european journal of operational research...m. guo et al. / european journal of operational research...

European Journal of Operational Research 273 (2019) 198–216

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European Journal of Operational Research

journal homepage: www.elsevier.com/locate/ejor

Production, Manufacturing, Transportation and Logistics

The single-period (newsvendor) problem under interval grade

uncertainties

Min Guo

a , ∗, Yu-wang Chen

b , Hongwei Wang

c , Jian-Bo Yang

b , Keyong Zhang

a

a College of Economics and Management, North University of China, Taiyuan, PR China b Alliance Manchester Business School, The University of Manchester, Manchester, UK c School of Management, Huazhong University of Science & Technology, Wuhan, PR China

a r t i c l e i n f o

Article history:

Received 24 April 2016

Accepted 30 July 2018

Available online 7 August 2018

Keywords:

Inventory

Decision analysis

Uncertainty expression

a b s t r a c t

Traditional stochastic inventory models assume to have complete knowledge about the demand probabil-

ity distribution. However, in reality it is often difficult to characterize demand precisely, especially with

limited historical data or through subjective forecasting. In this paper, we aim to develop a consistent

framework of formulating demand uncertainties in single-period (newsvendor) problems, where a set of

discrete assessment grades and/or grade intervals are used to represent complex uncertainties in both

quantitative and qualitative evaluations. In this uncertainty formulation framework, we use random set

theory to study optimal ordering policies for the newsvendor problem under optimistic, pessimistic, mini-

mum regret and maximum entropy criteria respectively. Numerical studies are conducted to illustrate the

effectiveness of the proposed approach.

© 2018 Elsevier B.V. All rights reserved.

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1. Introduction and literature review

This paper is concerned with the problem of controlling the in-

ventory of a single item with uncertain demand over a single pe-

riod, referred to as ‘the single-period problem’ (SPP, thereinafter)

or ‘newsvendor’ problem ( Khouja, 1999 ). Generally speaking, the

standard SPP has the following characteristics. Early each morn-

ing (i.e., at the beginning of a selling season), a newsvendor (i.e.,

retailer) must decide how many newspapers (i.e., a kind of perish-

able product which only keeps fresh in a time period) to purchase.

The procurement lead-time tends to be quite long compared to the

selling season, so the newsvendor cannot observe demand prior to

placing the order. In addition, there is no opportunity to replen-

ish inventory once the season has begun. Due to uncertainty, the

newsvendor can face the outcome of having either newspapers un-

sold or demand unmet. The unsold newspapers will incur a loss

because their recycle value is lower than the purchase price. The

unmet demand will also generate a cost from lost sales, such as

a penalty for the lost customer goodwill to the newsvendor. The

classical SPP is one of the fundamental models in inventory man-

agement, and it has a wide range of application areas, such as

revenue management, supply chain management, and accounting

management.

∗ Corresponding author.

E-mail address: [email protected] (M. Guo).

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https://doi.org/10.1016/j.ejor.2018.07.048

0377-2217/© 2018 Elsevier B.V. All rights reserved.

Traditional stochastic SPP models in which demand uncertainty

s formulated by stochastic variables have been studied extensively

nder various assumptions ( Khouja, 1999; Qin, Wang, Vakharia,

hen, & Seref, 2011 ). These stochastic SPP models often assume

o have full knowledge about the demand probability distribution.

owever, in reality it is often difficult to characterize demand pre-

isely, since historical data are not always available or reliable. This

s especially true in the circumstances where market turbulence or

echnological innovation takes place commonly ( Dalrymple, 1987 ),

nd the forecasting of demand depends on not only quantitative

ata but also experts’ subjective judgments, which can be in the

ormat of linguistic evaluation with uncertainties. Unfortunately,

hese uncertain and linguistic judgments are very complex in gen-

ral. For example, one expert may claim that the demand is ‘high’

r ‘very low’ or ‘very likely to be low’, where the term ‘very likely’

ossibly means 70% confidence; It is also common that experts’

valuation may contain incompleteness (or ignorance) due to their

nability of providing complete judgments or the lack of informa-

ion. The incompleteness can vary from totally unknown to locally

nknown. For example, if the actual demand varies from 10 to 90

nits, and an expert claims ‘I believe the demand is likely to be in

he range of 50–60 units’. This means that he cannot give a sin-

le forecast but an approximated interval due to his incomplete

nowledge to the future, which can be called as ‘local unknown’.

ut if he gives an interval as [10, 90], this actually means that the

xpert has no idea at all about the future demand, and is called

s ‘global unknown’. According to Knight (1921) , these kinds of


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M. Guo et al. / European Journal of Operational Research 273 (2019) 198–216 199

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ncertainty can be termed as ‘Knight uncertainty’, ‘complex uncer-

ainty’, or ‘ambiguity’, as opposed to ‘risk’ associated with decision

roblems where no objective probability distribution is given over

ystem states. How to formulate these kinds of complex uncertain-

ies and make better decision becomes a challenging problem in

oth SPP and other relevant inventory management areas.

In the literature, much effort has been devoted to the SPP

here the demand does not satisfy the classical assumptions of

aving a specific probability distribution. We can categorize these

esearch studies into two main directions according to their uncer-

ainty formulation methods.

1) Traditional probability theory

In the past decades, a large number of researchers put lots

f efforts to apply the traditional probability theory to the SPP

here the demand does not satisfy the classical assumption of

aving a specific probability distribution with known parameters.

here are essentially two different research streams in terms of

he availability of the information about the demand distribution

nd its parameters: (i) The form of the demand distribution func-

ion is known, but its parameters are unspecified, and (ii) Par-

ial information (e.g., mean, variance, mode, range) about the de-

and distribution is available, but the form of the distribution is

nknown. Within the first research stream, the Bayesian method

as been widely used for estimating the unknown parameters

Azoury, 1985; Iglehart, 1964; Scarf, 1959 ). The Bayesian method

eads to the sequential update of the demand parameter estimates

y means of a pair of prior and posterior density functions as soon

s an additional demand data become available. Some other re-

earchers employed the classical statistical theory for demand es-

imation ( Akcay, Biller, & Tayur, 2011; Chu, Shanthikumar, & Shen,

008; Hayes, 1969; Liyanage & Shanthikumar, 2005; Ritchken &

ankar, 1984; Rossi, Prestwich, Tarim, & Hnich, 2014; Silver & Rah-

ana, 1986; Weerahandi, 1987 ). It is apparent that in these re-

earch studies sufficient historical or revealed demand data should

e available in order to estimate the unknown parameters accu-

ately.

In the second stream of research, Scarf (1958) made one of the

ost important advances to the situation where only the mean

nd variance of demand are assumed to be known, or the demand

as assumed to follow a family of distributions having the same

ean and variance. In this case, the expected total profit or cost

s not a single value as in the traditional stochastic SPP models.

hus some exogenous decision criteria must be applied to ensure a

nique optimal solution. Scarf (1958) also derived the closed-form

ptimal order quantity which maximizes the total profit against

he worst possible distribution of demand, i.e., the conservative

aximin criteria. Since the solution obtained in Scarf’s approach is

ather pessimistic, alternative decision criteria were discussed later,

ncluding the minimax regret ordering rule ( Savage, 1951; Wang,

ebster, & Zhang, 2014; Yue, Chen, & Wang, 2006; Zhu, Zhang,

Ye, 2013 ) and the maximum entropy rule ( Andersson, Jörnsten,

onås, Sandal, & Ubøe, 2013; Eren & Maglaras, 2006; Jaynes, 1957,

003 ).

Following Scarf’s work, a number of researchers have also ex-

mined other families of demand distributions with given param-

ters. For instance, Reyniers (1991) developed a high-low search

lgorithm for SPP under uncertainty rather than risk, and the ac-

ual demand was assumed to be a constant with only lower and

pper bounds known. Similar demand assumptions can be found

n the literature such as Lin and Ng (2011) for a multi-market

PP and Vairaktarakis (20 0 0) for multi-item and budget constraint

ettings. Kumaran and Achary (1996) solved the SPP for demand

ith a generalized λ-type four-parameter class of distributions.

erakis and Roels (2008) considered the maximin and the min-

max regret decision criteria for the capacity allocation problem

n revenue management under general polyhedral uncertainty sets.

pecifically, the demand probability distribution was chosen from

set with the given mean and range (or support) of the ran-

om demand. Ben-Tal, Chung, Mandala, and Yao (2011) proposed

robust-optimization approach ( Bertsimas & Thiele, 2006 ) to both

ingle and multiple product SPP over a set of demand distribu-

ions which are defined within the so-called ‘ φ-divergences’ dis-

ance measure from a given single distribution. Similar approach

an be also found in Carrizosa, Olivares-Nadal, and Ramírez-Cobo,

016 .

In this research stream, how to obtain the required information

emains a question. Sometimes, estimating these parameters is to

ome extent more of an art than a science, especially when the

orecasting is done subjectively.

2) Fuzzy theory

Another research stream of managing demand uncertainties is

o use fuzzy models. Petrovic, Petrovic, and Vujosevic (1996) was

rstly proposed to deal with fuzzy uncertainty in SPP. Their

odels address two cases: (i) uncertain demand is represented

y a discrete fuzzy set on a finite discrete support set, and (ii)

oth demand and inventory costs are estimated as fuzzy numbers.

he concept of 2-level fuzzy set and the method of arithmetic

efuzzification were employed to generate an optimal order quan-

ity. Ishii and Konno (1998) introduced a fuzzy SPP model where

he shortage cost is given by a l -shape fuzzy number. As a result

he total expected profit function becomes a fuzzy number and an

ptimal order quantity is obtained in the sense of a fuzzy min/max

rder based on a kind of partial order preference of fuzzy num-

ers. This approach was further extended by Li, Kabadi, and Nair

2002) and Kao and Hsu (2002) by taking into consideration both

he left and right-shape membership functions. In the recent fuzzy

PP literature, Ryu and Yücesan (2010) considered the optimal

rder quantity and the coordination policies in a single-period

upply chain network case where several fuzzy parameters, e.g.,

he demand, the wholesale price, and the market sales price, were

onsidered in the model. Chen and Ho (2011) also proposed an

nalysis method for the SPP with fuzzy demands and incremental

uantity discounts. Zhang, Lu, Zhang, and Wen (2014) considered

he newsvendor-type supply chain coordination model with the

fuzzy random demand’, i.e., the demand is defined as a random

ariable that is valued as fuzzy numbers.

The basic premise of the Fuzzy SPP literature is that the mem-

ership function of each fuzzy variable must be given in advance.

owever, it is often a problem when information is limited to as-

essing an exact membership function. It has the same drawbacks

s the classical stochastic SPP with the assumption of a specific

istribution with known parameters.

In recent years, the Evidential Reasoning (ER) approach ( Yang,

001; Yang & Sen, 1994 ), which is the advance of the Dempster–

haffer (D–S) theory ( Dempster, 1967; Shafer, 1976 ), sheds new

ight on modelling complex uncertainty problems. The ER approach

ses a distributed expression framework, in which each uncer-

ainty factor is accessed using a set of collectively exhaustive and

utually exclusive assessment grades. Probabilistic uncertainty is

haracterized by a belief structure, which can model both pre-

ise data and various types of uncertainties, such as probabilities,

agueness, local and global ignorance in subjective judgments. For

xample, in order to estimate the demand, experts usually give

heir assessments by means of a set of evaluation grades, e.g.,

Very low, Low, Average, High, Very high}, where each grade can

e associated with a particular reference demand value. For exam-

le, the ‘Very low’ demand can be equivalent to 50 units, similarly

he demand values of 10 0, 20 0, 30 0, 50 0 may be related to the

Low’, ‘Average’, ‘High’, and ‘Very high’ demand respectively. Thus

200 M. Guo et al. / European Journal of Operational Research 273 (2019) 198–216

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according to the ER formulation framework, some evaluations of

demand given by experts can be expressed as follows.

Statement 1: The demand is evaluated to be 100 units of prod-

uct with a chance or belief degree of 80% and 200 units with a

chance of 20%.

The above is a typically probabilistic expression and can be for-

mulated exactly by the ER formulation framework as {(Low, 0.8);

(Average, 0.2)}.

