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Decision-making in a single-period inventory environment with fuzzy demand

Rung-Hung Su

a,*

, Dong-Yuh Yang

b

, W.L. Pearn

a

a

Department of Industrial Engineering and Management, National Chiao Tung University, Hsinchu, Taiwan, ROC

b

Institute of Information Science and Management, National Taipei College of Business, Taipei, Taiwan, ROC

a r t i c l e

i n f o

Keywords: Newsboy

Achievable capacity index Decision-making Fuzzy sets

Fuzzy hypothesis testing

a b s t r a c t

This paper first defines the profitability to be the probability of achieving a target profit under the optimal ordering policy, and introduces a new index (achievable capacity index; IA) which can briefly analyze the profitability for newsboy-type product with normally distributed demand. Note that since the level of profitability depends on the demand meanland the demand standard deviationrif the related costs, selling price, and target profit are given, the index IAis a function oflandr. Then, we assess level per-formance which examines if the profitability meets designated requirement. The results can determine whether the product is still desirable to order/manufacture. However,landrare always unknown, and the demand quantity is common to be imprecise, especially for new product. To tackle these prob-lems, a constructive approach combining the vector of fuzzy numbers is introduced to establish the mem-bership function of the fuzzy estimator of IA. Furthermore, a three-decision testing rule and step-by-step procedure are developed to assess level performance based on fuzzy critical values and fuzzy p-values. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The classical newsboy problem (single-period problem) deals with the purchasing inventory problem for short shelf-life prod-ucts with the uncertainty of demand. For such problems, the man-agers should determine ordering quantity at the beginning of each period. Products cannot be sold in the next period and need addi-tional cost (excess cost) to dispose it if the ordering quantity ex-ceeds actual demand. Therefore, the determination of the ordering quantity is critical in the classical newsboy problem. Sev-eral extensions to the newsboy problem have been proposed and solved in the literature. Among those extensions are alternative objective functions such as minimizing the expected cost ( Nah-mias, 1993), maximizing the expected profit (Khouja, 1995), max-imizing the expected utility (Ismail & Louderback, 1979; Lau, 1980), and maximizing the probability of achieving a target profit (Ismail & Louderback, 1979; Khouja, 1996; Lau, 1980; Sankarasubr-amanian & Kumaraswamy, 1983; Shih, 1979). In fact, these maxi-mum and minimaxi-mum values can be adopted to measure product’s capacity. For example, the maximum expected profit and maxi-mum probability of achieving the target profit can measure prod-uct’s profitability.

In this paper, we consider the newsboy-type product with nor-mally distributed demand and assume that the profitability is

defined to be the probability of achieving a target profit under the optimal ordering policy. Furthermore, in order to simplify the calculation, we develop a new index, which has a simple form and can correspond to the profitability, and so-called ‘‘achievable capacity index (ACI)”, and be denoted by IA. Note that since the

le-vel of profitability depends on the demand mean

l

and the de-mand standard deviation

r

if the related costs, selling price, and target profit are given, the index IAis a function of

l

and

r

. Then,

we assess level performance which examines if the profitability meets designated requirement. However,

l

and

r

are always un-known. To tackle this problem, one should collect the historical data of demand, and then implement the following hypothesis testing, H0:IA6C versus H1:IA>C, where C is a designated

requirement. Critical value of the test must be calculated to deter-mine the results. The results can deterdeter-mine whether the product is still desirable to order/manufacture. But in practice, especially for new product, the demand quantity is difficult to acquire due to lack of information and historical data. In this case, the demand quan-tity is approximately specified based on the experience. Some papers have dealt with this case by applying fuzzy theory.Petrovic, Petrovic, and Vujosevic (1996) first proposed a newsboy-type problem with discrete fuzzy demand. Dutta, Chakraborty, and Roy (2007)studied the newsboy problem with reordering opportu-nities under fuzzy demand.Zhen and Xiaoyu (2006)considered the multi-product newsboy problem with fuzzy demands under bud-get constraint.Kao and Hsu (2002) compared the area of fuzzy numbers to obtain the optimal order quantity. To the best of our knowledge, no researchers have investigated the fuzzy hypothesis testing for assessing level performance. In this study, we first use a

0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.07.123

* Corresponding author. Address: Department of Industrial Engineering and Management, National Chiao Tung University, 1001 University Road, Hsinchu, Taiwan 300, ROC. Tel.: +886 3 5731630; fax: +886 3 5268330.

