Optimal production decision for a risk‐averse manufacturer faced with random yield and stochastic demand

• Date: 01 May 2020
• DOI: http://doi.org/10.1111/itor.12483
• Author: Shiming Deng,Zhong Zheng
• Published date: 01 May 2020

Intl. Trans. in Op. Res. 27 (2020) 1622–1637

DOI: 10.1111/itor.12483

INTERNATIONAL

TRANSACTIONS

IN OPERATIONAL

RESEARCH

Optimal production decision for a risk-averse manufacturer

faced with random yield and stochastic demand

Shiming Deng and Zhong Zheng∗

School of Management, Huazhong University of Science and Technology, Wuhan 430074, China

E-mail: smdeng@hust.edu.cn [Deng]; zhongzhenghust@163.com [Zheng]

Received 28 August2016; received in revised form 8 October 2017; accepted 10 October 2017

Abstract

Consider a risk-averse manufacturer who produces a single product with a random yield to satisfy uncertain

market demand. The manufacturer maximizes its expected profit subject to a chance constraint that requires

the probability of the total profit below a target be less than a predetermined level. We show that due to

the profit target constraint in the presence of random yield and stochastic demand, the manufacturer can

neither produce too much nor too little irrespective of the predetermined probability level, depending solely

on the profit target level. Further, the special case of uniform yield and uniform demand is examined and we

obtain the manufacturer’s optimal production quantity. In addition, two special cases of random yield rate

and deterministic demand or deterministic yield rate and stochastic demand are considered. The opposite

impacts of random yield and stochastic demand are revealed:the random yield induces a minimum production

quantity that may cause the manufacturer to increase its production quantity, while the stochastic demand

induces a maximum production quantity thatmay cause the manufacturer to decrease its production quantity.

By comparing the solutions of the above-mentioned special cases with the case of both random yield and

stochastic demand, it is demonstrated that the existence of both random yield and stochastic demand results

in a more constrained production requirement for the manufacturer (a larger minimum production quantity

and a smaller maximum production quantity). That is, the opposite impacts of random yield and stochastic

demand will not offset, but enhance each other.

Keywords:random yield; stochastic demand; risk aversion; downside risk; profit target

1. Introduction

Many firms face the production planning problem before various types of supply chain uncer-

tainty are realized. The most prominent types of uncertainty are demand uncertainty and supply

uncertainty. On the demand side, if the demand exceeds or falls below the inventory level, the com-

pany will face shortage or overstock. On the supply side, supply uncertainty is also an important

∗Corresponding author.

C

2017 The Authors.

International Transactionsin Operational Research C

2017 International Federation ofOperational Research Societies

Published by John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St, Malden, MA02148,

USA.

S. Deng and Z. Zheng / Intl. Trans. in Op. Res. 27 (2020) 1622–1637 1623

randomness source that should not be neglected or underestimated. Long machine downtimes due

to unplanned maintenance, strikes and yield uncertainty in a production run, lack of raw material,

weather conditions, and rework are different reasons that may result in supply randomness. For

example, as reported in Norrman and Jansson (2004), a fire at a supplier’s plant disrupted the

supply of radio-frequency chips to Ericsson in 2001 resulting in a loss of $400 million. Chiron

Corporation, a major supplier of influenza vaccines made for consumption in the United States,

announced in October 2004 that the batch of Fluvirin vaccines for the 2004–2005 influenza season

was contaminated by bacterial infection. The subsequent closure of Chirons Liverpool, England,

production facility reduced the supply of flu shots available to American citizens by half (Whalen

et al., 2004). Therefore, taking into account the impact of yield uncertainty during the production

process for the decision makersof these display companies is of prime importance. Specifically, yield

uncertainty is characterized in the form of random yield in this paper.

In order to control the supply chain risk resulting from uncertainty, the staff who are responsible

for production decisions aregiven profit (or profit-related) targets fortheir performance assessment.

Deng and Yano(2016) state that “a number of retailer stuff numbers tend to ‘work backward’ from

the targets to determine order quantities, recognizing that they need to order enough to have any

chance of reaching the target, but cannot order too much, due to the cost of unsold inventory.”

Therefore, the profit target could be regarded as an important constraint for the decision maker.

A manufacturer faced with supply and demand uncertainty determines the optimal production

quantity, recognizing that it needs to produce enough to reach the profit target, but cannot yield

too much for the risk of overproduction. Note that the manufacturer might not reach the target

constraint, this is also called downside risk. A general definition of downside risk was introduced by

Fishburn (1977) in the form of lower partial moments operationalized as the probability-weighted

functions of deviations below some target return. However, to the best of our knowledge, the

impact of target constraint has only been investigated in the scenario of uncertain demand, not

under the circumstance of random yield. Motivated bythis, the research questions of this paper can

be summarized as follows:

rWhat is the risk-averse manufacturer’s optimal productionquantity when faced with both random

yield and stochastic demand?

rWhat is the difference between the impacts of random yield and stochastic demand on the risk-

averse manufacturer’s production quantity?

rHow do the impacts of random yield and stochastic demand affect each other?

In order to shed light on the abovequestions, we consider a risk-aversemanufacturer who produces

a single product with a random yield to satisfy uncertain market demand. The manufacturer

maximizes its expected profit subject to a chance constraint that requires the probability of the total

profit below a target to be less than a predetermined level. It is shown that due to the profit target

constraint in the presence of random yield and stochastic demand, the manufacturer can neither

produce too much nor too little.Further more, to isolate the impacts of random yield and stochastic

demand, two special cases of random yield rate and deterministic demand or deterministic yield rate

and stochastic demand are examined. The opposite impacts of random yield and stochastic demand

are revealed via these two special cases: whenthe manufacturer’s extent of risk aversion is moderate,

the random yield induces a minimum production quantity, which will cause the manufacturer

C

2017 The Authors.

International Transactionsin Operational Research C

2017 International Federation of OperationalResearch Societies