Statement 2: The demand is between 100 and 200 units.

It gives the bounds of the demand and can be expressed in the

ER formulation framework as {(Low-Average, 1.0)}, which means

that the belief degree of 100% can be assigned to either single

value in the grade interval ‘Low-Average’, or arbitrarily distributed

in the same grade interval.

Statement 3: It is certain with a confidence of 80% that the de-

mand is between 100 and 200 units, and certain with 20% that it

is 300 units.

In Statement 3, the belief degree of 80% is given to a grade

interval, which can be regarded as local ignorance, thus by the

ER formulation framework, statement 3 can be denoted as {(Low-

verage, 0.8); (High, 0.2)}.

Statement 4: We are approximately sure with a confidence of

80% that the demand is between 100 and 200 units, otherwise we

have no idea.

In Statement 4, the global unknown of 20% means the actual

demand can take an arbitrary value between the ‘Very low’ and the

‘Very high’ demand. Thus Statement 4 can be expressed as {(Low-

verage, 0.8); (Very low-Very high, 0.2)}.

It can be shown that the above distributed interval grade be-

lief assessment framework of ER has the following advantages in

representing uncertainty. (i) It can represent the quantitative prob-

ability estimation, where each evaluation grade and its reference

value can measure the exact demand value. (ii) It can represent

the subjective and judgmental expressions from experts, especially

when the evaluations must be given to qualitative attributes. (iii)

It has the ability to capture the whole continuum of information

availability from local unknown to global unknown.

In this paper, we aim to analyze SPP by applying the ER

formulation framework to represent the complex uncertainties

of demand assessment. The ER formulation framework advances

from the D–S theory and more broadly from the random set

theory ( Kendall, 1974; Matheron, 1975 ), which can be regarded

as a generalization of the traditional probability theory. Thus

according to the relationship between the random set theory

and the traditional probability theory, we demonstrate that the

demand assessment in term of the ER formulation framework, can

be equivalently transformed to a family of traditional probability

distributions and then the system performances can be derived

and the optimal order quantities can be discussed under differ-

ent decision criteria, including the pessimistic, optimistic, Arrow

and Hurwicz α-maxmin criteria, minimum regret and maximum

entropy criteria. It is very interesting to note that the proposed

uncertainty expression framework and the derived optimization

methodology are closely related to the second research stream

of the traditional probability theory in the literature discussed

above. We prove that the models under study in the paper are the

general forms of the existing approaches and can be applied to

represent and analyze more types of uncertainties.

The rest of the paper is organized as follows. In Section 2 ,

the basic concepts of the random set theory and the ER for-

mulation framework are introduced. In Section 3 , the SPP mod-

els are presented under the uncertainty formulation framework,

and the optimal order policies under various decision criteria are

discussed. Then some numerical examples are demonstrated in

Section 4 . Finally, the paper is concluded with further research

directions.
2
. Basic notions of random set theory and ER uncertainty

xpression framework

.1. Random set theory

In recent years, the advantages of random set theory have at-

racted lots of researchers in uncertainty analyses. Random set the-

ry was originally studied independently by Kendall (1974) and

atheron (1975) in connection with stochastic geometry, general-

zing the notion of a random variable to a random set. Random

et theory provides a general mechanism for handling interval-

ased measurements and discrete probability distributions ( Dubois

Prade, 1991; Hall & Lawry, 2004 ). If fuzzy sets are regarded

s nested families of sets, they can also be handled in ran-

om set theory, providing a convenient link between probabil-

ty theory, possibility theory and fuzzy sets ( Florea, Jousselme,

renier, & Bosséc, 2008; Goodman, 1982; Mahler, 1996 ). Random

et theory can also be linked to Dempster–Shafer theory of evi-

ence. In this section, we give a brief introduction to random set

heory.

Suppose observations are made of a uncertain system state vari-

ble u ∈ �, where � is a finite value set of u . To obtain the uncer-

ain u , instead of evaluate the exact probability on �, questions

uch as ‘if a hypothesis that u is in a subset A of � is true’ are to

e answered. If we let 2 � denote the set of all the subsets of �

i.e., the power set of �), a basic belief assignment function m can

e defined such that,

: 2

� → [0 , 1] , (1)

(φ) = 0 , (2)

∑

∈ 2 �m (A ) = 1 , (3)

here φ denotes the empty set. Thus for a subset A of �, m (A )

easures the basic belief degree that the hypothesis ‘the actual

alue of u is in the subset A ’ is true. If m (A ) > 0 , i.e., A has a posi-

ive belief degree, it is referred to as a focal element. We define the

et that consists all the focal element as U = { A | m (A ) > 0 , A ∈ 2 �} .Because of the presence of imprecision, it is hardly possible to

alculate the probability of a generic u ∈ �. Alternatively, we want

o know the probability P (A ) of a generic subset A ⊆ � (or A ∈ 2 �),

here P (A ) is defined as the probability if A is ‘randomly’ chosen

rom 2 �. According to the random set theory, this probability has

he following characteristics:

el(A ) ≤ P (A ) ≤ P l(A ) , (4)

el(A ) =

∑

B ∈ 2 �,B ⊆A

m (B ) , (5)

l(A ) =

∑

B ∈ 2 �,B ∩ A � = φm (B ) , (6)

here Bel(A ) is the lower bound of P (A ) and it is called belief

easure. Essentially, Bel(A ) can be calculated as the sum of the

asic belief degree of those focal elements contained in A . P l(A )

s the upper bound of P (A ) and it is called plausibility measure.

l(A ) can be calculated as the sum of the basic belief degree of

hose focal elements having some elements in common with A .

n the other sense, there exists a ‘random’ set variable U with

alues in 2 � admitting precisely m (. ) as its probability density

unction, and Bel(. ) as its distribution function, in the sense that

(A ) = P (U = A ) , and Bel(A ) = P (U ⊆ A ) (please refer to Nguyen,

006 for details).


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The above mentioned procedures are to answer the question

f ‘How to model the uncertainty affecting the information avail-

ble on a system state variable’. However, what we care about

ore in real problems is ‘How to propagate uncertainty to the

ystem response, given complex uncertain system state variables

s system inputs’. By using random set theory, this challeng-

ng problem can be solved in the following way (Tonon et al.,

004) .

Let y = y (u ) , y : � → �′ be a function of u , where �′ is an fi-

ite value set of y . One is interested in getting information about

given the uncertainty measure of u (if the uncertainty of u is

easured by a random set variable U with values in 2 �, the ba-

ic belief assignment function m (. ) and its focal element set U),

hat is to find a propagated random set variable Y , with values

n 2 �′ , the basic belief assignment function m

′ (. ) and its focal

lement set Y . This is accomplished by the following extension

rinciple:

= { Y i | Y i = y (A ) , A ∈ U} , (7)

′ ( Y i ) =

∑

A : Y i = y (A ) ,A ∈ U m (A ) , (8)

here Y i ∈ Y is the image of the focal point A ∈ U through the func-

ion y = y (u ) . According to the extension principle, m

′ ( Y i ) , which is

he belief degree of Y i , can be obtained by the sum of all the belief

egree of A ∈ U that have the same image of Y i .

.2. The ER uncertainty formulation framework

The Evidential Reasoning approach is originally developed for

ggregating multiple attributes based on a belief decision matrix

nd the revised evidence combination rule of D–S theory in the

ultiple attribute decision analysis (MADA) problems under uncer-

ainty. Here we only introduce the interval grade ER formulation

ramework which is an extension of the original ER ( Xu, Yang, &

ang, 2006 ).

Suppose an uncertain variable u is assessed using the a set of

individual assessment grades H i , i = 1 , . . . , N, which are mutually

xclusive and collectively exhaustive. For example, if N = 5, we can

enote H 1 , H 2 , . . . , H 5 as linguistic terms ‘Very low’, ‘Low’, ‘Average’,

High’ and ‘Very high’ respectively. In order to make effective as-

essment, we assume each assessment grade H i is associated with

reference value u i , thus when u = u i , we can evaluate u exactly

s grade H i . It is noted that the reference values u i , i = 1 , . . . , N

re not required to be evenly distributed, and for a given u , if

i < u < u i +1 , then the assessing belief masses are to be divided

etween grade H i and H i +1 (see Yang, 2001 for details). Without

oss of generality, it is assumed that grade H i +1 is preferred to

rade H i for i = 1 , . . . , N − 1 (It implies for a profit attribute that

he reference value u i +1 of grade H i +1 is higher than u i of H i . It is

he opposite for a cost attribute. For example, if we assess a car’s

uel economic performance as ‘Very poor, Poor, Average, Good and

xcellent’ according to its fuel consumption value, then a higher

eference value of fuel consumption will correspondent to a lower

ssessment grade.)

According to Xu et al. (2006) , because u can be assessed to an

ndividual grade or a grade interval, the complete set of all indi-

idual grades and grade intervals denoted by ˆ H can be represented

y

ˆ = { H i j , i = 1 , . . . , N, j = i, . . . , N} , r equivalently

ˆ =

⎧ ⎪ ⎪ ⎪ ⎪ ⎨

⎪ ⎪ ⎪ ⎪ ⎩

H 11 H 12 · · · H 1(N−1) H 1 N

H 22 · · · H 2(N−1) H 2 N

. . . . . .

. . . H (N−1)(N−1) H (N−1) N

H NN

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎭

, (9)

here H ii ( i = 1 , ... N) denotes an individual grade and it is equiv-

lent to the individual grade H i . H i j ( i = 1 , . . . , N, j = i + 1 , . . . , N)

enotes the grade interval which has the lowest individual grade

i and the highest individual grade H j .

In assessing the uncertainty of u according to a grade set ˆ H , we

efine m i j ≥ 0 , j ≥ i , as the belief degree exactly assessed to the

rade interval H i j , thus the overall assessment of u is given by a

elief distribution with respect to the grade set ˆ H and can be ex-

ressed as an upper triangle matrix { m i j , i = 1 , . . . , N, j = i, . . . , N} ,r equivalently

m 11 m 12 · · · m 1 ,N−1 m 1 N

m 22 · · · m 2 ,N−1 m 2 N

. . . . . .

. . . m N −1 ,N −1 m N−1 ,N

m NN

⎫ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎭

(10)

It is assumed that ∑

i =1 , ... ,N j= i, ... ,N

m i j = 1 . For notational simplicity, we

rite ∑

i =1 , ... ,N j= i, ... ,N

as ∑

i, j

henceforth. Similarly we write ∑

i, j: i =1 , ... ,N,

j= i, ... ,N,

...

as ∑

i, j: ...

nd

∑

i =1 , ... ,N

as ∑

i

and so on.

Based on the above assumption, the assessment of u is then

iven by

( H i j , m i j ) ; i = 1 , . . . , N, j = i, . . . , N} . (11)

Note that the above ER formulation framework is a special

ase of the general discrete random set since the evaluation

rade matrix ˆ H is a subset of the power set of individual grades

H 1 , H 2 , . . . , H N } .

. The ER and random set based SPP model

.1. Basic notations

Consider a newsvendor who sells newspapers to local cus-

omers. Early each morning, he needs to determine an ‘optimal’ or-

ering quantity Q from his supplier in order to satisfy customers’

emand denoted as D for the newspapers. The supplier is assumed

o operate with no capacity restrictions and zero lead time of sup-

ly. Thus an order placed by the newsvendor to the supplier in

he morning is immediately fulfilled. The unit purchase cost for

he newspapers is c and the unit selling price is r. The sales of

he newspapers occur during or at the end of the day and,

(1) if Q ≥ D , then Q − D newspapers are left over at the end of

the period and will have the recycle value s per unit; and,

(2) if Q < D , then D − Q newspapers are out of stock and will

have a shortage penalty cost l per unit.

To avoid trivial cases, we assume 0 ≤ s ≤ c ≤ r.

The actual end of period profit for the newsvendor is

(Q, D ) = −cQ + r min { Q, D } + s (Q − D ) + − l (D − Q ) +

= −(r − s ) (Q − D ) + − l (D − Q ) + + (r − c) Q, (12)

here (x ) + = max { x, 0 } .


