Available online at www.sciencedirect.com
Physics Procedia 25 (2012) 924 – 931
2012 International Conference on Solid State Devices and Materials Science
Random Fuzzy Demand Newsboy Problem*
Yuanji Xu1,Jinsong Hu2
1
Department of Management Science and Engineering
Zhejiang Sci-Tech University
Hangzhou, Zhejiang Province, China
2
Department of Management Science and Engineering
Qingdao University
Qingdao, Shangdong Province, China
Abstract
For the more realistic description of the uncertain demand, the newsboy model with random fuzzy demand that
maximizes the profit is developed. By the credibility measure of fuzzy event, the expected profit model is presented.
Moreover, the concavity of the profit function is revealed. A hybrid algorithm integrating the random fuzzy
simulation and simultaneous perturbation stochastic approximation is designed to obtain the optimal order quantity.
Finally, numeral examples are given to illustrate the effect of the hybrid algorithm.
2011 Published by Elsevier
ofof
[name
organizer]
© 2012
Elsevier Ltd.
B.V.Selection
Selectionand/or
and/orpeer-review
peer-reviewunder
underresponsibility
responsibility
Garry
Lee
Open access under CC BY-NC-ND license.
Keywords: Newsboy problem; Random fuzzy variable; Random fuzzy simulation; Simultaneous perturbation stochastic
approximation.
1. Introduction
The objective of the classical newsboy problem is to find the optimal order quantity by minimizing
expected cost or maximizing expected profit. In the real life, Newsboy problem plays an important role in
the decision making for perishable goods, spare parts and seasonal products. Therefore, the classical
newsboy problem and its various extensions are widely introduced Most of these extensions only considered the randomness aspect of uncertainty demand and random
inventory models are developed by utilizing the probability theory. However, in practical situations
*
This work is partially supported by National Natural Science Foundation of China under Grand #71071082 and #70671056 and
Shandong Province Natural Science Foundation under Grant #Y2008H07.
1875-3892 © 2012 Published by Elsevier B.V. Selection and/or peer-review under responsibility of Garry Lee
Open access under CC BY-NC-ND license. doi:10.1016/j.phpro.2012.03.179
Yuanji Xu and Jinsong Hu / Physics Procedia 25 (2012) 924 – 931
sometimes the probability distributions of the demands for products are difficult to acquire due to lack of
information and historical data. Thus, in inventory system, the demands are approximately estimated by
the experts depend on their experience and managerial subjective judgments in order to tackle the
uncertainties which always fit the real situations. As such, these characteristics are better described by the
use of fuzzy sets originally introduced by Zadeh [8] which encompass a specific range of values. Several
researchers have considered newsboy problem with fuzzy demand. Petrovic, Petrovic and Vujosevic [9]
considered two fuzzy models for the newsboy problem with discrete fuzzy demand and fuzzy costs and
obtained the optimal order quantity using centroid method. Li, Kabadi and Nair [10] developed two fuzzy
models, in one the demand is probabilistic while the holding and shortage cost are fuzzy and in the other
the costs are deterministic but the demand is a fuzzy number. Kao and Hsu [11] constructed a singleperiod inventory model for cases of fuzzy demand and determined the optimal order quantity using Liou
and Wang’s [12] ranking method. Dutta, Chakraborty and Roy [13] considered newsboy problem with
fuzzy demand and reordering strategy and employed the possibilistic mean value of a fuzzy number
(Carlsson and Full’er [14]) to determine the optimal order quantity that maximizes the profit function.
Dutta and Chakraborty [15] developed the fuzzy demand single-period inventory model for two-item with
upward substitution policy. Lu [16] applied a fuzzy newsvendor inventory management approach to
analyze optimal order policy based on probabilistic fuzzy sets with hybrid data so that the expected total
cost is minimized. And considering the cooperation between the supplier and the retailer, Xu and Zhai
[17] researched a model for fuzzy newsboy problem in the supply chain environment with one
manufacturer and one retailer. Xu and Zhai [18] extended the Xu and Zhai’s [17] model by designing an
efficient coordination mechanism from the manufacturer's point of view.
In all the above works, the authors addressed the demand as fuzzy number. However, in real-life
inventory system, randomness and fuzziness often appear simultaneously. For example approximately
estimate from experts about the demand varies randomly. In this case, it is more reasonable to
characterize the demand as a fuzzy random variable, which is consistent with the definition of fuzzy
random variable originally introduced by Kwakernaak [19, 20]. Dutta, Chakraborty and Roy [21]
investigated a single-period inventory model with discrete fuzzy random demand. The optimal order
quantity maximizing the fuzzy random profit is achieved by using a graded mean integration
representation . Hu et al.[22] presented two fuzzy random models for the newsboy problem with
imperfect quality in the decentralized and centralized systems and optimal policies were obtained by
using the signed distance (Yao and Wu [23]). Sometimes, it is difficult or impossible to get enough
statistics data for the parameters of the probability distributions of the demand. Such a parameter value is
determined subjectively and imprecisely. In this case, it is more reasonable to characterize the demand as
a random fuzzy variable. The concept of random fuzzy variable was firstly given by Liu [24].