E-mail address:runghung316@gmail.com(R.-H. Su).

Contents lists available atScienceDirect

Expert Systems with Applications

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new approach in fuzzy statistics to estimate the demand mean and variance parameters of normal distribution (Buckley, 2004, 2005a, 2005b). Then, a general method combining the vector of fuzzy numbers of sample mean ~x, and sample variance ~s2 is proposed

to derive the membership function of the fuzzy estimator of IA.

Fur-thermore, a three-decision testing rule for assessing level perfor-mance according to two different criteria, critical value and fuzzy p-value are proposed. Based on the test, we develop a step-by-step procedure for managers to use so that decisions made in examin-ing the profitability are more reliable. The rest of the paper is orga-nized as follows. In the next section, we calculate the profitability, and develop a new index IAto correspond profitability. Section3

discusses the statistical properties of estimation for IA based on

crisp data. In Section4, we present some basic definitions, nota-tions of fuzzy sets and the

a

-cuts of fuzzy estimation for IA. Section

5deals with implementing fuzzy hypothesis testing for assessing level performance. Following critical value and fuzzy p-value, deci-sion rules and testing procedures are developed. In Section 6, a numerical example is discussed to illustrate the procedure of solv-ing the problem. Some conclusions are given in the final section.

2. Profitability and achievable capacity index IA

The total profit function, Z, in the newsboy model depends on the demand quantity D and ordering quantity Q, and is formulated as

Z ¼ cpD  ceðQ  DÞ ¼ ðcpþ ceÞD  ceQ ; 0 6 D 6 Q; cpQ  csðD  QÞ ¼ csD þ ðcpþ csÞQ ; Q < D < 1;



where

cp the net profit per unit (selling price per unit minus

purchas-ing cost per unit),

ce the excess cost per unit (purchasing cost per unit plus

dis-posal cost per unit; cp> ce> 0),

cs the shortage cost per unit (cp> cs> 0),

k the target profit which is set according to the product prop-erty and the sales experience.

Note that if cpQ < k, then the profit impossibly achieve the target

profit, even the demand is large enough. Therefore, the order quan-tity should be at least k/cp. For Q P k/cp, Z increases for 0 6 D 6 Q

and decreases for D P Q, and has a maximum at point D = Q. The maximum value of Z is equal and higher than k, i.e., Z = cpD = cpQ P k. The target profit will be realized when D is equal

to either LAL(Q) or UAL(Q). So the target profit will be achieved in D 2 [LAL(Q), UAL(Q)], where

LALðQÞ ¼ceQ þ k cpþ ce

and UALðQÞ ¼ðcpþ csÞQ  k cs

;

are the lower and upper achievable limits, respectively, and both are the functions of Q.

Under the assumption that the demand is normally distributed, the probability of achieving the target profit is

Pr½Z P k ¼

U

UALðQÞ 

l

r

  

U

LALðQÞ 

l

r

  ¼

U

dðQÞ þ mðQÞ 

l

r

  

U

dðQ Þ þ mðQÞ 

l

r

  ; ð1Þ

where U() is the cumulative distribution function (CDF) of the standard normal distribution, d(Q) = [UAL(Q)  LAL(Q)]/2 is the half-length of the achievable interval [LAL(Q), UAL(Q)], and m(Q) = [UAL(Q) + LAL(Q)]/2 is the midpoint between the lower and upper achievable limits. Since the necessary condition for maximiz-ing Pr½Z P k is dPr½Z P k=dQ ¼ 0, we have

l

¼ mðQ Þ 

xr

2

2dðQÞ; ð2Þ

where

x

¼ ln½1 þ cpA=csce > 0 and A = cp+ ce+ cs. For Q P k/cp, the

optimal ordering quantity can be obtained by solving Eq.(2), i.e.:

Q¼k cp þcsðcpþ ceÞðcp

l

 kÞ cpðcpA þ 2cecsÞ þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi csðcpþ ceÞðcp

l

 kÞ cpðcpA þ 2cecsÞ  2 þ2c 2 sðcpþ ceÞ2

xr

2 cpAðcpA þ 2cecsÞ s > k cp : ð3Þ

In addition, the sufficient condition is also calculated as follows:

d2Pr½Z P k dQ2      Q ¼Q ¼ ðcpþ csÞe 1 2 UALðQ Þl r 2 ffiffiffiffiffiffiffi 2

p

p

r

3c2 sðcpþ ceÞ dðQÞðcpA þ 2cecsÞ þ cpA

xr

2 2dðQÞ   <0: ð4Þ

As a result, it leads to the conclusion that Q* is the optimal ordering quantity that maximizes the probability of achieving the target profit. By using Eq.(2)and substituting Eq.(3)into Eq.(1), the profitability, P, can be obtained as follows:

U

dðQ  Þ

r

þ

xr

2dðQ Þ   

U

dðQ  Þ

r

þ

xr

2dðQ Þ   : ð5Þ

2.1. Achievable capacity index IA

We develop a new index to express the product’s profitability. It is defined as: IA¼ dðQÞ

r

¼ Mðcp

l

 kÞ

r

þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mðcp

l

 kÞ

r

 2 þ cpM

x

s ; ð6Þ

where M = A/2(cpA + 2cecs), and call it ‘‘achievable capacity index”.

The numerator of IA provides the half demand range over which

the total profit will achieve the target profit under the optimal order quantity. The denominator gives demand standard deviation. Obvi-ously, it is desirable to have a IAas large as possible. From the Eq.

(5), P can be rewritten as follows:

U

IAþ

x

2IA   

U

IAþ

x

2IA   : ð7Þ

It is easy to see that P is the function of IA. Taking the first-order

derivative of P with respect to IA, we obtain

dP dIA¼ 1 ffiffiffiffiffiffiffi 2

p

p e 1 2 IAþ2IAx 2

x

2I2A ðex 1Þ þ exþ 1 " # >0: ð8Þ Table 1

The profitability for IA= 0.5(0.5)4.0 andx= 0.5(0.5)5.0.

x IA 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.5 0.3413 0.6677 0.8610 0.9528 0.9871 0.9972 0.9995 0.9999 1.0 0.2417 0.6247 0.8450 0.9477 0.9858 0.9969 0.9995 0.9999 1.5 0.1359 0.5586 0.8186 0.9391 0.9835 0.9964 0.9994 0.9999 2.0 0.0606 0.4773 0.7825 0.9270 0.9803 0.9957 0.9993 0.9999 2.5 0.0214 0.3891 0.7377 0.9111 0.9759 0.9948 0.9991 0.9999 3.0 0.0060 0.3023 0.6853 0.8914 0.9703 0.9936 0.9989 0.9998 3.5 0.0013 0.2236 0.6267 0.8677 0.9634 0.9920 0.9986 0.9998 4.0 0.0002 0.1573 0.5639 0.8400 0.9550 0.9901 0.9983 0.9998 4.5 0.0000 0.1051 0.4987 0.8083 0.9449 0.9877 0.9978 0.9997 5.0 0.0000 0.0666 0.4330 0.7728 0.9330 0.9848 0.9973 0.9996

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As a result, the large value of IA, the larger value of P. Therefore,

in order to simplify the calculation, one also observe the value of IA

to present the profitability. Table 1 displays the P for IA=

0.5(0.5)4.0 and

x

= 0.5(0.5)5.0.