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o

E

I

l

m

s

I

I

I

T

D

D

I

If the demand is given by a random variable D with the cumu-

lative distribution function �(·) , according to the traditional news-

boy problem analysis, the optimal order quantity Q

∗ can be deter-

mined as ( Axäter, 2015 )

�( Q

∗) =

r − c + l

r − s + l . (13)

In this paper, in order to handle various types of uncertainties

from different sources, demand is represented by the ER formula-

tion framework. In order to evaluate demand, we first define a set

of N individual assessment grades H i , i = 1 , . . . , N, and a set of N

reference values of demand { D 1 , D 2 , . . . , D N } can be defined corre-

spondingly. In other words, if the demand equals to the reference

value of demand D i , it is evaluated exactly as grade H i , i = 1 , . . . , N.

It is assumed that grade H i +1 is preferred to grade H i , and D i +1 > D i

for i = 1 , . . . , N − 1 . We also assume that the realized demand can

take any real number in the interval [ D 1 , D N ] .

According to Section 2.2 , in order to assess the uncertainty

of the demand, we define m i j ≥ 0 , j ≥ i , as the belief degree as-

sociated with the grade interval H i j , thus the overall assessment

of demand is given by a belief distribution { m i j , i = 1 , . . . , N, j =i, . . . , N} and we assume

∑

i, j

m i j = 1 .

Given the above mentioned assessment scheme of demand be-

lief distribution, we can derive an equivalent probability distri-

bution family using the random set theory. Firstly let the inter-

val [ D 1 , D N ] be split evenly into a series of K discrete values as

x 1 , x 2 , . . . , x K , where K is a sufficiently large number (For example

K = 10 0 0), thus in theory, any real value in the interval [ D 1 , D N ]

can be included in this series. Assume that all the reference de-

mand values in { D 1 , D 2 , . . . , D N } are also included in the series.

Then we can derive an equivalent family of probability distribu-

tions P , in which a particular (singleton) probability distribution

P = { p x k , k = 1 , . . . , K} ∈ P can be regarded as one of the realiza-

tions of the belief distribution matrix { m i j , i = 1 , . . . , N, j = i, . . . , N},

where p x k denotes the probability of the demand being equal to x k .

To do this, we assume that each m i j > 0 , i � = j, is divided into a se-

ries of belief degrees as m i j, x k ≥ 0 , for all x k such that D i ≤ x k ≤ D j ,

and

∑

k : k =1 , ... ,K, D i ≤x k ≤D j

m i j, x k = m i j , where m i j, x k

refers to the belief degree

that will be eventually assigned to x k . Note that the diagonal ele-

ments in matrix { m i j }, i.e., m ii can only be assigned to x k = D i . We

simply write ∑

k =1 , ... ,K

as ∑

k

and

∑

k : k =1 , ... ,K, ...

as ∑

k : ...

henceforth. Thus P

can be defined as

P = { P | P = { p x k =

∑

i, j: D i ≤x k ≤D j

m i j, x k , k = 1 , . . . , K} ,

s.t. ∑

k : D i ≤x k ≤D j

m i j, x k = m i j , m i j, x k

≥ 0 , f or all

i = 1 , . . . , N, j = i, . . . , N} (14)

It is easy to verify that for a singleton probability distribution

P = { p x k , k = 1 , . . . , K} ∈ P , ∑

k

p x k = 1 will hold since ∑

i, j

m i j = 1 .

3.2. The pessimistic, optimistic and Arrow and Hurwicz α-maxmin

criteria

The most conventional decision criteria is to maximize the ex-

pected profit, i.e., E P [ C(Q, D )] , if the demand D follows a given sin-

gleton probability distribution P . However, under the assessment

scheme of demand belief distribution in Section 3.1 , the expected

profit is somehow ambiguous since P can be arbitrarily chosen

from the family of distributions P defined by expression (14) . To

make it clear, we define the pessimistic and optimistic expected

rofit of the Random Set based SPP (RS-SPP) as

C (pe ) (Q ) = min

P∈ P E P [ C(Q, D )] = min

P∈ P

∑

k

[ p x k · C(Q, x k )] , (15)

C (op) (Q ) = max P∈ P

E P [ C(Q, D )] = max P∈ P

∑

k

[ p x k · C(Q, x k ) ] , (16)

here in Eq. (15) , the pessimistic expected profit of RS-SPP is de-

ned as the lowest expected profit for ∀ P ∈ P . While in Eq. (16) ,

he optimistic expected profit of RS-SPP is defined as the highest

xpected profit.

We can show that Eqs. (15) and (16) can be simplified as

he following equivalent expressions, whose proofs are given in

ppendix A :

C (pe ) ( Q ) =

∑

i, j

[ m i j · min

x = D i or D j C(Q, x )] , (17)

C (op) ( Q ) =

∑

i, j

[ m i j · max D i ≤x ≤D j

C(Q, x )] . (18)

In Eqs. (17) and (18) , both the pessimistic and optimistic ex-

ected profit functions are equivalently expressed by the belief de-

rees m i j , i = 1 , . . . , N, j = 1 , . . . , N. In Eq. (17) , in order to minimize

he profit function C(Q, x ) , x will be chosen between D i and D j .

hile in Eq. (18) , x is not necessarily equal to D i or D j and it

an be obtained by solving a general maximizing problem. The

onvexities of the above two expected profits can be proved at

roposition 1 .

roposition 1. Both the pessimistic and optimistic expected profit

unctions in Eqs. (17) and (18) are convex in Q .

Proof. See Appendix B .

Based on the pessimistic and optimistic expected profits, the

rrow and Hurwicz α-maxmin decision RS-SPP model can be de-

ned as

ax Q

[ α · E C (pe ) (Q ) + (1 − α) · E C (op) (Q )] , (19)

here the coefficient α lying between 0 and 1 is interpreted as

measure of the decision maker’s pessimism. When α = 1 , the

bjective function in expression (19) is analogous to E C (pe ) ( Q ) in

q. (17) , while it is analogous to E C (op) ( Q ) in Eq. (18) when α = 0 .

n fact, model (19) can be solved by the following scenario-based

inear programming model (proofs are given in Appendix C ):

ax z =

∑

i, j

m i j · [ α · T C (pe ) i, j

+ (1 − α) · T C (op) i, j

] (20)

.t. For all i = 1 , . . . , N, j = i, . . . , N and k = i, j,

P O

(pe ) i, j,k

≥ Q − D k , (21)

NE (pe ) i, j,k

≥ −Q + D k , (22)

P O

(pe ) i, j,k

≥ 0 , I NE (pe ) i, j,k

≥ 0 , (23)

C (pe ) i, j

≤ −(r − s ) · IP O

(pe ) i, j,k

− l · INE (pe ) i, j,k

+ (r − c) Q, (24)

L i, j ≥ D i , (25)

L i, j ≤ D j , (26)

P O

(op) i, j

≥ Q − D L i, j , (27)


I

I

T

w

I

n

c

(

d

t

d ∑

e

∑

w

t

b

p

t

2

[

t

d

0

E

m

w

(

m

w

d

(

Q

w

P

m

m

3

r

(

d

m

m

w

p

o

o

s

m

E

C

r

a

m

s

P

fi

q

w

i

t

a

m

d

o

t

D

f

Q

w

3

d

d

o

p

R

i

NE (op) i, j

≥ −Q + D L i, j , (28)

P O

(op) i, j

≥ 0 , I NE (op) i, j

≥ 0 , (29)

C (op) i, j

= −(r − s ) · IP O

(op) i, j

− l · INE (op) i, j

+ (r − c) Q, (30)

here subsidiary variables T C (pe ) i, j

, T C (op) i, j

, I P O

(pe ) i, j,k

, I NE (pe ) i, j,k

, I P O

(op) i, j

,

NE (op) i, j

, D L i, j are introduced in the model in order to transfer the

on-linear objective function in expression (19) to a set of linear

onstraints and an objective function.

It is evident that our proposed models defined by expression

19) and expressions ( 20 - 30 ) are the generalizations of the tra-

itional stochastic SPP model in which the demand is assumed

o follow a given singleton discrete probability distribution. To

emonstrate this statement, we can let m i j = 0 for all i � = j, thus

i

m ii = 1 , and both the right hand sides of Eqs. (17) and (18) are

qual to,

i

[ m ii · C(Q − D i )] ,

hich is consistent with the expected profit function of the tradi-

ional SPP model in which the demand follows a discrete proba-

ility distribution, and more specifically the demand is D i with the

robability being equal to m ii , i = 1 , 2 , . . . , N.

The proposed model defined by expression (19) with α = 1 is

he generalization of robust optimization approach ( Vairaktarakis,

0 0 0 ), in which demand is estimated within an interval, say

D L , D H ] , where D L , D H denote the lower and upper bounds of

he demand respectively. Illustratively, we can set the reference

emand values as D 1 = D L , and D N = D H , and let m 1 N = 1 , m i j = for all i, j except for i = 1 , j = N. In this setting, maximizing

q. (17) becomes

ax Q

E C (pe ) ( Q ) = max Q

· min

x = D 1 or D N C(Q, x ) , (31)

hich is consistent with the following robust optimization model

Vairaktarakis, 20 0 0 ) according to the proof in Appendix B ,

ax Q

E C (ro) ( Q ) = max Q

· min

D 1 ≤x ≤D N C(Q, x ) . (32)

here E C (ro) ( Q ) denotes the expected profit function given Q un-

er the robust optimization criterion.

It is easy to verify that both models defined by expression

31) and (32) will yield the ‘optimal robust solution’ as

(ro) =

(r − s ) D 1 + l D N

r − s + l =

(r − s ) D L + l D H

r − s + l .

Considering the optimistic model given by Eq. (19) with α = 0 ,

e have the following characteristic of the optimal order quantity.

roposition 2. The optimal solution of the optimistic RS-SPP

odel is Q = D n , where D n , 1 ≤ n ≤ N, is one of the reference de-

and values, and it satisfies the following condition,

∑

i, j: i ≤n −1 , j≤n −1

m i j − l ∑

i, j: i ≥n, j≥n

m i j ≤r − c

r − s and

∑

i, j: i ≤n, j≤n

m i j − l ∑

i, j: i ≥n +1 , j≥n +1

m i j ≥r − c

r − s .

(33)

Proof. See Appendix D .

.3. The minimax regret criterion

Assuming the decision maker is regret aversion, the minimax

egret criterion is to minimize the worst-case regret. From Savage

1951) , given an order quantity Q , the ‘regret’ of measuring the ad-

itional profit can be obtained with full information about the de-

and probability distribution P ∈ P , i.e.,

ax q

E P [ C(q, D )] − E P [ C(Q, D )] ,

here q is the order quantity to be chosen to maximize the ex-

ected profit given the probability distribution P .

The maximum regret can be seen as the maximum price, and

ne would have to know the exact demand distribution. Thus the

ptimal order quantity of RS-SPP under the minimax regret deci-

ion criterion is to solve the following problem,

in

Q max

P∈ P { max

q E P [ C(q, D )] − E P [ C(Q, D )] } . (34)

Since the actual end of period profit for the newsvendor in

q. (12) can be reformulated as

(Q, D ) = (r − s + l)[ min (Q, D ) − c − s

r − s + l Q − l

r − s + l D ] . (35)

Now let β =

c−s r−s + l . Then the problem given by Eq. (34) can be

eformulated as follows, by inverting the order of maximization

nd minimization:

in

Q · max

q ρ(q, Q ) (36)

.t. ρ(q, Q ) = max P∈ P

E P [ min (D, q ) − min (D, Q )] + β(Q − q ) . (37)

roposition 3. The optimal order quantity Q

(re ) of the model de-

ned in Eqs. (36) and (37) , satisfies the condition that q 1 ≤ Q

(re ) ≤ 2 and,

E P 1 [ min (D, Q

(re ) )] − E P 2 [ min (D, Q

(re ) )] = E P 1 [ min (D, q 1 )]

−E P 2 [ min (D, q 2 )] + β( q 2 − q 1 ) . (38)

here P 1 is a singleton probability distribution obtained by assign-

ng the belief degree m i j > 0 for all i = 1 , . . . , N and j = i, . . . , N to-

ally to the demand reference value D i , and P 2 is a singleton prob-

bility distribution obtained by assigning m i j > 0 totally to the de-

and value D j , q 1 and q 2 denote the optimal order quantities (as

efined in Eq. (13) ) under P 1 and P 2 respectively.

