run time

Here we are assuming that process mean is not machine-related characteristic while the standard deviation is machine related. Moreover we assume that we have a wide range of machines available in the market varying in their accuracy (standard deviation); we need to determine the one with the optimal standard deviation which will minimize the total cost. Also we assume that all available machines are sharing the same breakdown distribution since they are all new and provided by qualified suppliers and they only differ in their accuracy. For instance, two identical CNC machines can differ in their accuracy due to different toolkits used.

The number of production cycles the new machine can serve is given by

The total cost in this case is given by

And again by the renewal theory, the average cost per unit time is given by

Now we solve an example to find the optimal production cycle length, the optimal process mean, and the optimal process standard deviation simultaneously. We set

From the results shown in Table

Dependent of the optimal policy (

1.005 | 2.575 | 254.254 | 1.464 | 1119.39 |

1.01 | 2.558 | 253.476 | 1.483 | 1226.58 |

1.02 | 2.499 | 252.537 | 1.581 | 1574.04 |

1.03 | 2.454 | 254.031 | 2.319 | 1880.18 |

1.04 | 2.440 | 255.392 | 1.941 | 1914.69 |

1.005 | 2.581 | 254.263 | 1.480 | 1194.87 |

1.01 | 2.563 | 253.486 | 1.502 | 1300.15 |

1.02 | 2.503 | 252.558 | 1.625 | 1641.17 |

1.03 | 2.464 | 254.02 | 2.347 | 1939.33 |

1.04 | 2.444 | 255.395 | 1.961 | 1974.51 |

1.005 | 2.584 | 254.272 | 1.496 | 1268.16 |

1.01 | 2.566 | 253.496 | 1.521 | 1371.57 |

1.02 | 2.507 | 252.578 | 1.668 | 1706.18 |

1.03 | 2.473 | 254.008 | 2.375 | 1996.69 |

1.04 | 2.448 | 255.397 | 1.982 | 2032.53 |

On the other hand, under a fixed value of

And finally we notice that the average cost per unit time increases when any of the two parameters (

In this paper, we have investigated the joint effect of process deterioration and machine breakdown on production lot-sizing and process-targeting decisions. The model is formulated under general failure and corrective repair times, while numerical examples are solved under exponential failure time and uniform repair (preventive and corrective) time distributions. We have considered two models: the first one for determining the optimal production cycle length

Nonenegative random variable denoting time to machine breakdown

Time to breakdown probability density function pdf

Failure rate (parameter for

Random variable denoting the time taken by the machine to shift from “in-control” state to “out-of-control” state

p.d.f. of the shift time (from in-control to out-of-control state)

Production cycle length

Nonenegative random variable denoting corrective repair time

Corrective repair time probability density function p.d.f.

The upper bound on

None-negative random variable denoting preventive repair time

Preventive repair time probability density function p.d.f.

The upper bound on

The expected time while process is in-control

The expected time while process is out of control

Demand rate

Production rate

Set up cost for each production run

Corrective repair cost per unit time

Preventive repair cost per unit time

Inventory holding cost per unit product per unit time

Shortage penalty cost per unit product

Cost incurred due to production of a nonconforming item with

Cost incurred due to production of a nonconforming item with

Total cost of nonconforming items per cycle

Taguchi loss function

Total loss cost per cycle

Expected total cost per production-inventory cycle

Expected length of a production-inventory cycle

The average cost per unit time

Cost of process standard deviation

Expected number of production cycles a new machine can serve

Expected life of the machine

The cost incurred to reduce

The maximum allowed value for

Random variable denoting the quality characteristic of process under consideration

p.d.f. of the quality characteristic

In-control process mean

Out-of-control process mean

Process deterioration factor

Process standard deviation

Lower specification limit on quality characteristic

Upper specification limit on quality characteristic

Proportion of nonconforming while process is in-control with

Proportion of nonconforming while process is in-control with

Proportion of nonconforming while process is out-of-control with

Proportion of nonconforming while process is out-of-control with