From literature survey, there are few literatures dealing with the random fuzzy demand newsboy
problem. On the other hand, the possibility measure or necessary measure is employed by almost all of
the papers and the objective function is to maximize the profit or minimize the total cost. However, Liu
[24] point out that corresponding to probability measure, the credibility measure instead of possibility
measure is more appropriate. In this paper, based on credibility measure, we attempt to develop the model
for newsboy problem with random fuzzy demand.
2. Preliminaries
A. Fuzzy variable
Definition 1 [25] A possibility space is defined as a triplet (4, 5(4), Pos) , where 4 is a nonempty set,
5(4) the power set of 4 , and Pos a possibility measure satisfying the four axioms :Axiom 1
925
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Yuanji Xu and Jinsong Hu / Physics Procedia 25 (2012) 924 – 931
Pos{4} 1 ;Axiom 2 Pos{‡} 0 , where ‡ denotes the empty set; Axiom 3 Pos{‰i Ai } supi Pos{ Ai } for
any collection { Ai } in 5(4) .
Definition 2 [25]A fuzzy variable [ is defined as a function from a possibility space (4, 5(4), Pos) to the
set of real numbers ƒ .
Definition 3 [24] Let [ be a fuzzy variable, and D  (0,1] , then [DL inf{r | Pos{[ d r} t D } and
[DU sup{r | Pos{[ t r} t D } are defined as the D pessimistic value and D optimistic value of [ .
Definition 4 [26] Let (4, 5(4), Pos) be possibility space. For each A  5(4) , the credibility measure of A
1
(Pos{ A} Nec{ A}) ,where Nec{ A} is the necessity measure of fuzzy event A , and
2
Nec{ A} 1 Pos{ Ac } , in which Ac denotes the complement of A .
Definition 5 [26] Let [ be a fuzzy variable. The expected value of [ , denoted by E[[ ] , is defined
is defined as Cr{ A}
as E[[ ]
³
f
0
0
Cr{[ t r}dr ³ Cr{[ d r}dr ,
f
providing that at least one of the two integrals is finite.
Property1 [27] Let [ be a fuzzy variable and its expected value is finite. Then E[[ ]
1 1 L
([D [DU )dD .
2 ³0
B. Random fuzzy variable
Definition 6 [24] Let [ be a function from possibility space (4, 5(4), Pos) to the set of random variable, we
say [ is called random fuzzy variable.
Property 2 [28] Let [ be a random fuzzy variable on possibility space (4, 5(4), Pos) . If for each T  4 , the
expected value E[[ (T )] of random variable [ (T ) is finite, and then E[[ (T )] is a fuzzy variable, denoted
as E[[ (˜)] .
Definition 7 [27] Let [ be a random fuzzy variable on possibility space (4, 5(4), Pos) , then its expected
value is defined as
E[[ ]
³
f
0
0
Cr{T  4 | E[[ (T )] t r}dr ³ Cr{T  4 | E[[ (T )] d r}dr
f
Property 3 [27] let [ be a random fuzzy variable, then
1 1
E[[ ]
{E[[ (˜)]DL E[[ (˜)]UD }dD , where E[[ (˜)]DL and E[[ (˜)]UD are the D pessimistic value
2 ³0
and D optimistic value of fuzzy variable E[[ (˜)] respectively.
3. Random fuzzy newsboy problem
The random fuzzy newsboy problem statement is: given a random fuzzy demand D , constant unit
sale price, unit purchase cost c and unit salvage value s , determine the order quantity Q* which will
maximize the total profit. The total profit is given as follows:
S (Q, [ ) p min{[, Q} cQ s max{0, [ Q} h max{0, Q [}
(1)
where p is the selling price per item, c is the purchasing cost per item ( p ! c) , s is the salvage value per
item (c ! s) , the demand [ is assumed to be a random fuzzy variable on probability space (4, 5(4), Pos) ,
while for any T  4 , [ (T ) is continuous random variable on probability space (:, P(:), Pr) , Q is the
order quantity.
Since the demand is random fuzzy, the profit associated with each inventory policy, i.e., the quantity
to be ordered Q , is random fuzzy as well. Since our aim is to find the optimal order quantity maximizing
the expected profit.