Proof. See Appendix E .

It is easy to verify that our proposed result is the generalization

f the interval SPP model. If we assume that the demand lies in

he interval [ D L , D H ] , we can set the reference demand grade as

1 = D L , and D N = D H . Let m 1 N = 1 and m i j = 0 for all i, j except

or i = 1 , j = N. Applying our result yields

( D L − D L ) + β(Q − D L ) = ( D H − Q ) + β(Q − D H ) .

Thus we have the optimal order quantity as

= βD L + (1 − β) D H ,

hich is consistent with the result of Perakis and Roels (2008) .

.4. The maximum entropy criterion

Entropy maximization is not a criterion for decision making un-

er uncertainty. Instead, it is a criterion for selecting a probability

istribution as an input to a stochastic model. Laplace’s principle

f insufficient reason states that, with no information available, all

ossible outcomes should be considered as equally likely ( Luce &

aiffa, 1957 ). Jaynes (1957; 2003 ) generalized this principle by tak-

ng into consideration the distribution that maximizes the entropy


(

Fig. 1. Distribution of the sales.

i

t

2

d

t

a

b

s

o

(

format as {(10–90, 1.0)}.

over the set of distributions. The entropy of a probability distri-

bution represents the amount of uncertainty associated with the

distribution. The distribution that maximizes the entropy is thus

a good prior distribution because it is the ‘maximally noncommit-

tal with regard to missing information’ ( Jaynes, 1957 ). The RS-SPP

under the maximum entropy criterion can be expressed as

max P∈ P

(

−∑

k

p x k · ln p x k

)

, (39)

where P , P and x k , k = 1,…, K , are defined in Section 3.1 .

Under the maximum entropy criterion formulate above, the op-

timal singleton probability distribution P (me ) ∈ P is chosen to be

distributed as evenly as possible, and it has the following property:

Proposition 4. The optimal probability distribution of RS-SPP un-

der the maximum entropy criterion has the following property. In

each interval ( D i , D i +1 ) , i = 1 , . . . , N − 1 , all the demand values x k(i.e., D i < x k < D j ) have the equal probability value.

Proof. See Appendix F .

Proposition 4 means that we need only consider the belief

degree assigned to each of the single reference demands D i , i =1 , . . . , N, and the intervals ( D i , D i +1 ) , i = 1 , . . . , N − 1 , thus we can

simplify the calculations during the use of the algorithms to calcu-

late maximum entropy probability densities ( Meyerowitz, Richman,

& Walker, 1994 ).

4. Numerical examples and applications

As an example, we consider a company in China, which pro-

duces and sells various functional costumes. Now the company has

developed a new swimming suit. The sales of the suit are strongly

seasonal, mainly from June to September. After this period, since

the costs related to the logistics process (i.e., transportation, sales

promotion, etc.) are high compared with the production costs (in-

cluding material price, manufacturing costs, etc.), the leftover suits

will be sold at a clearance price to improve the company’s cash

flow. Every year, the company has to purchase raw materials and

arrange manufacturing in advance, and there is no opportunity

to replenish inventory once the season has begun. As a conse-

quence, the company must forecast the future demand and decide

the quantity to produce before the selling season. Thus it can be

formulated as a typical newsvendor problem. In this example, the

unit product cost is c = 20 CNY (China Yuan), the unit selling price

r = 30 CNY, the clearance or recycle value per unit s = 10 CNY, and

the shortage penalty cost per unit l = 30 CNY. We assume a set of

assessing grades with N = 9 (more grades can be implemented in

a similar way), i.e., { H 1 , H 2 , . . . , H 9 }, which is correspondent to the

demand reference values (10, 20,…, 90) (multiplied by ten thou-

sand units). Note for illustrative purpose the data presented in this

section have been altered, however, the resulting qualitative rela-

tionships and insights drawn from this example are the same as

they would be from using the actual data.

Comparing with the most existing SPP models with non-

probabilistic uncertainties, our proposed methodology is quiet gen-

eral, i.e., do not need any assumption of the demand, thus it is ap-

plicable to the problems with very complex demand uncertainties.

It is apparently we cannot analyze all these complex cases because

of the limitation of space although we try to address the most typ-

ical situation in the real application.

In general, the information for demand forecasting can be col-

lected from several sources, for example,

1) Source 1: Since it is a fully new product, there is no historical

sales data. However if it is believed that the new product has

some common characteristics with a similar product. We can

make use of the historical sales data of this similar product al-

though we know that the lost sales were not recorded in the

sales. To do this, we firstly divide the entire range of demand

values into a series of intervals (called bins) and then count

how many demand values fall into each interval. For example, if

we have 10 0 0 samples of the historian sales, and there are 100

samples with the sales value within the interval [15, 25] (mul-

tiplied by ten thousand units), then the frequency of this bin

is 0.1 as shown in Fig. 1 . Thus the historical sales of this sim-

ilar product can be represented by ER formulation format as:

{(20, 0.1); (30, 0.1); (40, 0.3); (50, 0.4); (60, 0.1)}, where (20,

0.1) means there are 10% chances or belief degrees that the de-

mand value is 20 ten thousand units and so on. Note hereafter

for clarity we use the demand reference values instead of the

grades in the ER format as defined in expression (11) .

In order to simplify our analysis, we set the center point of each

nterval/bin exactly equal to one of the reference demand value of

he grade, for example, the second bin is set as the interval [15,

5], which is correspondent to the grade H 2 with the reference

emand value as 20. However, this setting will lead to some ex-

ra errors. In fact, a better ‘binning’ method, for example, can set

n interval/bin such as [10, 20], then the frequency between this

in can be assigned to a grade interval H 12 . It is not within the

cope of the paper to explore which ‘binning’ method is the best

r more appropriate for a specific situation.

1) Source 2: Experts can give their demand forecasting from their

different perspectives, and their forecasting values apparently

may be very complex and disperse. For example, if there are

two experts, Expert 1 mainly considers the sales in terms of

the characteristics of the new product, and believes that this

new product completely meets a group of customers’ prefer-

ence. Thus the sales are believed to be high and the value will

be in the range of (60–90) ten thousand units. Besides this, Ex-

pert 1 also thinks that the sales value may not be uniformly

distributed in this minimum and maximum bounds if consider-

ing the possibility, he thinks the most possible (more than 50%)

sales value is 80 ten thousand units, while he cannot give any

other concrete information. Thus according to Expert 1, his as-

sessment can be given by the ER format as {(80, 0.5); (60–90,

0.5)}. While Expert 2 argues that the future sales of the new

product is totally unclear since the economic trends is highly

fluctuating, and thus his assessment is formulated by the ER


I

w

W

E

0

(

a

p

p

4

p

p

a

s

m

l

0

s

B

(

0

S

a

t

a

b

a

P

w

(

v

v

b

s

t

b

s

e

c

g

g

e

P

} )

Fig. 2. The profit function C(Q, D ) with D = 60.

i

w

e

a

b

H

m

m

l

n

b

E

t

t

f

e

�

m

t

i

m

l

o

c

s

2

t

s

d

s

m

p

m

a

v

a

t

Thus the final forecasting should combine the above sources.

f we assume the combining weight of Source 1 is 0.6, and the

eight of each of the two expert’s judgments in Source 2 is 0.2.

e can obtain the final demand forecasting assessments by the

R combination algorithms as: {(20, 0.0762); (30, 0.0762); (40,

.2286); (50, 0.3048); (60, 0.0875); (80, 0.0635); (60-90, 0.0635);

10-90, 0.1015)}. Apparently, this combined assessment is complex

nd also very informative about the future demand. The detailed

rocedure of the ER combination algorithm is not the focus of this

aper, and it can be found in Yang (2001) and Xu et al. (2006) .

.1. Numerical analyze of a simplified example

In order to explain the computational details of the above pro-

osed methodology systematically, we consider a simplified exam-

le in this subsection, and a more complicated example will be

dopted in the next subsection in order to further analyze the sen-

itivity properties of the proposed models. We assume the final de-

and forecasting is obtained from two sources: Source 1, the simi-

ar product’s historical sales data {(20, 0.1); (30, 0.1); (40, 0.3); (50,

.4); (60, 0.1)} with a weight of 0.6, and Source 2, we only con-

ider the opinion of Expert 2 {(10-90, 1.0)} with a weight of 0.4.

y the ER combination algorithm, the final forecasting assessment

denoted as S 19) is {(20, 0.0789); (30, 0.0789); (40, 0.2368); (50,

.3158); (60, 0.0789); (10-90, 0.2105)} or,

19 = { m 22 = 0 . 0789 ; m 33 = 0 . 0789 ; m 44 = 0 . 2368 ;m 55 = 0 . 3158 ; m 66 = 0 . 0789 ; m 19 = 0 . 2105 ;the other m i j unmentioned are zeros } . (40)

It is worth noting in expression (40) that there is approximately

belief degree of 21% assigned to the grade interval H 19 . According

o the random set theory, if we assume the realized demand values

re chosen from the set {10, 20,…, 90}, then this expression can

e equivalently transformed to a family of random distributions P

s,

= { P | P = { p 10 = m 19 , 1 , p 20 = m 22 + m 19 , 2 , p 30 = m 33 + m 19 , 3 ,

p 40 = m 44 + m 19 , 4 , p 50 = m 55 + m 19 , 5 , p 60 = m 66 + m 19 , 6 ,

p 70 = m 19 , 7 , p 80 = m 19 , 8 , p 90 = m 19 , 9 | 9 ∑

k =1

m 19 ,k = m 19 ,

m 19 ,k ≥ 0 , k = 1 , . . . , 9 } (41)

here p 10 denotes the probability of the demand value being 10

ten thousand units), p 20 denotes the probability of the demand

alue being 20,…, and so on, and m 19, k , k = 1,…,9 is an auxiliary

ariable that denotes a portion of the belief degree m 19 which will

e eventually reassigned to single grade H k if S 19 is realized to a

pecific probability distribution. According to this transformation,

he belief degrees expressed by the ER formulation framework can

e interpreted by this family of probability distributions. In expres-

ion (40) , if the belief degree is evaluated to a grade interval, for

xample, m 19 = 0.2105, it means that the same value of probability

an be redistributed to any subset of this grade interval, such as

rade intervals H 12 , H 13 , . . . , H 89 etc., and finally to any individual

rade sets, such as H 11 , H 22 , . . . , H 99 etc. It is also easy to rewrite

xpression (41) as follows,

= { P | P = { p 10 , p 20 , . . . , p 90 | 0 ≤ p 10 ≤ 0 . 2105 ,

0 . 0789 ≤ p 20 ≤ 0 . 2894 , 0 . 0789 ≤ p 30 ≤ 0 . 2894 ,

0 . 2368 ≤ p 40 ≤ 0 . 4473 , 0 . 3158 ≤ p 50 ≤ 0 . 5263 ,

0 . 0789 ≤ p 60 ≤ 0 . 2894 , 0 ≤ p 70 ≤ 0 . 2105 ,

0 ≤ p 80 ≤ 0 . 2105 , 0 ≤ p 90 ≤ 0 . 2105 , p 10 + p 20 + ... + p 90 = 1

(42

From expression ( 41 - 42 ), it is evident that if the belief degree

s exactly assigned to a single grade, for example m 22 = 0.0789

hich is exactly assigned to grade H 22 with the demand ref-

rence value of 20, it actually means that the minimum prob-

bility of this demand value is 0.0789. Since the maximum

elief degree of m 19 , 2 , which is finally assigned to the grade

22 , is equal to m 19 = 0.2105, thus the maximum of p 20 is

22 + m 19 = 0.0789 + 0.2105 = 0.2894. In the similar way, the mini-

um and the maximum of p 10 , p 30 , p 40 , . . . , p 90 can also be calcu-

ated as shown in expression (42) .