3. Crisp estimation for IA

The formula of IAcan be easy to understand and straightforward

to apply. But in practice, the demand mean

l

and the demand stan-dard deviation

r

are usually unknown. Thus, we should collect the historical data of demand to estimate actual IA. If a demand sample

of size n is given as {x1, x2, . . . , xn}, the natural estimator bIAis

ob-tained by replacing

l

and

r

by their estimator X ¼Pn

i¼1xi=n and

s ¼ Pni¼1 xi X

2

=ðn  1Þ

h i1=2

, respectively. We have the following result bIA¼ M cpX  k s þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M cpX  k s " #2 þ cpM

x

v u u t : ð9Þ

Under the consideration that D is a normally distributed, D  N(

l

,

r

2), we have X  Nð

l

;

r

2=nÞ. Let us define Y = T/V, where

T ¼ M cpX  k

=

r

and V ¼pffiffiffiffiffiffiffiffiffiffiffiffis2=

r

2. It is well known that T 

N Mðcp

l

 kÞ=

r

;c2pM 2=n

, so the probability density function (PDF) of T can be expressed in the following:

fTðtÞ ¼ 1 cpM ffiffiffiffi 2p n q exp  t  MðcplkÞ r 2 2c2 pM2 n 2 6 4 3 7 5; 1 < t < 1:

Since the random variable (n  1)s2/

r

2follows the Chi-squared

distribution with n  1 degree of freedom, V2= s2/

r

2

C((n  1)/ 2, 2/(n  1)). By using the technique of change-of-variable, the PDF of V can be derived as follows:

fVð

v

Þ ¼ 2

v

n2en1 2v 2

C

n1 2 2 n1 n1 2 ;

v

>0:

Because T and V are independent continuous random variables, we easily obtain the PDF of Y by the convolution formula

fYðyÞ ¼ Z 1 0 fTð

v

yÞfVð

v

Þj

v

jd

v

¼ Z 1 0 2

v

n1 2 n1 n1 2 cpM

C

n12 ffiffiffiffi2p n q exp 

v

y  I2 AcpMx 2IA 2 2c2 pM2 n n  1 2

v

2 2 6 4 3 7 5d

v

;  1 < y < 1: Subsequently, we define R ¼ bIA¼ Y þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Y2 þ cpM

x

q . The CDF and PDF of R can be derived, respectively, as follows:

FRðrÞ ¼ Z r2 cp Mx 2r 0 fYðyÞdy; 0 < r < 1 and fRðrÞ ¼ BðrÞ Z 1 0 gð

v

;rÞd

v

; 0 < r < 1; where BðrÞ ¼ r 2þ c pM

x

r2c pM ffiffiffiffi 2p n q

C

n1 2 2 n1 n1 2 and gð

v

;rÞ ¼

v

n1exp 

v

r2c pMx 2r  I2 AcpMx 2IA 2 2c2 pM2 n ðn  1Þ

v

2 2 2 6 4 3 7 5:

Fig. 1 shows the CDF and PDF plots of R with cp= 10, ce= 5,

cs= 3, IA= 3.0(0.5)4.0 for n = 30, 50, 100 and 200 (from bottom to

top in plots). FromFig. 1, the PDF plots of R reveal the following features:

(1) The larger the value of IA, the larger the variance of

R ¼ bIA;

(2) The distribution of R is unimodal and is rather symmetric to IAeven for small sample sizes;

(3) The larger the sample sizes n, the smaller the variance of R, which is certain for all sample estimators.

4. Fuzzy estimation for IA

4.1. Definitions and notations for fuzzy set theory

In the traditional precise set, the degree of an element belongs to a set is either one or zero. In order to deal with the imprecise data,Zadeh (1965)proposed the fuzzy set theory. The definitions and notations are shown as follows:

R the universal set, e

A the fuzzy set,

g

e

A the membership function of eA;

g

eA:R! ½0; 1,

e

A½a the

a

-cuts of fuzzy number eA (the set of elements that

belong to the fuzzy set eA at least to the degree of member-ship

a

, eA½a¼ fxj

g

eAP

a

;x 2 Rg),

LeAð

a

Þ the lower bound of the closed interval of eA ½a,

UeAð

a

Þ the upper bound of the closed interval of eA ½a.

