Optimization of a condition-based duration-varying preventive maintenance policy for the stockless production system based on queueing model

Jianyu Cao; Weixin Xie

doi:10.3934/jimo.2018085

Article Contents

2019, Volume 15, Issue 3: 1049-1083. Doi: 10.3934/jimo.2018085

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Optimization of a condition-based duration-varying preventive maintenance policy for the stockless production system based on queueing model

Jianyu Cao^, and
Weixin Xie

College of information engineering, Shenzhen University, Shenzhen 518060, China

^* Corresponding author: Jianyu Cao
^* Corresponding author: Jianyu Cao

Received: March 31, 2016

Revised: February 28, 2018

Early access: June 2018

Published: April 2019

This work is supported by the National Natural Science Foundations of China (Grant Nos. 61401286, 61702341 and 61771319) and the Research Project of Shenzhen Technology University (Grant No. 201727).

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Abstract

A stockless production system is considered, in which the products are not produced until the orders are accepted. Due to this character, the duration of the preventive maintenance has an influence on the lead time. In addition, in this stockless production system, the cost of the preventive maintenance depends on its duration; if the lead time exceeds to the quoted lead time, some penalty cost should be considered; and the non-conforming products can still be sold by a discount. A condition-based duration-varying preventive maintenance policy is designed for the stockless production system, by making a tradeoff among the duration of the preventive maintenance, the time for the machine continuously producing, and the lead time of the order. According to the characters of the stockless production system with the designed preventive maintenance policy, it can be modeled by a BMAP/G/1 infinite-buffer queueing model with gated service and queue-length dependent vacation. Based on this queueing model, the stationary probability distributions of four performance measures for the stockless production system are analyzed, including, the number of the products produced in a production cycle, the number of the unfulfilled orders at arbitrary time, the time required to fulfill the tasks present at arbitrary time, and the lead time of the order accepted at arbitrary time. Moreover, based on some information of these performance measures, a profit function, which represents the average profit of the manufacture in a production cycle, is constructed to optimize the designed preventive maintenance policy according to specific conditions. Finally, given an example with the purchasers having different sensitivities to the lead time, some numerical experiments are carried out; and from the numerical experiments, some general results can be inferred for the stockless production system with the designed preventive maintenance policy.

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Mathematics Subject Classification: Primary: 90B30, 60K25; Secondary: 68M20.

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Figure 1. ${{K''}_t}\left\{ {\left. {\left( {i,v} \right) \times [0,x]} \right|\left( {{i_0},{v_0}} \right)} \right\}$ for ${{i}_{0}} = 0$, provided that at time $t$ $\left( t<{{T}_{1}} \right)$, a virtual customer arrives at the queue, while the server does not attend to the queue.

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Figure 2. ${{K''}_t}\left\{ {\left. {\left( {i,v} \right) \times [0,x]} \right|\left( {{i_0},{v_0}} \right)} \right\}$ for ${{i}_{0}}\ge 1$, provided that at time $t$ $\left( t<{{T}_{1}} \right)$, a virtual customer arrives at the queue, while the server does not attend to the queue.

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Figure 3. ${K''_t}\left\{ {\left. {\left( {i,v} \right) \times [0,x]} \right|\left( {{i_0},{v_0}} \right)} \right\}$ for $1\le {{i}_{0}}\le i$, provided that at time $t$ $\left( t<{{T}_{1}} \right)$, a virtual customer arrives at the queue, while the server is attending to the queue.

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Figure 4. ${K''_t}\left\{ {\left. {\left( {i,v} \right) \times [0,x]} \right|\left( {{i_0},{v_0}} \right)} \right\}$ for ${{i}_{0}}\ge 2$ and ${{i}_{0}}>i>0$, provided that at time $t$ $\left( t<{{T}_{1}} \right)$, a virtual customer arrives at the queue, while the server is attending to the queue.

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Figure 5. Schematic diagrams of ${\varphi }(n)$, ${M_{c}}(x)$ and ${W_{c}}(x)$.

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Figure 6. $\eta(q)$ vs. $q$ for the purchasers with different sensitivities to the lead time.

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Figure 7. ${{\sigma }_{m}}$ vs. $q$.

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Table 1. The correspondence between performance measures of the stockless production system and the queueing model

The stockless production system	The queueing model
The number of the products produced in a production cycle	The queue length just after the server travels to the queue
The number of the unfulfilled orders at arbitrary time	The queue length (including the customer in service) at arbitrary time
The time required to fulfill the tasks ¹ present at arbitrary time	The virtual waiting time² at arbitrary time
The lead time of the order accepted at arbitrary time	The actual waiting time³ at arbitrary time
¹ The tasks contain the future or remaining machine set-up, machine close-down as well as PM (or the idle period) in the current production cycle, and the present unfulfilled orders. ² The definition of the virtual waiting time is given in Section 5. ³ The definition of the actual waiting time is given in Section 6.

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