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January  2021, 17(1): 467-484. doi: 10.3934/jimo.2020043

An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts

 1 School of Economics and Management, Sanming University, Sanming, Fujian 365004, China 2 Department of Mechanical and Industrial Engineering, Louisiana State University, Baton Rouge, LA 70803, USA 3 Department of International Business, National Chengchi University, Taipei 11605, Taiwan (R.O.C)

* Corresponding author: Chien-Jui Lin (j698102@gmail.com)

Received  July 2018 Revised  August 2019 Published  March 2020

The purpose of this paper concentrates on an economic production quantity model with the factors of imperfect quality and quantity discounts, in which the inspection action occurs during the production stage. There is specific consideration of there being a finite production rate, and the quantity discounts offered by the supplier serves the purpose of stimulating buying greater quantities. This is in contrast to EPQ models that do not take these added factors into consideration. The objective of this paper is to determine the setup cost reduction, which is a function of capital investment, and inventory lot size. An alternative solution procedure was developed that does not employ the Hessian Matrix concavity in the expected total profit. We develop an algorithm to determine the optimal solution for this model. Theoretical results are discussed and a numerical example is proposed. Managerial insights are also examined.

Citation: Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial & Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043
References:

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References:
The behavior of the inventory level per cycle
The expected total profit
Procurement cost structure for the manufacture
 $r$ $Q_{r-1} \sim Q_{r}$ $c_{r}$ 1 $0 < Q < 150$ $c_{1} = 20.05$ 2 $150 \leq Q < 400$ $c_{2} = 20.04$ 3 $400 \leq Q < 800$ $c_{3} = 20.03$ 4 $800 \leq Q < 1250$ $c_{4} = 20.02$ 5 $Q \geq 800$ $c_{5} = 20.01$
 $r$ $Q_{r-1} \sim Q_{r}$ $c_{r}$ 1 $0 < Q < 150$ $c_{1} = 20.05$ 2 $150 \leq Q < 400$ $c_{2} = 20.04$ 3 $400 \leq Q < 800$ $c_{3} = 20.03$ 4 $800 \leq Q < 1250$ $c_{4} = 20.02$ 5 $Q \geq 800$ $c_{5} = 20.01$
The values of $Q^{*}, S^{*}$ and $E T P U^{*}$ corresponding to 32 combinations of $\sigma, f_{g} M, i, U(d)$
 $\sigma$ $f_{g}$ $M$ $i$ $U(d)$ $Q^{*}$ $S^{*}$ $E T P U^{*}$ 9600 0.05 192000 0.2 0.04 858.9 175.3 274724.7 0.056 866.3 175.4 275046.5 0.28 0.04 910.9 200 274036.9 0.056 918.8 200 274358.8 268800 0.2 0.04 846.1 163.3 274694.8 0.056 853.5 172.8 275016.6 0.28 0.04 897.4 200 274004.6 0.056 905.2 200 274326.4 0.07 192000 0.2 0.04 800 163.3 274570.9 0.056 800 162 274892.8 0.28 0.04 858.9 200 273906.7 0.056 866.4 200 274228.7 268800 0.2 0.04 800 163.3 374599.6 0.056 800 162 274862.7 0.28 0.04 846 200 273872.3 0.056 853.5 200 274194.3 13440 0.05 192000 0.2 0.04 877.4 128 385689.5 0.056 885 128.1 386139.7 0.28 0.04 930.5 190 384839 0.056 938.6 190.1 385289.5 268800 0.2 0.04 858.9 125.2 385646.9 0.056 866.3 125.3 386097 0.28 0.04 910.9 186 384779.3 0.056 918.8 186.1 385229.8 0.07 192000 0.2 0.04 812.5 118.4 385535.8 0.056 819.6 118.5 385986.1 0.28 0.04 877.4 179.1 384674.4 0.056 885.1 179.2 385125.1 268800 0.2 0.04 800 116.7 385493.1 0.056 800 115.7 385943.3 0.28 0.04 858.9 175.3 384614.7 0.056 866.4 175.4 385065.3
 $\sigma$ $f_{g}$ $M$ $i$ $U(d)$ $Q^{*}$ $S^{*}$ $E T P U^{*}$ 9600 0.05 192000 0.2 0.04 858.9 175.3 274724.7 0.056 866.3 175.4 275046.5 0.28 0.04 910.9 200 274036.9 0.056 918.8 200 274358.8 268800 0.2 0.04 846.1 163.3 274694.8 0.056 853.5 172.8 275016.6 0.28 0.04 897.4 200 274004.6 0.056 905.2 200 274326.4 0.07 192000 0.2 0.04 800 163.3 274570.9 0.056 800 162 274892.8 0.28 0.04 858.9 200 273906.7 0.056 866.4 200 274228.7 268800 0.2 0.04 800 163.3 374599.6 0.056 800 162 274862.7 0.28 0.04 846 200 273872.3 0.056 853.5 200 274194.3 13440 0.05 192000 0.2 0.04 877.4 128 385689.5 0.056 885 128.1 386139.7 0.28 0.04 930.5 190 384839 0.056 938.6 190.1 385289.5 268800 0.2 0.04 858.9 125.2 385646.9 0.056 866.3 125.3 386097 0.28 0.04 910.9 186 384779.3 0.056 918.8 186.1 385229.8 0.07 192000 0.2 0.04 812.5 118.4 385535.8 0.056 819.6 118.5 385986.1 0.28 0.04 877.4 179.1 384674.4 0.056 885.1 179.2 385125.1 268800 0.2 0.04 800 116.7 385493.1 0.056 800 115.7 385943.3 0.28 0.04 858.9 175.3 384614.7 0.056 866.4 175.4 385065.3
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