A novel single-period inventory problem with uncertain random demand and its application

Applied Mathematics and Computation 269 (2015) 133–145

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

A novel single-period inventory problem with uncertain random demand and its application Dan Wang a, Zhongfeng Qin b,∗, Samarjit Kar c a

National Institutes for Food and Drug Control, Beijing 100050, China School of Economics and Management, Beihang University, Beijing 100191, China c Department of Mathematics, National Institute of Technology, Durgapur 713209, India b

a r t i c l e

i n f o

Keywords: Uncertainty modelling Single-period inventory problem Uncertainty theory Uncertain random variable Chance theory

a b s t r a c t In many real inventory situations of short life-cycle products the decision maker often has to provide a subjective estimate of new demand distribution due to the lack of historical data. Thus, both randomness and uncertainty simultaneously appear in a single-period inventory (newsboy) problem. In this paper, we develop both single-item and multi-item single-period inventory models when market demands are assumed to be uncertain random variables. The objective of this study is to provide theoretical analysis of the models that attains optimality when demand information availability in subjective judgments leading to uncertainty along with random variation. The uncertain random models are transferred to equivalent deterministic forms by considering expected profit and providing more information of chance distributions. Finally, the numerical examples of ordering pharmaceutical reference standard materials are presented to illustrate the models. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Decision making in inventory management is always accompanied by uncertainties especially when dealing with demand related to new innovative products, seasonal products, sport goods or fashion goods etc. The characteristic of short selling season and long lead times for such products provide only one order opportunity and hence inventory problem can be solved properly as single-period decision-making setting. Thus, single-period inventory problem (popularly known as the Newsboy problem) is often formulated as a classical production/inventory management problem, which was primarily studied by Hadley and Whitin [4]. Most of the extensions of single-period inventory problem have been made in the probabilistic framework, in which the uncertainty of demand is characterized by the random demands. After Hadley and Whitin’s pioneering work, both single-item case and multi-item case with constraints have been widely studied in the area of production/inventory management. By assuming random demand, Nahmias and Schmidt [17] considered a multi-item single-period inventory problem subject to linear and deterministic constraints on space or budget. Moon and Silver [16] continued considering the multi-item issue subject to not only a budget constraint on the total value of the replenishment quantities but also fixed costs for non-zero replenishment. Vairaktarakis [28] focused on how to describe uncertainty and discussed the multi-item problem with budget constraint by using interval and discrete demand scenarios. Olzer et al. [18] utilized value-at-risk as the risk measure and investigated the

∗

Corresponding author. Tel.: +86 10 82339735. E-mail addresses: [email protected] (D. Wang), [email protected] (Z. Qin), [email protected] (S. Kar).

http://dx.doi.org/10.1016/j.amc.2015.06.102 0096-3003/© 2015 Elsevier Inc. All rights reserved.

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D. Wang et al. / Applied Mathematics and Computation 269 (2015) 133–145

