Annals of Operations Research

, Volume 271, Issue 2, pp 445–467

# Optimization of sample size and order size in an inventory model with quality inspection and return of defective items

• Naoufel Cheikhrouhou
• Biswajit Sarkar
• Baishakhi Ganguly
• Asif Iqbal Malik
• Rafael Batista
• Young Hae Lee
Original Paper

## Abstract

To ensure all products as perfect, inspection is essential, even though it is not possible to inspect all products after producing them like some special type products as plastic joint for the water pipe. In this direction, this paper develops an inventory model with lot inspection policy. With the help of lot inspection, all products need not to be verified still the retailer can decide the quality of products during inspection. If retailer founds products as imperfect quality, the products are sent back to supplier. As it is lot inspection, mis-clarification errors (Type-I error and Type-II error) are introduced to model the problem. Two possible cases are discussed for sending back products as defective lots are immediately withdrawn from the system and send back to supplier with retailer’s payment and for second case, retailer sends defective products during receiving next lot from supplier with supplier’s investment, like in food industry or in hygiene product industry. The model is solved analytically and results indicate that optimal order size and sample size are intrinsically linked and maximize the total profit. Numerical examples, graphical representations, and sensitivity analysis are given to illustrate the model. The results suggest that sending defective products maintaining the first case is the more profitable than the second case.

## Keywords

Production Imperfect quality Sampling inspection Inspection errors Nonlinear optimization

## Decision variables

y

Order quantity size (units)

n

Number of items inspected per lot (integer)

## Parameters

$$c_{l}$$

Unit inspection cost ($/unit) $$c_{U}$$ Lot purchase cost ($/lot)

$$c_{a}$$

Cost of falsely accepting a defective lot ($/lot) $$c_{r}$$ Cost of falsely rejecting a non-defective lot ($/lot)

D

Demand rate (units/year)

h

Holding cost of a good quality item ($/unit/year) $$h_{d}$$ Holding cost of a defective item kept in inventory (for the second subcase model) ($/unit/year)

K

Ordering cost ($/order) $${\mathrm {K}}_{d}$$ Transport cost of defective lots back to supplier (for the first subcase model) ($)

L

Number of items per lot

$$m_{1}$$

Random variable representing the Type I error

$$m_{2}$$

Random variable representing the Type II error

P

Unit selling price (\$/unit)

$${\uprho }$$

Fraction of defective lots supplied

X

Screening rate (units/year)

E[.]

Expected value of a random variable

$$t_{l}$$

Time to screen a shipment (years) $$=\frac{y}{X}$$

$${{\mathrm {HC}}}_{n}$$

Holding costs of the subcase n

$${\mathrm {IC}}$$

Inspection costs

IEC

Inspection error costs

$${{\mathrm {PC}}}_{n}$$

Purchase costs of the subcase n

$${{\mathrm {TP}}}_{n}$$

Total profit of the subcase n

$${\mathrm {T}}_{0}$$

Cycle time (years)

$$y^{*}$$

Optimal order quantity size (units)

## Notes

### Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Project Number: 2017R1D1A1B03033846).

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## Authors and Affiliations

• Naoufel Cheikhrouhou
• 4
• Biswajit Sarkar
• 2
• Baishakhi Ganguly
• 3
• Asif Iqbal Malik
• 2
• Rafael Batista
• 1
• Young Hae Lee
• 2
1. 1.Laboratory of Management and Production ProcessesSwiss Federal Institute of Technology, Lausanne (EPFL)LausanneSwitzerland
2. 2.Department of Industrial & Management EngineeringHanyang UniversityAnsanSouth Korea
3. 3.Department of Mathematics & StatisticsBanasthali VidyapithVanasthaliIndia
4. 4.Haute école de gestion de Genève, HES-SOUniversity of Applied Sciences Western SwitzerlandGenevaSwitzerland