4.2.

a

-Cuts of a fuzzy estimation for IA

An important fact that the demand mean and the demand stan-dard deviation are usually unknown. One should estimates these two parameters from observations. However, there is a great deal of uncertainty in the model due to these parameters not being known precisely. According toBuckley and Eslami (2004), we con-sider additional uncertainty in the unknown parameters,

l

and

r

2,

by using fuzzy estimators. The fuzzy estimators, ~x and ~s2, are

con-structed from a set of confidence intervals. As the triangular fuzzy numbers with

a

-cuts are constructed, the fuzzy numbers are given as follows: ~  x½a¼ L ~  xð

a

Þ; U~xð

a

Þ ½  ¼ x  ta=2;n1 s ffiffiffi n p ; x þ ta=2;n1 s ffiffiffi n p   ; for

8

a

2 ð0; 1; ð10Þ ~s2½a¼ L ~s2ð

a

Þ; U~s2ð

a

Þ ½  ¼ ðn  1Þs 2

v

2 a=2;n1 ;ðn  1Þs 2

v

2 1a=2;n1 " # ; for

8

a

2 ð0; 1; ð11Þ

where ta/2,n1is the upper

a

/2 quantile of the t distribution with

n  1 degrees of freedom,

v

2

a=2;n1 and

v

21a=2;n1are the upper

a

/2

and 1 

a

/2 quantiles of the chi-square with n  1 degrees of free-dom, respectively.Figs. 2 and 3display the membership functions of fuzzy estimation for ~x and ~s2, respectively, x ¼ 25; s2¼ 4 and

k = 200 with n = 30, 50, 100, and 200.

In order to obtain an

a

-cuts of fuzzy number bIA, let

l

2 ~x½aand

(4)

ebI½a A ¼ Le bIA ð

a

Þ; Ue bIA ð

a

Þ " # ¼ M cpL~xð

a

Þ  k ffiffiffiffiffiffiffiffiffiffiffiffiffiffi U~s2ð

a

Þ p " þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M cpL~xð

a

Þ  k ffiffiffiffiffiffiffiffiffiffiffiffiffiffi U~s2ð

a

Þ p " #2 þ cpM

x

v u u t ;M cpU~xð

a

Þ  k ffiffiffiffiffiffiffiffiffiffiffiffiffi L~s2ð

a

Þ p þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M cpU~xð

a

Þ  k ffiffiffiffiffiffiffiffiffiffiffiffiffi L~s2ð

a

Þ p " #2 þ cpM

x

v u u t 3 7 5: ð12Þ

Fig. 4plots the membership function of fuzzy estimation ebIA

with bIA¼ 2:5705 for n = 30, 50, 100 and 200.

5. Fuzzy hypothesis testing for assessing level performance

To test whether the profitability meets the designated require-ment, we consider the following testing hypothesis, procedure with the null hypothesis H0:IA6C, versus the alternative

H1:IA>C, where C is a designated requirement. Based on CDF of

R, given designated requirement C, sample size n, and level of Type I error h, the critical value c0can be calculated by solving the

fol-lowing equation PrfR P c0jIA¼ C; ng ¼ 1  Z c2 0cpMx 2c0 0 fYðyÞdy ¼ h: ð13Þ

(5)

On the other hand, the p-value is generally used for making decisions in hypothesis testing. The p-value corresponding to a specific value of bIAcalculated from the sample data, c*, can be

ob-tained from the following equation

p-value ¼ Pr R P cjI A¼ C; n f g ¼ 1  Z c2 cpMx 2c 0 fYðyÞdy: ð14Þ

If bIA>c0or p-value < h, we reject the null hypothesis, and

con-clude that the profitability is better than requirement with signif-icance level h in non-fuzzy statistics. For the imprecise data, the testing problem must extend the three-decision byFilzmoser & Viertl (2004) & Neyman & Pearson (1933), that is: (a) accept H0

and reject H1, (b) reject H0and accept H1, (c) both H0and H1are

neither accepted nor rejected. The final decision will become to de-pend on the relationship between fuzzy set ebIAand c0.