multi-product newsvendor problem under a value-at-risk constraint. As a fundamental management tool, single-period inventory problem has actually always attracted more attentions. Some recent developments are devoted to the extensions of the basic model. For example, Kamburowski [6] provided new theoretical foundations for analyzing the case under incomplete information about probability distribution of random demand, Rossi et al. [24] introduced a novel strategy to address the issue of demand estimation by combining confidence interval analysis and inventory optimization, Wu et al. [31] studied a risk-averse situation with quantity competition and price competition based on conditional value-at-risk criterion and Sayın et al. [25] considered both random demand and random supply and provided the optimal ordering policy and optimal portfolio at the same time. For the case with partial information, several authors such as Qiu and Shang [23], Turgay et al. [27] and Wang et al. [29] also apply robust optimization to handle the distribution uncertainty of probability parameters in the newsvendor or inventory problems. However, randomness is not the unique uncertainty to deal with the real world inventory problems. Sometimes the probability distributions of the demands for products are difficult to acquire due to lack of information and historical data. In such a case, the demands are approximately specified based on the experience and subjective judgments of decision makers/experts. The decision maker has to provide the belief degree to describe the market demand. The uncertainty theory, founded by Liu [8] is a feasible tool that deals with such uncertainties or belief degrees. Based on this framework, Liu [9] further proposed uncertain programming for solving optimization problems involving uncertain variables. In this connection one may refer to the recent works on different applications of uncertainty theory such as risk analysis [10], shortest path problem [3], portfolio optimization problem [5,15], insurance risk model [11], facility location-allocation [30], new product development [32], p-hub center location [19] and so on. Particularly, Qin and Kar [20] first extended the single-period inventory model by considering market demand as an uncertain variable. On this basis, Ding [1] extended Qin and Kar’s work to multi-product case by adding chance constraint. No matter random models (Hadley and Whitin [4]) or uncertain models (Qin and Kar [20]) only handles single-fold uncertainty in the single-period inventory problem. However, in many real situations, randomness and uncertainty often exist simultaneously in a complex system. To describe such a system, Liu [12] first proposed the concept of uncertain random variable and a new chance measure to measure the possibility of a hybrid event. Correspondingly, Liu [12] also presented the operational law of uncertain random variable and introduced the related concepts such as chance distribution, expected value, variance and so on. As a general theoretical framework to model the practical problems with unknown parameters, uncertain random programming was introduced by Liu [13] and then extended to uncertain random multi-objective programming [33], uncertain random multilevel programming [7] and uncertain random goal programming [21]. As applications, Sheng and Gao [26] considered arc capacities of a network as random variables or uncertain variables and then derived the chance distribution of the maximum flow. Qin [22] presented a mean-variance model for portfolio optimization problem with mixture of random and uncertain returns. Liu and Ralescu [14] defined a risk index to quantify the risk of a system with uncertain random parameters, and Ding [2] introduced single-product newsboy problem with the demand consisting of random variable and uncertain variable. In some real-life inventory problems, demand parameter is uncertain and diversity of events may cause the lack of data. So we have to invite some experts to evaluate their degree of belief that each event will occur. For instance, the demands for different pharmaceutical reference standard materials are determined by the decision makers through integrating their subjective judgments and the historical demands for similar goods. Motivated by this point, we assume that the demands are described by uncertain random variables and develop novel single-period inventory models for such decision environments. The main contribution of this paper is to provide a more general framework for single-period inventory problem by considering single-item and multiple items with a budget constraint, respectively. The proposed models capture both uncertain and random behavior of the demands and cover not only the random case but also the single-fold uncertain case. They are more suitable for determining an unambiguous optimal order quantity when the demands are estimated by combining experts’ point of view and some small amount of historical data. The rest of the paper is organized as follows. In Section 2, we review the necessary preliminaries related to uncertain measure, uncertain variable and uncertain random variable. In Section 3, an uncertain random single-period inventory model is constructed for single item and some analytical findings are provided related to the optimal order quantity. Section 4 develops the multi-item situation by introducing a budget constraint and the equivalent deterministic forms are given for simple and mixed uncertain random demands. The numerical examples of ordering pharmaceutical reference standard materials are presented and analyzed in Section 5. Finally, we conclude the paper in Section 6. 2. Preliminaries In this section, we review some preliminaries about uncertain measure, uncertain variable and uncertain random variable. The first concept is uncertain measure, which was proposed by Liu [8] to indicate the possibility that a possible event happens. Let  be a nonempty set and L a σ -algebra on it. A set function M : L → [0, 1] is called an uncertain measure by Liu [8] if it satisfies: (1) M{ } = 1 for the universal set  ; (2) M{} + M{c } = 1 for any event  ∈ L; (3) For every countable sequence ∞  of events 1 , 2 , . . . , we have M{ ∞ i=1 i } ≤ i=1 M{i }. The triple ( , L, M) is called an uncertain space. If (k , Lk , Mk ) are uncertainty spaces for k = 1, 2, . . . , then Liu [9] defined the product uncertain measure M as an uncertain measure satisfying



M

∞ 

k=1



k

=

∞ 

Mk {k }

k=1

where k are arbitrarily chosen events from Lk for k = 1, 2, . . . , respectively.