In comparison, we firstly consider a traditional stochastic

ewsvendor model in which the demand forecasting is solely given

y Source 1:{(20, 0.1); (30, 0.1); (40, 0.3); (50, 0.4); (60, 0.1)} and

xpert 2 ′ s opinion of Source 2 is completely ignored. According to

he standard newsvendor analysis, if the demand follows a tradi-

ional probability distribution with continuous cumulative density

unction �(·) , an optimal order quantity Q

∗ satisfies the following

quation:

( Q

∗) =

r − c + l

r − s + l = 0 . 80 .

Thus according to the above formula, we can obtain this opti-

al order quantity as Q

∗ = 50 (ten thousand units). It is obvious

hat this result is imperfect, since all the uncertainties contained

n Expert 2 ′ s opinion have not been considered.

Now we consider the problem under the setting of the final de-

and assessment given by S 19. Given an order quantity Q , the be-

ief distribution of the total profit C(Q, D ) can also be expressed by

ur proposed ER uncertainty formulation scheme, and can be cal-

ulated using the extension theorem of the random set theory as

hown in Eqs. (7) and (8) .

In order to assess the profit, we firstly construct a set of

1 evaluation grades, and their profit reference values are set

o − 10 0 0, − 90 0,…, − 10 0, 0, 10 0,…, 10 0 0 ten thousand CNY, re-

pectively. Then, each demand reference value and its correspon-

ent profit value according to the function C(Q, D ) is calculated as

hown in Fig. 2 (where Q = 60), and each grade interval (or a de-

and reference value interval) and its image of the function map-

ing can be derived. Note that we only need to consider the de-

and reference values or intervals with positive masses.

For example, in S 19, m 22 = 0.0789 means that the demand is ex-

ctly equal to 20 with a mass value of 0.0789. Since this demand

alue D = 20 has a definite profit value as − 200 ten thousand CNY

ccording to the function C(Q, D ) when Q = 60. This means the ac-

ual profit is exactly − 200 ten thousand CNY with the same mass


Fig. 3. The mass density, belief and plausibility distributions of the profit.

C

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value of 0.0789. Similarly, m 33 = 0.0789, means that the demand

value D = 30, and its profit C(Q, D ) when Q = 60 is equal to 0 CNY,

means the profit is exactly 0 CNY with the same mass value of

0.0789. For m 44 = 0.2368, we can derive that the profit is exactly

200 ten thousand CNY with the same mass value of 0.2368, and

so on, for m 55 = 0.3158, the profit is exactly 400 ten thousand CNY

with the same mass value of 0.3158; for m 66 = 0.0789, the profit is

exactly 600 ten thousand CNY with the same mass value of 0.0789.

While for m 19 = 0.2105, this means the demand is between an in-

terval [10, 90], since the function C(Q, D ) is continuous with D ,

thus we know the profit is also within the interval [ C min, C max],

where

min = min

10 ≤D ≤90 , Q=60

C(Q, D ) = −400 ,

max = max 10 ≤D ≤90 , Q=60

C(Q, D ) = 600 .

Thus for m 19 = 0.2105, we know the profit C(Q, D ) takes values

in [ − 400, 600] with the mass value of 0.2105.

In summary, when the demand is evaluated as S 19 and Q = 60,

the mass density distribution of the profit can be evaluated by the

ER uncertainty formulation framework, which are {( − 200, 0.0789);

(0, 0.0789); (200, 0.2368); (400, 0.3158); (600, 0.0789); ([ − 400,

600], 0.2105)}. In this example for simplicity, the profit reference

values are carefully chosen that the resulting profit C(Q, D ) is ex-

actly equal to one of the profit reference values when the demand

takes one demand reference value. However, this is not necessarily

in general. If the resulting profit is not equal to one of the profit

reference values, the mass belonging to this profit can be redis-

tributed between the two adjacent profit reference values linearly.

The mass distribution of the profit is important, especially

when the decision maker wants to know the complete vision of

the final results if there are complex uncertainties. In the above

calculation, the uncertain parts of the demand (i.e., m 19 = 0.2105

in S 19) will definitely result in the propagated uncertainties in the

total profit. Thus, according to our methodology, these uncertain-

ties can also be evaluated and expressed by the above ER formu-

lation framework. The belief and plausibility distributions of the

profit can also be calculated by Eqs. (5) and (6) and are shown in

Fig. 3 , the decision maker may be able to give more scientific de-

cisions given these complete visions of the resulting profit without

losing some important scenarios.

It is apparent that the decision is not necessarily unique if the

problem is associated with ‘deep’ uncertainties. In our proposed

RS-SPP models we try to help the decision maker to choose the

ost suitable solution, and the following decision criteria can be

pplied.

1) The pessimistic, optimistic and Arrow and Hurwicz α-maxmin

criteria

In this example, the expected profits of our proposed pes-

imistic and optimistic models can be seen as the upper and lower

ounds of the cases as shown in Fig. 4 , when the demand belief

istribution contains ‘less uncertainty’. For example, the following

wo demand belief distributions, S 11 and S 99, can be regarded as

he ‘realized’ demand belief distributions of S 19 in expression (40) ,

here in S 11 the total unknown m 19 = 0.2105 is assigned to H 1 (or

he demand is realized to D 1 ) and in S 99 it is assigned to H 9 (or

he demand is realized to D 9 ):

11 = { m 11 = 0 . 2105 ; m 22 = 0 . 0789 ; m 33 = 0 . 0789 ;m 44 = 0 . 2368 ; m 55 = 0 . 3158 ; m 66 = 0 . 0789 ;the other m i j unmentioned are zeros } , (43)

99 = { m 22 = 0 . 0789 ; m 33 = 0 . 0789 ; m 44 = 0 . 2368 ;m 55 = 0 . 3158 ; m 66 = 0 . 0789 ; m 99 = 0 . 2105 ;the other m i j unmentioned are zeros } . (44)

Although S 11, S 99 and S 19 both mean families of single prob-

bility distributions, it is easy to verify that the families of sin-

le probability distributions defined by S 11, S 99 are ‘included’ in

he family of S 19, thus S 11, S 99 can be regarded as the ‘realized

ess uncertain’ belief distributions of S 19. Thus the expected prof-

ts of S 11 and S 99 must always lie between the pessimistic and

ptimistic expected profits of S 19.

More demand belief distributions that are ‘included’ in S 19

uch as S 22 , S 33 , S 44 , S 55 , S 66 , S 77 , S 88 (which have the sim-

lar definitions to S 11 and S 99) can be considered. For example,

33 = { m 22 = 0.0789; m 33 = 0.0789 + 0.2105 = 0.2894; m 44 = 0.2368;

55 = 0.3158; m 66 = 0.0789; the other m i j unmentioned are zeros},

hen we fix the order quantity Q = 50, both the pessimistic and

ptimistic expected profits of the case S 33 are 265.77 (ten thou-

and CNY), and they lie between the interval of the pessimistic

nd optimistic expected profits of case S 19 as shown in Fig. 5 . And

nother example, S 46 = { m 22 = 0.0789; m 33 = 0.0789; m 44 = 0.2368;

55 = 0.3158; m 66 = 0.0789; m 46 = 0.2105; the other m i j unmen-

ioned are zeros}, compared with S 19, ( m 19 = 0.2105) is replaced

y ( m 46 = 0.2105), thus S 46 can also be regarded as one of the ‘re-

lized’ demand belief distributions of S 19, thus the pessimistic and

ptimistic expected profits of S 46 must lie between the interval


Fig. 4. The expected profits with respect to Q .

S11 S22 S33 S44 S55 S66 S77 S88 S99 S12 S13 S14 S15 S16 S17 S18 S19

Exp

ecte

d Pr

ofit

Optimistic Pessimistic Average

S11 S22 S33 S44 S55 S66 S77 S88 S99 S23 S45 S56 S46 S57 S37 S39 S79 S19

Exp

ecte

d P

rofit

Optimistic Pessimistic Average

Fig. 5. The optimistic and pessimistic expected profits with Q = 50 and different demand belief distributions.

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or S 19. Other cases such as S 13 , S 14 , S 15 , S 17 , S 45 , S 56 , S 57 , S 37

nd S 39 can also be found in Fig. 5 . Furthermore, Fig. 5 also shows

ore general relationships between one demand belief distribution

nd its ‘realized’ and ‘included’ belief distributions. For example,

he pessimistic and optimistic expectation profits of S 46 are the

pper and lower bounds of those of S 44, S 55, S 66, S 45 and S 56.

From the calculation of the optimal order quantity in case of

19, it is apparent that our proposed models under the optimistic,

essimistic and Arrow and Hurwicz α-maxmin decision criteria

iffer from the traditional stochastic SPP model. Under the op-

imistic criterion, we can show that the optimal order quantity

s 50, which is equal to one of the demand reference values, as

e have proved in Proposition 2 . But it is noted that the op-

imal solution cannot be obtained simply by assigning the un-

nown ( m 19 = 0.2105) to either demand grade D 1 or D 9 as shown

n Fig. 4 (where these two cases are denoted by S 11 and S 99 re-

pectively). While under the pessimistic criterion, the optimal so-

ution is not necessarily equal to one of the demand reference val-

es. In the same example, when Q is given, ( m 19 = 0.2105) will be

ssigned to one of the endpoints in [ D 1 , D 9 ] in the calculation of

aximum T C (pe ) under the pessimistic criterion, but the resulting

ptimal order quantity is 58. We note that when Q = 58 , the ex-

ected profit under the probability distribution S 11 where m 19 is

otally assigned to D 1 is equal to that of S 99 where m 19 is assigned

o D 9 .

Thus according to our example, in our RS-SPP model, unlike

he traditional stochastic SPP models where this unknown is com-

letely ignored, the existing unknown ( m 19 = 0.2105), can be ex-

licitly expressed and the ‘optimal’ order quantity is carefully con-

idered under various decision preferences/criteria. For example, if


Fig. 6. The optimal order quantities of the Arrow and Hurwicz α-maxmin RS-SPP model.

(

Fig. 7. The computation of the optimal order quantity under the Minimax Regret

Criterion.

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the decision maker is optimistic, he only considers to maximize

the highest possible profit ( α = 0 , i.e., the optimistic case as shown

in Fig. 4 ), thus the resulting order quantity is Q = 50. While if he is

a pessimistic decision maker, what he consider is the lowest pos-

sible profit, the problem turns to maximize this lowest possible

profit, thus the optimal order quantity according to his attitude is

Q = 58 ( α = 1 , i.e., the pessimistic case as shown in Fig. 4 ). Other

cases such as (α = 0 . 5) can also be found in Fig. 4 .

Further experiments (as shown in Fig. 6 , where we consider

cases with l = 0, 10, 20,…, 100 respectively) also show that the op-

timal order quantity under the Arrow and Hurwicz α- maxmin cri-

terion always equals to one of the solutions under the pessimistic

and optimistic criteria.

2) The minimax regret criterion

Given the belief distribution of the demand as S 19 in expres-

sion (40) , we can find the two extreme distribution as S 11 where

the unknown m 19 = 0.2105 is assigned to D 1 , and S99 where m 19

is assigned to D 9 . Thus according to the results of the traditional

stochastic SPP model, we have q 1 = 50 and q 2 = 90.

In order to find the optimal order quantity Q of the RS-SPP

model under the minimax regret criterion, we assume that Q and

q take values as 10, 11,…, 90 ten thousand units. Table 1 shows

part of the calculation table of ρ(q, Q ) . As proved in Appendix

E , considering the lower-left triangular part of matrix ρ(q, Q ) ,

each item in the row q = q 2 = 90 is the maximum of its same col-

umn; while considering the upper-right triangular part of ρ(q, Q ) ,

each item in the row q = q 1 = 50 is the maximum of its same col-

umn. Thus in order to solve max q ρ(q, Q ) , we need only com-

pare these two rows. From Table 1 , it can be observed that in

the gray part of the q = q 1 row, the value of ρ(q, Q ) is increasing

from 0 to 7.211, while in the grey part of the q = q 2 row, it is de-

creasing from 1.2090 to 0, and we observe that ρ(50 , 55)= 0.6055,

ρ(50 , 56)= 0.7266, ρ(90 , 55)= 0.7620, ρ(90 , 56)= 0.6726, thus the

optimal order quantity which is the solution of min Q · max q ρ(q, Q )

must be between 55 and 56. Computational experiments with

more densely distributed Q and q serial values in Table 1 b)

indicate this optimal value is Q

(re ) ≈55.74, and ρ( q 1 , Q

(re ) ) =ρ( q 2 , Q

(re ) ) ≈ 0 . 695 .