5.1. Decision-making by critical value

In this subsection, we use the finite interval Leb

IA ð

a

Þ; Ue bIA ð

a

Þ " # to

test with critical value c0, and make the decision according to the

three-decision testing rule:

(a) IF LebI

A

ð

a

Þ > c0THEN reject H0and accept H1;

(b) IF UebI

A

ð

a

Þ < c0THEN accept H0and reject H1;

0 0.2 0.4 0.6 0.8 1 24.00 24.20 24.40 24.60 24.80 25.00 25.20 25.40 25.60 25.80 26.00 n=30 n=50 n=100 n=200

Fig. 2. The membership functions of fuzzy estimation for x with n = 30, 50, 100, 200.

0 0.2 0.4 0.6 0.8 1 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 n=30 n=50 n=100 n=200

Fig. 3. The membership functions of fuzzy estimation for s2

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(c) IF Leb

IA

ð

a

Þ 6 c06Ue

bIA

ð

a

Þ THEN both H0 and H1 are neither

accepted nor rejected.

(Note: IF LebI

A

ð

a

Þ ¼ Ue

bIA

ð

a

Þ THEN the hypothesis testing is reduced to crisp data with a binary testing rule.)

A simple step-by-step procedure for judging whether the prof-itability meets the designated requirement based on critical value, is summarized as follows:

Step 1: Determine the designated requirement C, the target profit k, the sample size n, the significance level h and the user-approved degree of imprecision

a

on sample data. Step 2: Calculate the

a

-cuts of fuzzy numbers ~x and ~s2by Eqs.(10)

and (11).

Step 3: Calculate the

a

-cuts of fuzzy number ebIA; ebI½Aa¼

LebI A ð

a

Þ; Ue bIA ð

a

Þ " # by Eq.(12).

Step 4: Find the critical value c0from Eq.(13).

Step 5: Conclude that the profitability is better than requirement IA> C if Le

bIA

ð

a

Þ > c0. Conclude that the profitability is lower

than requirement IA6C if Ue

bIA

ð

a

Þ < c0. Conclude that the

profitability is not proven to be better or lower than requirement, and that further study is needed, if Leb

IA

ð

a

Þ 6 c06Ue

bIA

ð

a

Þ.

For example, if the bIA¼ 2:5705 ðx ¼ 25; s2¼ 4; k ¼ 200 and

n ¼ 100Þ, C = 2.0, h = 0.05,

a

= 0.7, we obtain Le

bIA

ð0:7Þ ¼ 2:4871 > c0¼ 2:1966. One can conclude that the profitability is better than

designated requirement, IA> 2.0. If the designated requirement

C = 2.5, we obtain UebI

A

ð0:7Þ ¼ 2:6442 < c0¼ 2:7713. One can

con-clude that the profitability is lower than designated requirement, IA< 2.5.Fig. 5displays the membership function

g

ebIA

ðbIAÞ. The

a

-cuts

ebI½a¼0:7

A is also exhibited in theFig. 5.Fig. 6depicts the PDF plots of

bIAfor IA= C = 2.0, 2.5 and n = 100 associated with the critical values

c0with the significance level h = 0.05.

5.2. Decision-making by fuzzy p-value

The p-value is also widely used for making decisions in hypothesis testing. Therefore, we employ the finite interval

0 0.2 0.4 0.6 0.8 1 1.50 2.00 2.50 3.00 3.50 4.00 n=30 n=50 n=100 n=200

I

A

Fig. 4. The membership functions of fuzzy estimation for bIAwith n = 30, 50, 100, 200.

Fig. 5. The membership functiong~^I

A

^IA

and thea-cuts~^I½a¼0:7 A .