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The second concept is an uncertain variable, which is defined by Liu [9] as a measurable function from an uncertainty space to the set of real numbers. That is, for any Borel set B of real numbers, if the set

 {ξ ∈ B} = {γ ∈   ξ (γ ) ∈ B}

is an event, then ξ is called an uncertain variable on an uncertainty space ( , L, M). For x ∈ , (x) = M{ξ ≤ x} is called the uncertainty distribution of ξ . An uncertainty distribution is said to be regular if it is a continuous and strictly increasing function with respect to x at which 0 < (x) < 1, and

lim (x) = 0,

x→−∞

lim (x) = 1.

x→+∞

The inverse function −1 (α) is called the inverse uncertainty distribution of ξ if it exists and is unique for each α ∈ (0, 1). The commonly used uncertainty distributions are all regular such as linear, zigzag, normal and lognormal uncertainty distributions. In addition, inverse uncertainty distribution also plays a crucial role in the operations of independent uncertain variables. Next we give the definition of independence in the sense of uncertain measure. Definition 1. (Liu [9]) The uncertain variables ξ1 , ξ2 , . . . , ξn are said to be independent if



M



n

{ξi ∈ Bi } =

i=1

n 

M{ξi ∈ Bi }

i=1

for any Borel set B1 , B2 , . . . , Bn of real numbers. Lemma 1. (Liu [9]) Let ξ1 , ξ2 , . . . , ξn be independent uncertain variables with regular uncertainty distributions 1 , 2 , . . . , n , respectively. If f (x1 , x2 , . . . , xn ) is strictly increasing with respect to x1 , x2 , . . . , xn , then ξ = f (ξ1 , ξ2 , . . . , ξn ) is an uncertain variable with an inverse uncertainty distribution −1 −1 −1 (α) = f (−1 1 (α), 2 (α), . . . , n (α)).

Definition 2. (Liu [9]) The expected value of an uncertain variable ξ is defined by

Eu [ ξ ] =



+∞ 0

M{ξ ≥ x}dx −



0 −∞

M{ξ ≤ x}dx

provided that at least one of the two integrals exists. Lemma 2. (Liu [9]) Let ξ be an uncertain variable with a regular uncertainty distribution . If the expected value Eu [ξ ] exists, then

Eu [ ξ ] =



1 0

−1 (α)dα .

The third concept is uncertain random variable, which is employed to describe a complex system with not only uncertainty but also randomness. Let ( , L, M) be an uncertainty space, and ( , P, Pr) a probability space. Then the product ( , L, M) × ( , P, Pr) is called a chance space. Definition 3. (Liu [12]) Let ( , L, M) × ( , P, Pr) be a chance space, and let  ∈ L × P. Then the chance measure of uncertain random event  is defined as

Ch{} =



0

1

Pr{ω ∈ |M{γ ∈  |(γ , ω) ∈ } ≥ r}dr.

(1)

The chance measure Ch is proved by Liu [12] to be monotone increasing and self-dual, i.e., for any event , we have Ch{} + Ch{c } = 1. For any  ∈ L and A ∈ P, we have Ch{ × A} = M{} × Pr{A}. Especially, Ch{∅} = 0 and Ch{ × } = 1. Moreover, the chance measure is also subadditive. That is, for any countable sequence of events 1 , 2 , . . . , we have ∞  Ch{ ∞ i=1 i } ≤ i=1 Ch{i }. Definition 4. (Liu [12]) An uncertain random variable is a function ξ from the chance space ( , L, M) × ( , P, Pr) to the set of real numbers, i.e., ξ ∈ B is an event in L × P for any Borel set B. It follows from Definition 4 that random variables and uncertain variables are special cases of uncertain random variables. If η is a random variable and τ is an uncertain variable, then the sum η + τ and the product ητ are both uncertain random variables. Lemma 3. (Liu [12]) Let f: n →  be a measurable function, and ξ1 , ξ2 , . . . , ξn uncertain random variables on the chance space ( , L, M) × ( , P, Pr). Then ξ = f (ξ1 , ξ2 , . . . , ξn ) is an uncertain random variable determined by