The logic of the above mentioned calculation procedure can

also be explained as follows. In this case where the demand is

19, under the Minimax Regret Criterion, the maximum of the re-

ret is to be minimized. From Proposition 3 , in order to maximize

he regret in the family of the singleton probability distributions

s defined by Eqs. (41) and (42) , we need only consider the two

xtreme single probability distributions S 11 and S 99 as defined in

qs. (43) and (44) :

ax { E P 1 [ C( q 1 , D )] − E P 1 [ C(Q, D )] , E P 2 [ C( q 2 , D )] − E P 2 [ C(Q, D )] } (45)

here P 1 in this case is S 11, and P 2 is S 99.

Then, the above expression is to be minimized to find the op-

imal quantity Q = Q

(re ) . For illustration purposes, the function

P 1 [ C( q 1 , D )] − E P 1 [ C(Q, D )] and E P 2 [ C( q 2 , D )] − E P 2 [ C(Q, D )] with Q

re shown in Fig. 7 , it is easy to observe that the two functions

ntersect at Q = Q

(re ) where their maximum function defined in

xpression (45) is minimized.


Table 1

ρ(q, Q ) .

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3) The maximum entropy criterion

We firstly define a series of demand values x k as 10, 11, …, 90,

.e., K = 81. According to Proposition 4 , for the demand belief dis-

ribution S 19 under the maximum entropy criterion, the unknown

m 19 = 0.2105) will be distributed among the above demand series

s uniformly as possible, thus the selected probability distribution

(me ) = {(20, 0.0789); (30, 0.0789); (40, 0.2368); (50, 0.3158); (60,

.0789); other x k is with a probability 0.2105/(81 − 5) = 0.00277}.

ccording to the results of the traditional stochastic SPP model,

he optimal order quantity under the maximum entropy criterion

s Q

(me ) = 50.

.2. Numerical analysis for the more complicated example

In this subsection, we analyze the more complicated example

s mentioned in the beginning of this section. i.e., the demand is

valuated as {(20, 0.0762); (30, 0.0762); (40, 0.2286); (50, 0.3048);

60, 0.0875); (80, 0.0 635); (60-90, 0.0 635); (10-90, 0.1015)} which

s combined from Source 1 and Source 2 with the two expert’s

udgments. In this mass distribution, besides the mass assigned

xactly to the six reference demand values, there are two appar-

nt demand unknowns, one is between 60-90 ten thousand units

ith mass value of 6.35%, the other is between 10-90 ten thousand

nits, which is a global unknown and with mass value of 10.15%.

According to the same procedures as shown in Section 4.1 , it is

asy to derive the mass distribution of the output profit when Q is

iven. For this example, (20, 0.0762) means m 22 = 0.0762, the de-

and is exactly 20 ten thousand units with a mass value of 0.0762.

ince this demand value D = 20 has a definite profit value as − 200

en thousand CNY according to the function C(Q, D ) when Q = 60.

his means the actual profit is exactly − 200 ten thousand CNY

ith the same mass value of 0.0762. Similarly, (30, 0.0762) means

= 0.0762, the demand D = 30 and its correspondent profit is
33
qual to 0 CNY with the same mass value, and so on. (80, 0.0635)

eans m 88 = 0.0635, the demand D = 80 and its correspondent

rofit is equal to 0 CNY with the mass value of 0.0635.

Now we consider the two unknowns. (60-90, 0.0635) means

69 = 0.0635, the demand is between an interval [60, 90] (mul-

iplied by ten thousand units), since the function C(Q, D ) is con-

inuous with D , thus we know the resulting profit is also between

n interval [ − 30 0, 60 0] (multiplied by ten thousand CNY). (10-90,

.1015) means m 19 = 0.1015 and the resulting profit is also between

n interval [ − 400, 600].

In the above calculation, only when the demand D = 30 and

= 80, their resulting profits are same. Thus according to the ex-

ension principle defined in Eqs. (7) and (8) , the mass value of the

rofit reference value 0 CNY is equal to the sum of those of D = 30

nd D = 80, i.e., 0.0762 + 0.0635 = 0.1397.

In summary, the mass density distribution of the profit can be

valuated by the ER uncertainty formulation framework as {( − 200,

.0762); (0, 0.1397); (200, 0.2286); (400, 0.3048); (600, 0.0875);

[ − 30 0, 60 0], 0.0635); ([ − 40 0, 60 0], 0.1015)}.

Fig. 8 shows the sensitive analysis of the optimal order quanti-

ies under the Arrow and Hurwicz α-maxmin Criterion. From these

umerical analyses, the optimal order quantity when α= 0.5 is al-

ays between the quantities for α= 0.0 and α= 1.0. Unlike the tra-

itional SPP model as shown in Eq. (13) , the optimal order quantity

n our proposed models is more complicated and not necessarily

onotonously increasing or decreasing with any of the parameters.

The sensitivity analysis of the optimal order quantities under

he Minimax Regret Criterion are performed as shown in Fig. 9 ,

here Q

(re ) denotes the optimal order quantities under the Mini-

ax Regret Criterion, q 1 denotes the optimal order quantity calcu-

ated by Eq. (13) if the demand follows a singleton probability dis-

ribution P 1 that is obtained by assigning the belief degree m i j > 0

or all i = 1 , . . . , N and j = i, . . . , N total to the demand reference


Fig. 8. Sensitivity analysis of the optimal order quantities under the Arrow and Hurwicz α-maxmin Criterion.

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value D i , and q 2 denotes that of P 2 , i.e., a singleton probability

distribution obtained by assigning m i j > 0 totally to the demand

value D j . From these numerical analyses, it is easy to verify that

q 1 ≤ Q

(re ) ≤ q 2 , and Q

(re ) does not monotonously increasing or de-

creasing with any of these parameters.

5. Conclusions

Demand uncertainty is one of the most important factors affect-

ing the effectiveness of an inventory management system. Dealing

with various types of uncertainty has been the focus of a wide

range of research in inventory management. Many researchers

have concentrated on the randomness aspect of uncertainty and

developed a lot of stochastic modelling techniques under certain

probability assumptions. But in reality, it is not always possible for

the decision-maker to gather enough historical data. In fact, exter-

nal demand is often estimated imprecisely based on experts’ judg-

ments, experience and intuitions. So it is necessary to explore the

methodology to handle various uncertainties in the decision prob-

lems.

In this paper, we proposed an ER based interval grade belief

uncertainty expression framework of SPP, This framework is quite

general and can be applied to represent a variety of complex un-

ertainties, including probability, interval as well as global and lo-

al ignorance. Unlike most of the traditional SPP models, this un-

ertainty expression framework is basically non-parametric, i.e., it

akes fewer assumptions about the uncertainty, and it is espe-

ially applicable to the environment which is complex, ambiguous,

r lack of statistical data.

Our proposed RS-SPP model aims to understand and measure

he propagation of these uncertainties. In order to do so, we em-

loyed the random set theory to calculate and express the uncer-

ainty of output in the same ER based interval grade belief uncer-

ainty expression framework, thus the decision maker can have a

omplete vision of the systems output (e.g. the total profit in our

umerical example) given the uncertain inputs (e.g. the uncertain

emand in our example). As a result a good and reliable decision

an be made under this complete vision of uncertainties.

In this paper, we also investigated the ‘optimal’ order quanti-

ies under various decision conditions such as the optimistic, pes-

imistic, Arrow and Hurwicz α-maxmin, minimax regret and max-

mum entropy criteria. We proved that some of these ‘optimal’ or-

er quantities have the close form results in our RS-SPP model.

The work discussed in this paper provides a fundamental step

o solve more complex inventory problems, such as multiple pe-

iod periodic review inventory problems or multi-echelon systems.


Fig. 9. Sensitive analysis of the optimal order quantities under the Minimax Regret Criterion.

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here is no question that there remain many uncertain resources

o be considered in our model, including lead time, cost, supply

nd uncertain relationships between demand and price. Further re-

earch will be conducted to address these issues.

ppendices: Proofs

ppendix A: Proof of Eqs. (17 ) and (18)

According to the relationship between the random set theory

nd the probability theory, we rewrite Eqs. (15) and (16) and

qs. (17) and (18) in terms of the probability family P defined in

q. (14) .

From Eq. (14) , a single distribution P = { p x k , k = 1 , . . . , K} ∈ P

an be constructed as

p x k =

∑


m i j, x k , k = 1 , . . . , K (A-1)

By substituting Eq. (A-1) , the pessimistic expectation profit

unction in Eq. (15) can be written as,

C (pe ) (Q ) = min

P∈ P

∑

k

[ p x k · C(Q, x k )]

= min

∑

k

∑


[ m i j, x k · C(Q, x k )]

= min

∑

i, j

∑


[ m i j, x k · C(Q, x k )] (A-2)

Now we consider the term

∑


[ m i j, x k · C(Q, x k )] , since the

onstraint ∑


m i j, x k = m i j is independently affecting the dis-

ribution of m i j, x k , its value at ( i = i 1, j = j 1) is independent to the

ne at ( i = i 2, j = j 2), where i 1 and i 2 are two given values of i , j 1

nd j 2 are two given values of j , and i 1 � = i 2 or j 1 � = j 2. Thus we can

ewrite Eq. (A-2) as

C (pe ) (Q ) =

∑

i, j

min

∑

k : D i ≤x ≤D j

[ m i j, x k · C(Q, x k )] .

Now we consider the minimum of ∑

k : D i ≤x ≤D j

[ m i j, x k · C(Q, x k )]

iven i and j as follows.


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1) When Q < D i , ∑


[ m i j, x k · C(Q, x k )]

=

∑


m i j, x k [ −(r − s ) (Q − x k )

+ − l ( x k − Q ) + + (r − c) Q]

=

∑


m i j, x k [ −l( x k − Q ) + (r − c) Q]

Since ∑

k : D i ≤x ≤D j

m i j, x k = m i j , it is obvious that m i j [ −l( D j − Q ) +

(r − c) Q] is the minimum of the above term, i.e., under the pes-

simistic criterion all m i j is assigned to p x k = D j .

2) When Q > D j , ∑


m i j, x k C(Q − x k )

=

∑


m i j, x k [ −(r − s ) (Q − x k )

+ − l ( x k − Q ) + + (r − c) Q]

=

∑


m i j, x k [ −(r − s )(Q − x k ) + (r − c) Q]

It is easy to know that m i j [ −(r − s )(Q − D i ) − (r − c) Q] is the

minimum, i.e., all m i j is assigned to p x k = D i .

3) When D i ≤ Q ≤ D j , ∑


m i j, x k C(Q, x k )

=

∑


m i j, x k [ −(r − s ) (Q − x k )

+ − l ( x k − Q ) + + (r − c) Q]

≤ min { m i j [ −(r − s )(Q − D i ) + (r − c) Q] ,

m i j [ −l(D j − Q ) + (r − c) Q] } . This means that all m i j is assigned to either p x k = D i or p x k = D j .

Therefore, min

∑


m i j, x k C(Q, x k ) can always be written as

min

x k = D i , or D j

[ m i j · C(Q, x k )] .

Substituting x k with x completes the proof.

Similarly, for the optimistic profit function in Eq. (16)

E C (op) (Q ) = max ∑

P∈ P

∑

k

[ p x k · C(Q, x k )]

= max ∑

k

∑


[ m i j, x k · C(Q, x k )]

= max ∑

i, j

∑


[ m i j, x k · C(Q, x k )]

=

∑

i, j

max ∑


[ m i j, x k · C(Q, x k )]

1) When Q < D i , ∑


[ m i j, x k C(Q, x k )]

=

∑


m i j, x k [ −(r − s ) (Q − x k )

+ − l ( x k − Q ) + + (r − c) Q]

=

∑

m i j, x k [ −l( x k − Q ) + (r − c) Q]


It is easy to verify that m i j [ −l( D i − Q ) + (r − c) Q] is the maxi-

um, i.e., all m i j is assigned to p x k = D i .