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LebI A ð

a

Þ; Ue bIA ð

a

Þ " #

for defining the corresponding interval of

fuzzi-ness of ~p. The

a

-cuts of ~p can be represented as follows:

~ p½a¼ ½L ~ pð

a

Þ; U~pð

a

Þ ¼ Pr R P Ueb IA ð

a

ÞjIA¼ C; n ( ) ;Pr R P Le bIA ð

a

ÞjIA¼ C; n ( ) " # for

8

a

2 ð0; 1: ð15Þ

Then the decision is made according to the three-decision test-ing rule:

(a) IF U~pð

a

Þ < h THEN reject H0and accept H1;

(b) IF L~pð

a

Þ > h THEN accept H0and reject H1;

(c) IF L~pð

a

Þ 6 h 6 U~pð

a

Þ THEN both H0 and H1 are neither

accepted nor rejected.

(Note: IF Lp~ð

a

Þ ¼ Up~ð

a

Þ THEN the hypothesis testing is reduced

to crisp data with a binary testing rule.)

A simple step-by-step procedure for judging whether the prof-itability meets the designated requirement based on p-value, is summarized as follows:

Step 1: Determine the designated requirement C, the target profit k, the sample size n, the significance level h and the user-approved degree of imprecision

a

on sample data. Step 2: Calculate the

a

-cuts of fuzzy numbers ~x and ~s2by Eqs.(10)

and (11).

Step 3: Calculate the

a

-cuts of fuzzy number ebIA; ebI½Aa¼

LebI A ð

a

Þ; Ue bIA ð

a

Þ " # by Eq.(12).

Step 4: Calculate the

a

-cuts of fuzzy number ~p; ~p½a¼ ½L~

a

Þ; U~pð

a

Þ

by Eq.(15).

Step 5: Conclude that the profitability is better than requirement IA> C if U~pð

a

Þ < h. Conclude that the profitability is lower

than requirement IA6C if L~pð

a

Þ > h. Conclude that the

profitability is not proven to be better or lower than requirement, and that further study is needed, if Lp~ð

a

Þ 6

h 6U~pð

a

Þ.

For the above numerical example, if the designated requirement C = 2.0, we obtain U~pð0:7Þ ¼ 1:6387  104<h¼ 0:05. One can

conclude that the profitability is better than designated require-ment, IA> 2.0. If C = 2.5, we obtain L~pð0:7Þ ¼ 0:1836 > h ¼ 0:05.

One can conclude that the profitability is lower than designated requirement, IA< 2.5.

6. Numerical example

This section considers the following case taken from a publisher selling certain weekly magazine with stocks and investment. The kind of this magazine has the following features:

(1) The magazine cannot be sold in the next week; (2) The net profit of this magazine is $10 per unit;

(3) The unsold quantity need the transportation cost to dispose it, ce= $5 per unit;

(4) The unsatisfied demand will lost the opportunity cost, cs= $3

per unit.

Suppose that the target profit and the designated requirement for this magazine are set to k = 200 and C = 2.5 by the managers, respectively. To test the profitability meets the designated require-ment, one must determine whether the profitability meets IA> 2.5,

which is equivalent to having the maximum value of the probabil-ity of achieving the target profit larger than 0.9752. The historical date of magazine demand volume per week with sample size n = 100 has collected. One first uses the Kolmogorov–Smirnov test for the historical data to confirm if the data is normally distributed. The test result in p-value > 0.05, which means that data is normally distributed.

Since the data given by the retailer has some degrees of impre-cision, managers suggest fuzzy inference to assess the profitability with imprecise data. The sample mean, sample standard deviation and sample estimator are calculated as X ¼ 26:0316; s ¼ 2:0311 and bIA¼ 2:9216, respectively. If the user-approved degree of

imprecise on the sample data and significance level are set as

a

= 0.8 and h = 0.05, respectively. We execute the computer soft-ware (Mathematica 4.0) with n = 100 and IA= C = 2.5 to calculate

ebI½a¼0:8 A ¼ Le bIAð0:8Þ; UebIA ð0:8Þ " # ¼ ½2:8549; 2:9744 and c0= 2.7713.