ξ (γ , ω) = f (ξ1 (γ , ω), ξ2 (γ , ω), . . . , ξn (γ , ω)) for all (γ , ω) ∈  × .

(2)

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Definition 5. (Liu [12]) Let ξ be an uncertain random variable. Then its chance distribution is defined by (x) = Ch{ξ ≤ x}, and its expected value is defined by

E[ξ ] =



+∞ 0

Ch{ξ ≥ r}dr −



0 −∞

Ch{ξ ≤ r}dr

provided that at least one of the two integrals is finite. Lemma 4. (Liu [12, 13]) Let η1 , η2 , . . . , ηm be independent random variables with probability distribution functions 1 , 2 , . . . , m , respectively, and let τ1 , τ2 , . . . , τn be independent uncertain variables. Then the uncertain random variable ξ = f (η1 , . . . , ηm , τ1 , . . . , τn ) has a chance distribution

(x) =



m

M{ f (y1 , . . . , ym , τ1 , . . . , τn ) ≤ x}d 1 (y1 ) . . . d m (ym ),

x∈

(3)

and has an expected value

E[ξ ] =



m

Eu [ f (y1 , . . . , ym , τ1 , . . . , τn )]d 1 (y1 ) . . . d m (ym )

(4)

where Eu [ f (y1 , . . . , ym , τ1 , . . . , τn )] is the expected value of the uncertain variable f (y1 , . . . , ym , τ1 , . . . , τn ) for any real numbers y1 , …, ym . 3. Single-item single-period inventory problem In this section, we consider a single-item single-period inventory problem for a decision maker without initial inventory. The decision maker will purchase goods before a selling period at a unit ordering cost w and then sell these goods for the following period at a unit selling price p. The unsold goods are handled at a unit salvage value s at the end of the selling period. Without loss of generality, we assume that p > w > s > 0 to avoid the trivial situation. Let us denote the order quantity by y and the demand by ξ . As stated before, in this study we describe the demand ξ by an uncertain random variable (Definition 4). By the notations, the total profit of the decision maker is given by

h(y, ξ ) = p · min{ξ , y} + s(y − ξ )+ − wy = ( p − w)y − ( p − s)(y − ξ )+

(5) y − (y − ξ )+ .

= max (a, 0) and the second equality holds due to the identity min{ξ , y} = where Note that the total profit h(y, ξ ) is an uncertain random variable since it is a measurable function of an uncertain random variable ξ . Thus, it is meaningless to directly maximize the profit since we cannot compare two uncertain random variables. A basic criterion is to compare their expected values and the one with bigger expected value is called larger than the other with smaller expected value. In such criterion, we formulate the following expected value model to look for the optimal order quantity, a+

max H (y) ⇐⇒ max E[h(y, ξ )] ⇐⇒ max E[( p − w)y − ( p − s)(y − ξ )+ ]. y≥0

y≥0

(6)

y≥0

Denote the optimal order quantity by y∗ . Then if y∗ exists, we have H (y∗ ) = maxy≥0 H (y). Theorem 1. Let ξ be an uncertain random variable defined on chance space ( , L, M) × ( , A, Pr). Then we have



H (y) = ( p − s)

y (w−s)y

Ch{ξ ≥ x}dx −



p−s



(w−s)y p−s

Ch{ξ ≤ x}dx .