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Yuanji Xu and Jinsong Hu / Physics Procedia 25 (2012) 924 – 931
Therefore, our problem is expressed as
­max{E[S (Q, [ )]} E{ p min{[, Q} cQ s max{0, [ Q}
°°
h max{0, Q [}}
®
°s.t. Q t 0
°̄
(2)
Theorem 1 For a given order quantity Q , the expected value of random fuzzy profit S (Q, [ ) is
1 1 Q
E[S(Q,[)] ³ ³ ( p h)x[ f (x;TDc ) g(x;TDc )]dxdD
200
Q
1 1 Q
³ ³ (c h)Q ³ [ f ( x;TDc ) g ( x;TDcc )]dxdD
0
2 0 0
1 1 f
³ ³ ( p c s )Q[ f ( x;TDc ) g ( x;TDcc )]dxdD
2 0 Q
1 1 f
³ ³ sx[ f ( x;TDc ) g ( x;TDcc )]dxdD
2 0 Q
c
where f ( x;TD ) and g ( x;TDcc ) are the distribution density function associated with random
variable [ (T c ) and [ (T cc ) .
D
D
Proof. Since [ is a random fuzzy variable, then for each T  4 , [ (T ) and S (Q, [ (T )) are both random
variables. Denote the distribution density function of [ (T ) as I ( x;T ) . Therefore, the expected value of
random variable S (Q, [ (T )) is
E[S (Q, [ (T ))] E[ p min{[ (T ), Q}] cQ sE[max{0, [ (T ) Q}]
E[h max{0, Q [ (T )}]
From Property 2, E[S (Q, [ (˜))] : T o R is a fuzzy variable on the possibility space (4, P(4), Pos) .
Without loss of generality, we assume the image set of fuzzy variable E[S (Q, [ (˜))] is closed. Note that
the D pessimistic value E[S (Q, [ (˜))]DL and the D optimistic value E[S (Q, [ (˜))]UD of fuzzy variable
E[S (Q, [ (˜))] are both continuous functions of D almost everywhere. Hence, for any continuity point D
of E[S (Q, [ (˜))]L and E[S (Q, [ (˜))]U , there must exist T c ,T cc  4 , such that
D
D
E[S (Q, [ (˜))]DL
E[S (Q, [ (˜))]U
D
D
D
E[S (Q, [ (TDc ))]
E[S (Q, [ (T cc )]
D
Assumed that f ( x;TDc ) and g ( x;TDcc ) are the distribution density
function associated with random variable [ (TDc ) and [ (TDcc ) .Then
E[S (Q, [ (˜))]DL and E[S (Q, [ (˜))]UD are calculated as follows
Q
E[S (Q, [ (˜))]DL
³
E[S (Q, [ (˜))]UD
³
0
Q
0
f
[ px cQ h(Q x)] f ( x;TDc )dx ³ [ pQ cQ s ( x Q )] f ( x;TDc )dx
Q
f
[ px cQ h(Q x)]g ( x;TDcc )dx ³ [ pQ cQ s( x Q)]g ( x;TDcc )dx
Q
From Property 3 and the two above equations, we have
E[S (Q, [ )]
1 1
{E[S (Q, [ (˜))]DL E[S (Q, [ (˜))]UD }dD
2 ³0
Q
1 1 Q
{ ( p h) x[ f ( x;TDc ) g ( x;TDcc )]dx (c h)Q ³ [ f ( x;TDc ) g ( x;TDcc )]dx}dD
0
2 ³0 ³0
f
1 1 f
³ {³ ( p c s)Q[ f ( x;TDc ) g ( x;TDcc )]dx ³ sx[ f ( x;TDc ) g ( x;TDcc )]dx}dD
Q
2 0 Q
The proof is completed.
Theorem 2 E[S (Q, [ )] is a strictly concave function of Q .
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Yuanji Xu and Jinsong Hu / Physics Procedia 25 (2012) 924 – 931
Proof.
dE[S (Q, [ )]
dQ
Q
1 1
(c h) ³ [ f ( x;TDc ) g ( x;TDcc )]dxdD
³
0
0
2
(Q, [ )]
d 2 E[3
1 1 f
( p c s )[ f ( x;TDc ) g ( x;TDcc )]dxdD
³
³
dQ 2
2 0 Q
Since f (Q;TDc ) ! 0 and g (Q;TDcc ) ! 0 , it is clear that
1 1
( p h s )[ f (Q;TDc ) g (Q;TDcc )]dD
2 ³0
d 2 E[S (Q, [ )]
0 , which implies that E[S (Q, [ )] is a
dQ 2
strictly concave function of Q .The proof is completed.
4. SPSA algorithm based on random fuzzy simulation
In Model 2, the objective function is the expected value of random fuzzy profit S (Q, [ ) . In fact, it is
difficult to estimate E[S (Q, [ )] through analytical method. Hence, we introduce a procedure based on
random fuzzy simulation proposed by Liu [24] to calculate E[S (Q, [ )] . For any given order quantity Q , the
processes are given as follows.