Based on the three-decision testing rule, H0is rejected at the

signif-icance level h = 0.05 since LebI

A

ð0:8Þ ¼ 2:8549 > c0¼ 2:7713. One

can conclude that the profitability is higher than designated requirement IA> 2.5. On the other hand, the

a

-cuts of p-value also

leads to the same conclusion as above since U~pð0:8Þ ¼ 0:0180 <

h¼ 0:05.

7. Conclusion

In this paper, we proposed a method to calculate the IAindex

when the precise demand quantity cannot be identified. The fuzzy set theory was applied to tackle this problem. It is important for practical decision-making based on statistical hypothesis testing.

I

A

(8)

In this case, we described the three-decision testing rule and pro-vided a step-by-step procedure to assess the profitability by two fuzzy inference criteria, the critical value and the fuzzy p-value. Using fuzzy inference to assess the profitability with imprecise de-mand quantity under non-normality would be worthy of further investigation.

References

Buckley, J. J. (2004). Uncertain probabilities III: The continuous case. Soft Computing, 8, 200–206.

Buckley, J. J. (2005a). Fuzzy statistics: Hypothesis testing. Soft Computing, 9, 512–518.

Buckley, J. J. (2005b). Fuzzy systems. Soft Computing, 9, 757–760.

Buckley, J. J., & Eslami, E. (2004). Uncertain probabilities II: The continuous case. Soft Computing, 8, 193–199.

Dutta, P., Chakraborty, D., & Roy, A. R. (2007). An inventory model for single-period products with reordering opportunities under fuzzy demand. Computers and Mathematics with Applications, 53, 1502–1517.

Filzmoser, P., & Viertl, R. (2004). Testing hypotheses with fuzzy data: The fuzzy p-value. Metrika, 59, 21–29.

Ismail, B., & Louderback, J. (1979). Optimizing and satisfying in stochastic cost– volume–profit analysis. Decision Sciences, 10, 205–217.

Kao, C., & Hsu, W. K. (2002). A single period inventory model with fuzzy demand. Computers and Mathematics with Applications, 43, 841–848.

Khouja, M. (1995). The newsboy problem under progressive multiple discounts. European Journal of Operational Research, 84, 458–466.

Khouja, M. (1996). The newsboy problem with multiple discounts offered by suppliers and retailers. Decision Science, 27, 589–599.

Lau, H. (1980). The newsboy problem under alternative optimization objectives. Journal of the Operational Research Society, 31, 525–535.

Nahmias, S. (1993). Production and operations management. Boston: Irwin. Neyman, J., & Pearson, E. S. (1933). The theory of statistical hypotheses in relation to

probabilities a priori. Proceedings of Cambridge Philosophical Society, 29, 492–510.

Petrovic, D., Petrovic, R., & Vujosevic, M. (1996). Fuzzy models for the newsboy problem. International Journal of Production Economics, 45, 435–441.

Sankarasubramanian, E., & Kumaraswamy, S. (1983). Optimal order quantity for pre-determined level of profit. Management Science, 29, 512–514.

Shih, W. (1979). A general decision model for cost–volume–profit analysis under uncertainty. The Accounting Review, 54, 687–706.

Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.

Zhen, S., & Xiaoyu, J. (2006). Fuzzy multi-product constraint newsboy problem. Applied Mathematics and Computation, 180, 7–15.

數據

Fig. 1 shows the CDF and PDF plots of R with c p = 10, c e = 5,
Fig. 1. The CDF and PDF plots of R for sample sizes n = 30, 50, 100, 200 (from bottom to top in plots).
Fig. 2. The membership functions of fuzzy estimation for  x with n = 30, 50, 100, 200.
Fig. 4. The membership functions of fuzzy estimation for bI A with n = 30, 50, 100, 200.
+2

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