2) When Q > D j , ∑


m i j, x k C(Q, x k )

=

∑


m i j, x k [ −(r − s ) (Q − x k )

+ − l ( x k − Q ) + + (r − c) Q]

=

∑


m i j, x k [ −(r − s )(Q − x k ) + (r − c) Q]

It is known that m i j [ −(r − s )(Q − D j ) + (r − c) Q] is the maxi-

um, i.e., all m i j is assigned to p x k = D j .

3) When D i ≤ Q ≤ D j ,

max ∑


m i j, x k [ −(r − s ) (Q − x k )

+ − l ( x k − Q ) + + (r − c) Q]

= m i j [ −(r − s ) (Q − x k ) + − l ( x k − Q ) + + (r − c) Q]

∣∣x k = Q

= m i j (r − c) Q

The maximum will be achieved exactly when all m i j > 0 is as-

igned to p x k = Q , then −(r − s ) (Q − x k ) + − l ( x k − Q ) + is equal to

ero. Note here we assume K is sufficiently large, thus x k can sat-

sfy x k = Q exactly when Q can take an arbitrary real value in the

nterval ( D i , D j ) . Finally, we can substitute x k with x in the above

ormula to get,

C (op) (Q) =

∑

i, j

{ m i j · max D i ≤x ≤D j

[ − (r − s ) (Q − x ) +

−l (x − Q ) + + (r − c) Q] } . ppendix B: Proof of Proposition 1

B.1 Each term in the objective function of the pessimistic

odel in Eq. (17) , i.e., m i j · min x = D i ,or D j [ C(Q, x )] , can be rewritten as

i j · min [ C(Q, x ) | x = D i , C(Q, x ) | x = D j ] , thus it is the minimum of the

wo separated functions, which are both convex. Thus E C (pe ) (Q) is

onvex.

B.2 For each term in Eq. (18) , i.e., m i j · max D i ≤x ≤D j [ −(r −

) (Q − x ) + − l (x − Q ) + + (r − c) Q] ,

When Q ≤ D i , stock-outs occur, the above term is increasing

ith x , therefore its maximum can be achieved at x = D i .

When Q ≥ D j , unsold stocks occur, the term is decreasing with

, therefore the maximum can be achieved at x = D j .

When D i ≤ Q ≤ D j , the minimization operator will ensure x =, therefore the term is always equal to zero.

Combining all of the above cases, we know E C (op) (Q) is convex.

ppendix C: Proof of the equivalence of the model defined by

q. (19) and the model by Eqs. (20) –(30)

C.1 We firstly prove that for a given Q , the model defined by

q. (19) will have the same objective value as that of model de-

ned by Eqs. (20) –(30) .

Since the optimistic and pessimistic model are independent on

ach other, we only need to prove that for a given Q:

min

= D i or D j C(Q, x ) = max T C (pe )

i, j with constraints ( 21 −24 ) (C-1)

nd

max i ≤x ≤D j

C(Q, x ) = max T C (op) i, j

· with constraints ( 25 −30 ) (C-2)

For (C-1) , we consider the following three cases:


(

I

−

c

I

t

c

(

I

−

c

I

t

c

T

(

l

−

c

I

t

c

T

t

(

w

D

(

w

D

x

t

(

A

l

E

E

w

E

o

t

(

x

t

1) When Q < D i , constraints (21) –(24) become,

INE (pe ) i, j,k

≥ −Q + D k ,

I P O

(pe ) i, j,k

≥ 0 , I NE (pe ) i, j,k

≥ 0 ,

T C (pe ) i, j

≤ −(r − s ) · IP O

(pe ) i, j,k = i − l · INE (pe )

i, j,k = i + (r − c) Q, (C-3)

T C (pe ) i, j

≤ −(r − s ) · IP O

(pe ) i, j,k = j − l · INE (pe )

i, j,k = j + (r − c) Q (C-4)

With the above constraints, max [ −(r − s ) · IP O

(pe ) i, j,k = i − l ·

NE (pe ) i, j,k = i + (r − c) Q] will make IP O

(pe ) i, j,k = i = 0 and INE

(pe ) i, j,k = i =

Q + D i , therefore in this case IP O

(pe ) i, j,k = i = (Q − D i )

+ and INE (pe ) i, j,k = i =

( D i − Q ) + . Similarly, max [ −(r − s ) · IP O

(pe ) i, j,k = j − l · INE

(pe ) i, j,k = j + (r −

) Q] will make IP O

(pe ) i, j,k = j = 0 and INE

(pe ) i, j,k = j = −Q + D j , that is,

P O

(pe ) i, j,k = j = (Q − D j )

+ and INE (pe ) i, j,k = j = ( D j − Q ) + .

The constraints (C-3) and (C-4) ensure that T C (pe ) i, j

will choose

he minimum one between −(r − s ) · IP O

(pe ) i, j,k = i − l · INE

(pe ) i, j,k = i + (r −

) Q and −(r − s ) · IP O

(pe ) i, j,k = j − l · INE

(pe ) i, j,k = j + (r − c) Q .

Thus, min

x = D i or D j

C (Q, x ) = max T C (pe ) i, j

.

2) When D i ≤ Q < D j , constraints (21) –(23) become,

I P O

(pe ) i, j,k = i ≥ Q − D i , I P O

(pe ) i, j,k = j ≥ 0 ,

I NE (pe ) i, j,k = i ≥ 0 , I NE (pe )

i, j,k = j ≥ −Q + D j ,

With the above constraints, max [ −(r − s ) · IP O

(pe ) i, j,k = i − l ·

NE (pe ) i, j,k = i + (r − c) Q] will make IP O


(pe ) i, j,k = i =


(pe ) i, j,k = i = (Q − D i )


( D i − Q ) + . Similarly, max [ −(r − s ) · IP O

(pe ) i, j,k = j − l · INE

(pe ) i, j,k = j + (r −

) Q] will make IP O



P O

(pe ) i, j,k = j = (Q − D j )



will choose


(pe ) i, j,k = i − l · INE

(pe ) i, j,k = i + (r −

) Q and −(r − s ) · IP O

(pe ) i, j,k = j − l · INE

(pe ) i, j,k = j + (r − c) Q .

hus , min

x = D i or D j C(Q, x ) = max T C (pe )

i, j .

3) When Q ≥ D j , constraints (21)–(23) become,

I P O

(pe ) i, j,k

≥ Q − D k , I NE (pe ) i, j,k

≥ 0 , k = i, j.

Thus with the above constraints, max [ −(r − s ) · IP O

(pe ) i, j,k = i −

· INE (pe ) i, j,k = i + (r − c) Q] will make IP O


(pe ) i, j,k = i =


(pe ) i, j,k = i = (Q − D i )


( D i − Q ) + .Similarly, max [ −(r − s ) · IP O

(pe ) i, j,k = j − l · INE

(pe ) i, j,k = j + (r −

) Q] will make IP O



P O

(pe ) i, j,k = j = (Q − D j )



will choose


(pe ) i, j,k = i − l · INE

(pe ) i, j,k = i + (r −

) Q and −(r − s ) · IP O

(pe ) i, j,k = j − l · INE

(pe ) i, j,k = j + (r − c) Q .

hus , min

x = D i or D j C(Q, x ) = max T C (pe )

i, j .

For (C-2) , we rewrite x as D L i, j , thus Eqs. (25) and (26) are equal

o D i ≤ x ≤ D j . Now we consider the following two cases:

1) When D L i, j ≤ Q , constraints (27) –(29) become,

I P O

(op) i, j

≥ Q − D L i, j , I NE (op) i, j

≥ 0 ,

Thus max T C (op) i, j

= −(r − s ) · IP O

(op) i, j


+ (r − c) Q

ill make IP O

(op) i, j

= Q − D L i, j and INE (op) i, j

= 0 , therefore in this case

max i ≤x ≤D j


.

2) When D L i, j > Q , constraints (27)–(29) become,

I P O

(op) i, j

≥ 0 , I NE (op) i, j

≥ D L i, j − Q,

Thus max T C (op) i, j

= −(r − s ) · IP O

(op) i, j


+ (r − c) Q

ill make IP O

(op) i, j

= 0 and INE (op) i, j

= D L i, j − Q , therefore in this case

max i ≤x ≤D j


.

C.2 From Proposition 1 , we know both max D i ≤x ≤D j

C(Q, x ) and

min

= D i or D j

C(Q, x ) are convex in Q , thus the optimal Q is unique,

hus from Section C.1 , we can conclude that model defined by Eqs.

19) and model by Eqs. (20) –(30) yield the same result. QED.

ppendix D: Proof of Proposition 2

In general, the optimal order quantity Q must satisfy the fol-

owing two conditions,

C (op) (Q ) ≥ E C (op) (Q + ) , (D-1)

C (op) (Q ) ≥ E C (op) (Q − ) , (D-2)

here > 0 is a sufficiently small positive real number, and

C (op) (Q ) =

∑

i, j


[ −(r − s ) (Q − x ) +

−l (x − Q ) + + (r − c) Q] } . The optimal order quantity Q must be either exactly equal to

ne of the reference demand values or lying in the interval of these

wo reference demand values.

D.1 Without loss of generality, we first assume the optimal Q ∈( D n , D n +1 ) . Thus the left side of the inequality sign in condition

D-1) can be separated into the following three parts,

E C (op) (Q )

=

∑

i, j: i ≤n, j≤n


[ −(r − s ) (Q − x ) + − l (x − Q ) + + (r − c) Q] }

+

∑

i, j: i ≥n +1 , j≥n +1


[ −(r − s ) (Q − x ) + − l (x − Q ) + + (r − c) Q] }

+

∑

i, j: i ≤n, j≥n +1

m i j · max D i ≤x ≤D j

[ −(r − s ) (Q − x ) + − l (x − Q ) + + (r − c) Q] } ,

Consider the first term above, since the realized demand

is always less than Q , thus the term can be replaced by∑

i, j: i ≤n, j≤n

max D i ≤x ≤D j

[ − ( r − s ) (Q − x ) + (r − c) Q] m i j , after the maximiza-

ion operation, it becomes ∑

i, j: i ≤n, j≤n

[ − ( r − s ) (Q − D j ) + (r − c) Q] m i j .


i

u

e

t

e

{

p

c

E

t

c

i

T

p

l

p

Consider the second term above, since x is always larger than Q ,

it becomes ∑

i, j: i ≥n +1 , j≥n +1

max D i ≤x ≤D j

[ −l(x − Q ) + (r − c) Q] m i j , after the maxi-

mization operation, it becomes ∑

i, j: i ≥n +1 , j≥n +1

[ −l( D i − Q ) + (r − c) Q] m i j .

To attain the maximum for the third term, x = Q must be sat-

isfied, thus this term equals to ∑

i, j: i ≤n, j≥n +1

m i j (r − c) Q .

Similarly, since Q + ∈ ( D n , D n +1 ] , the right side of the in-

equality sign in condition (D-1) can also be separated into three

parts, i.e.,

E C (op) (Q + )

=

∑

i, j: i ≤n, j≤n


[ −(r − s ) (Q + − x ) + − l (x − Q − ) +

+(r − c)(Q + )] } +

∑

i, j: i ≥n +1 , j≥n +1


[ −(r − s ) (Q + − x ) + − l (x − Q − )

+

+(r − c)(Q + )] } +

∑

i, j: i ≤n, j≥n +1


[ −(r − s ) (Q + − x ) + − l (x − Q − )

+

+(r − c)(Q + )] } ,

where the first term above can be reformulated as ∑

i, j: i ≤n, j≤n

[ −(r −

s )(Q + − D j ) + (r − c)(Q + )] m i j , the second term becomes∑

i, j: i ≥n +1 , j≥n +1

[ −l( D i − Q − ) + (r − c)(Q + )] m i j , the third term be-

comes ∑

i, j: i ≤n, j≥n +1

[(r − c)(Q + )] m i j .