Step 1. Initialize parameters p , c , h and s , and set e 0 .
Step 2. Randomly generate T k from the possibility space 4 , such that Pos{T k } t H , where k 1, 2,3", M ,
and H is a sufficiently small positive number, M is the number of the sample in possibility space 4 and
a sufficiently large positive number.
Step 3. Generate a sequence of random numbers {[ j (T k ), j 1, 2,", N } according to the probability
distribution of [ (T k ) .
Step 4. If Q ! [ j (T k ) , set S (Q, [ j (T k ))
Otherwise, S (Q, [ j (T k ))
Step 5. Set E[S (Q, [ (T k )]
p[ j (T k ) cQ h(Q [ j (T k )) .
pQ cQ s ([ j (T k ) Q ) , j 1, 2,", N .
1 N
¦ S (Q,[ j (T k )) .
N j1
In fact, from Steps 3-5, it is a process of Monte Carlo simulation for the excepted value of the
random variable.
M
M
k 1
k 1
Step 6. Set l min{E[S (Q, [ (T k )]} and u max{E[S (Q,[ (T k )]} .
Step 7. Set vk pos{T k } .
Step 8. Randomly generate r from [l , u ] .
Step9. If r ! 0 , set
L
1
{max{vk || E[S (Q, [ (T k )] t r} min{1 vk || E[S (Q, [ (T k )] r}}
k
2 k
and e e L .
Otherwise, set
L
1
{max{vk || E[S (Q, [ (T k )] d r} min{1 vk || E[S (Q, [ (T k )] ! r}}
k
2 k
and e e L .
Step 10. Repeat 8-9th steps for M times.
Step 11. Set E[S (Q, [ )] l › 0 u š 0 e(u l ) / M .
Step 12. Return E[S (Q, [ )] .
Because it is difficult to get the gradient information of the expected value of random fuzzy
profit S (Q, [ ) , we produce a procedure, which combines the SPSA algorithm (see, Spall [29]) with
random fuzzy simulation to obtain the optimal order quantity Q* .
Yuanji Xu and Jinsong Hu / Physics Procedia 25 (2012) 924 – 931
Step 1. Determine a0 , c0 , A , D 0 and J 0 , which are the nonnegative coefficients adopted in SPSA, and set
counter index k 1 . Then determine the maximum allowable number of iterations n .
Step 2. Randomly generate Q1 from a suitable interval to be the initial guess.
Step 3. Set ak a0 /( A k )D and ck c0 / k J .
Step 4. Randomly generate ' k based on the distribution of Bernoulli r1 .
Step 5. Estimate E[S (Q, [ )] with parameters Q Qk ck ' k and Q Qk ck ' k , respectively.
0
Step 6. Set g k (Qk )
0
E[S (Qk ck ' k , [ )] E[S (Qk ck ' k , [ )]
and Qk 1
2ck ' k
Qk ak g k (Qk ) .
Step 7. If the maximum allowable number of iterations has been reached (i.e., k n ) or there is little
change in several successive iterates, terminate the procedure and return Qk as the optimal value of Q .
Otherwise, set k k 1 , and repeat Steps 3-7.
5.Numerical example
In order to illustrate the effectiveness of the proposed algorithms, we consider a newsboy problem
with random fuzzy parameters as follows: p =6, c =4, h =2, s =10, and assume that the demand [ is a
random fuzzy variable, [ N ( P ,V ) , where P (55,60,65) , V 2 . The choice of the parameters of SPSA
has been discussed by Spall (1998). We set a0 7 , A 50 , c0 7 , D 0.602 and J 0.101 in the numerical
example.
To illustrate that the SPSA algorithm based on random fuzzy simulation is applicable and effective
to solve the models in this paper, we estimate optimal order quantities and maximum profits with six
initial guesses Q0 of order quantity (700 cycles for each initial guess). The initial guess, optimal order
quantity and maximum profit are shown in Table 1.
TABLE I
Optimal order quantity and maximum profit to different initial guess
Q*
E[S (Q* , [ ]
Q0
Q*
E[S (Q* , [ ]
30
61.744
98.9048
60
61.742
99.2660
40
61.738
98.7799
70
61.770
99.1753
50
61.732
98.9623
80
61.754
99.2080
Q0
6. Conclusions
In this paper, we consider the demand as random fuzzy variable, we analyses the optimal order
quantity in terms of the profit incurring. The expected value of random fuzzy profit is presented through
credibility measure of fuzzy event, and the concavity of the expected profit function is revealed. Since it
is difficult to obtain the gradient information of the expected profit function, then a hybrid algorithm
integrating random fuzzy simulation and SPSA is designed to produce the optimal order quantity and the
maximum profit. Finally, a numerical example is provided to illustrate the effectiveness of the proposed
algorithm.
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