Thus condition ( D-1 ) can be derived as

(c − s ) ∑

i, j: i ≤n, j≤n

m i j + (c − r − l) ∑

i, j: i ≥n +1 , j≥n +1

m i j − (r − c) ∑

i, j: i ≤n, j≥n +1

m i j ≤ 0 .

(D-3)

Now consider condition ( D-2 ), since Q − ∈ [ D n , D n +1 ) , both

its right and the left side can be separated into three parts,

i.e., ∑

i, j: i ≤n, j≤n

.. +

∑

i, j: i ≥n +1 , j≥n +1

.. +

∑

i, j: i ≤n, j≥n +1

.. . In particular, the right side is

∑

i, j: i ≤n, j≤n

[ −(r − s )(Q − − D j ) + (r − c)(Q − )] m i j +

∑

i, j: i ≥n +1 , j≥n +1

[ −l( D i −

Q + ) + (r − c)(Q − )] m i j +

∑

i, j: i ≤n, j≥n +1

(r − c)(Q − ) m i j .

Similar to the derivation of condition ( D-1 ), the condition ( D-2 )

finally becomes,

(c − s ) ∑

i, j: i ≤n, j≤n

m i j + (c − r − l) ∑

i, j: i ≥n +1 , j≥n +1

m i j − (r − c) ∑

i, j: i ≤n, j≥n +1

m i j ≥ 0 .

(D-4)

Combining the results of (D-3) and (D-4), we have,

(c − s ) ∑

i, j: i ≤n, j≤n

m i j + (c − r − l) ∑

i, j: i ≥n +1 , j≥n +1

m i j − (r − c) ∑

i, j: i ≤n, j≥n +1 ,

m i j = 0 .

This means the optimal Q ∈ ( D n , D n +1 ) is generally not existent,

.e., the optimal Q must be chosen from the reference demand val-

es, until the above condition is exactly satisfied, this is when the

xpected profit is indifferent among all values of the order quan-

ity Q in the interval ( D n , D n +1 ) .

D.2 Now we consider the case when Q / ∈ ( D n , D n +1 ) , i.e., Q is

xactly equal to one of the reference demand values in the set

D 1 , D 2 , . . . , D N } . Without loss of generality, we assume Q = D n .

The left side in condition ( D-1 ) can be separated into the three

arts as: E C (op) (Q ) =

∑

i, j: i ≤n, j≤n

.. +

∑

i, j: i ≥n +1 , j≥n +1

.. +

∑

i, j: i ≤n, j≥n +1

.. . The right side in

ondition ( D-1 ) can also be separated into the three parts as:

C (op) (Q + ) =

∑

i, j: i ≤n, j≤n

.. +

∑

i, j: i ≥n +1 , j≥n +1

.. +

∑

i, j: i ≤n, j≥n +1

.. .

By applying the same procedures in Section D.1, condi-

ion ( D-1 ) becomes (c − s ) ∑

i, j: i ≤n, j≤n

m i j + (c − r − l) ∑

i, j: i ≥n +1 , j≥n +1

m i j − (r −

) ∑

i, j: i ≤n, j≥n +1

m i j ≥ 0 .

Since 1 − ∑

i, j: i ≤n, j≤n

m i j −∑

i, j: i ≥n +1 , j≥n +1

m i j =

∑

i, j: i ≤n, j≥n +1

m i j , we have,

∑

i, j: i ≤n, j≤n

m i j − l ∑

i, j: i ≥n +1 , j≥n +1

m i j ≥r − c

r − s . (D-5)

Similarly, the left side in condition ( D-2 ) can also be separated

nto the three parts as: E C (op) (Q) =

∑

i, j: i ≤n −1 , j≤n −1

.. +

∑

i, j: i ≥n, j≥n

.. +

∑

i, j: i ≤n −1 , j≥n

.. .

he right side in condition (2) can be separated into the three

arts as: E C op (Q − ) =

∑

i, j: i ≤n −1 , j≤n −1

.. +

∑

i, j: i ≥n, j≥n

.. +

∑

i, j: i ≤n −1 , j≥n

.. .

Thus the condition ( D-2 ) becomes ( c − s ) ∑

i, j: i ≤n −1 , j≤n −1

m i j + ( c − r −

) ∑

i, j: i ≥n, j≥n

m i j − (r − c) ∑

i, j: i ≤n −1 , j≥n

m i j ≤ 0 , and then

∑

i, j: i ≤n −1 , j≤n −1

m i j − l ∑

i, j: i ≥n, j≥n

m i j ≤r − c

r − s . (D-6)

Combining the result of Section D.1 with (D-5) and (D-6) com-

lete the proof. QED.


A

ρ .

ρ

==

=

i

w

g

k

m

ρ

k

(

i

c

t

(

x

c

D

0

0

0

0

q

Q

q=q1

Q=Q(re) Q=Qb

0

0

Decreasing in q

Increasing in q

0 Increasing in Q( )reρ

0q=q2 Decreasing in Q ( )reρ

Q=Qa

Fig. E1. Table ρ(q, Q ) .

v

a

w

p

s

t

p

t

(

Q

u

o

t

(

Q

c

o

t

ρ

ρ

a

i

ρ

c

m

ρ

p

ppendix E: Proof of Proposition 3

E.1 First of all, we prove that

(q, Q ) =

{E P 1 [ min (D, q ) − min (D, Q )] + β(Q − q ) , Q ≥ q E P 2 [ min (D, q ) − min (D, Q )] + β(Q − q ) , Q ≤ q

(E-1)

According to Eqs. (14) and (37) ,

(q, Q ) = max P∈ P

E P [ min (D, q ) − min (D, Q )] + β(Q − q )

max ∑

P∈ P

∑

k

{ p x k [ min (D, q ) − min (D, Q )] } + β(Q − q )

max ∑

k

∑


m i j, x k [ min ( x k , q ) − min ( x k , Q )] + β(Q − q )

max ∑

i, j

∑


m i j, x k [ min ( x k , q ) − min ( x k , Q )] + β(Q − q )

Now consider the term

∑


m i j, x k [ min ( x k , q ) − min ( x k , Q )] ,

ts value at ( i = i 1, j = j 1 ) is independent to that at ( i = i 2, j = j 2),

here i 1 and i 2 are two given values of i , j 1 and j 2 are two

iven values of j , and i 1 � = i 2 or j1 � = j2 . Note that the constraint∑

: D i ≤x k ≤D j

m i j, x k = m i j is independently affecting the distribution of

i j, x k . Thus we can rewrite this term as

(q, Q ) =

∑

i, j

max ∑


m i j,x [ min ( x k , q ) − min ( x k , Q )]

+ β(Q − q ) .

In the following we need only consider the maximization of∑

: D i ≤x k ≤D j

m i j, x k [ min ( x k , q ) − min ( x k , Q )] given i and j .

1) When q ≤ Q

In this case, min ( x k , q ) − min ( x k , Q ) =

{

0 x k ≤ q

q − x k q ≤ x k ≤ Q

q − Q Q ≤ x k This means the function min ( x k , q ) − min ( x k , Q ) is decreasing

n x k .

Thus the maximum of ∑


m i j, x k [ min ( x k , q ) − min ( x k , Q )]

an be achieved when the belief degree m i j > 0 is totally assigned

o D i . If we denote P 1 as this particular distribution, we have,

ρ(q, Q ) =

∑

i, j

m i j [ min ( D i , q ) − min ( D i , Q )] + β(Q − q )

= E P 1 [ min (D, q ) − min (D, Q )] + β(Q − q )

2) When Q ≤ q

In this case, min ( x k , q ) − min ( x k , Q ) =

{

0 x k ≤ Q

x k − Q Q ≤ x k ≤ q

q − Q q ≤ x k This means the function min ( x k , q ) − min ( x k , Q ) is increasing in

k .

Thus the maximum of ∑


m i j, x k [ min ( x k , q ) − min ( x k , Q )]

an be achieved when the belief value m i j is totally assigned to

j . If we denote P 2 as this particular distribution, we have,

ρ(q, Q ) =

∑

i, j

m i j [ min ( D j , q ) − min ( D j , Q )] + β(Q − q )

= E P 2 [ min (D, q ) − min (D, Q )] + β(Q − q )

E.2 Proof of the optimal condition in Eq. (38) .

Firstly we assume the order quantity Q can take a series of NQ

alues as Q 1 , Q 2 , . . . , Q NQ , let Q 1 = D 1 , Q NQ = D N , and Q i < Q i +1 for

ll i = 1 , . . . , NQ − 1 ,where NQ is a given integer and NQ N , and

e assume that this series is densely enough to include all the

ossible values of Q in the interval [ D 1 , D N ] , and q will take the

ame series of values. Then we can use this two series to calculate

he values of ρ(q, Q ) in a NQ by NQ table as shown in Fig. E1 . Ap-

arently each diagonal element in the table is zero. And it is easy

o verify that q 1 < q 2 .

Now we consider a column Q in matrix ρ(q, Q ) .

1) When Q < q 1 In this case, when q > q 1 , the maximum element of the column

is in the row q = q 2 ; when q ≤ q 1 , the values of ρ(q, Q ) in the col-

mn Q is increasing in q . Thus in summary, the maximum element

f the column Q is always in the row q = q 2 (This case is shown as

he column Q = Qa in Fig. E1 ).

2) When Q > q 2 In this case, when q < q 2 , the maximum element of the column

is in the row q = q 1 ; when q ≥ q 2 , the values of ρ(q, Q ) in the

olumn Q are decreasing in q . In summary, the maximum element

f the column Q is always in the row q = q 1 (This case is shown as

he column Q = Qb in Fig. E1 ).

Now we consider the row q = q 2 . When Q < q 2 , the values of

(q, Q ) in the row q 2 are decreasing in Q , until at ( Q = q 2 , q = q 2 )

(q, Q ) equals to zero.

Consider the row q = q 1 , ρ(q, Q ) equals zero at ( Q = q 1 , q = q 1 ),

nd when Q > q 1 , the values of ρ(q, Q ) in the row q 1 are increasing

n Q .

In summary, we can expect that the min-max value ρ(re ) of

(q, Q ) will be attained at an element in the row q = q 2 and the

olumn Q = Q

(re) , and ρ(re ) be attained simultaneously at the ele-

ent ( q = q 1 , Q = Q

(re) ) as shown in Fig. E1 . Thus we have

( q 1 , Q

(re ) ) = ρ( q 2 , Q

(re ) ) ,

It is also apparently that q 1 ≤Q

(re) ≤ q 2 .

Substituting Eq. (E-1) and rewriting the expression above com-

lete the proof. QED.


J

K

K

K

K

L

L

L

MM

NP

P

R

R

S

S

S

V

W

W

X

Y

Y

Y

Z

Z

Appendix F: Proof of Proposition 4

If we assume that the total belief degrees over the interval

( D i , D i + 1 ) is m̄ i,i +1 , and there are a series of possible demand val-

ues denoted by x k , k = 1 ,…,K 1, included in the above interval, i.e.,

D i < x k < D i +1 , and m̄ i,i +1 =

∑

k =1 , ... ,K1

m̄ i,i +1 , x k , where m̄ i,i +1 , x k

is one

of the divisions from m̄ i,i +1 to be assigned to x k . In order to find

the maximum entropy belief distribution over the above demand

series, we can formulate the following objective function by use of

the Lagrangian multiplier λi ,

max G = −∑

k =1 , ... ,K1

( m̄ i,i +1 , x k · ln m̄ i,i +1 , x k

)

+ λi

( ∑

k =1 , ... ,K1

m̄ i,i +1 , x k − m̄ i,i +1

)

.

For a given k , we have

∂G

∂ m̄ i,i +1 , x k

= − ln m̄ i,i +1 , x k − 1 + λi = 0 ,

Thus · m̄ i,i +1 , x k = e λi −1 .

This implies m̄ i,i +1 is evenly distributed into the above series

of demand values, each of which has the belief value of m̄ i,i +1 , x k =

m̄ i,i +1

K1 . QED.

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