−∞

(7)

Proof. It follows from the definition of expected value of uncertain random variable (Definition 5) that

H (y) = E[( p − w)y − ( p − s)(y − ξ )+ ]

+∞

 = Ch ( p − w)y − ( p − s)(y − ξ )+ ≥ r dr − 0

=



+∞

(y − ξ ) ≤ +

Ch 0

( p − w)y − r p−s



dr −



0 −∞

 ( p − w)y − ( p − s)(y − ξ )+ ≤ r dr −∞   ( p − w)y − r + Ch (y − ξ ) ≥ dr 0

Ch

p−s

By changing of variable r to u = [( p − w)y − r]/( p − s), we obtain

H (y) = ( p − s)

( p−w)y/( p−s) 0



= ( p − s)

Ch{(y − ξ )+ ≤ u}du − ( p − s)

( p−w)y/( p−s) 0

Ch{ξ ≥ y − u}du −





+∞

( p−w)y/( p−s)

+∞

( p−w)y/( p−s)

Ch{(y − ξ )+ ≥ u}du



Ch{ξ ≤ y − u}du .

By changing of variable u to x = y − u, the desired result follows and the theorem is proved. 

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137

Theorem 2. Let ξ be an uncertain random variable with a continuous chance distribution (x). Then y∗ is the optimal order quantity of model (6) if and only if

(y∗ ) =

p−w . p−s

(8)

Proof. The assumption of continuous chance distribution implies that Ch{ξ < x} = Ch{ξ ≤ x} = (x). It follows from the selfduality of chance measure that



H (y) = ( p − s)

y (w−s)y

(1 − Ch{ξ < x})dx −



p−s



(w−s)y p−s

−∞

Ch{ξ ≤ x}dx



y

(w − s)y − = ( p − s) y − Ch{ξ ≤ x}dx − (w−s)y p−s

= ( p − w)y − ( p − s)

p−s

y −∞



(w−s)y p−s

−∞

Ch{ξ ≤ x}dx

(x)dx.

Taking derivatives on both sides, we have H (y) = p − w − ( p − s)(y). Since (y) is an increasing function, H (y) is a decreasing function which indicates that H(y) is a concave function. Setting H (y) = 0 yields

(y) =

p−w p−s

which is the sufficient and necessary condition satisfied by y∗ . The proof is completed.  y Remark 1. From the proof of Theorem 2, the expected total profit is H (y) = ( p − w)y − ( p − s) −∞ (x)dx when ξ has a continuous chance distribution (x). As an uncertain random demand, ξ represents a more general description which covers both uncertain and random cases. If ξ degenerates into an uncertain variable, then model (6) becomes uncertain singleperiod inventory problem investigated by Qin and Kar [20]. Denote the uncertainty distribution of ξ by ϒ . Then we have y H (y) = ( p − w)y − ( p − s) −∞ ϒ (x)dx. If ξ degenerates into a random variable, then model (6) becomes the classical newsboy model in stochastic environment. Denote the probability distribution of ξ by . Then we have

E p [(y − ξ )+ ] =



+∞

−∞

(y − x)+ d (x) =



y

−∞

(y − x)d (x)

where Ep is the expected value operator of random variable. Further, we get

H (y) = ( p − w)y − ( p − s)



y −∞

(y − x)d (x)

which is consistent with Theorem 2. Actually, by the integral by parts formula and (−∞) = limx→−∞ (x) = 0, we can obtain that

H (y) = ( p − w)y − ( p − s)(y − x) (x)|y−∞ + ( p − s)



y −∞

(x)d(y − x) = ( p − w)y − ( p − s)



y −∞

(x)dx.

Simple uncertain random variable Let τ1 , τ2 , . . . , τm be uncertain variables with continuous uncertainty distributions ϒ1 , ϒ2 , . . . , ϒm , respectively. An uncertain random variable ξ is called simple if it has the following form,

⎧ τ1 , ⎪ ⎪ ⎨τ , 2 ξ= . . . ⎪ ⎪ ⎩ τm ,

where

m j=1

with probability π1 with probability π2

(9)

with probability πm

π j = 1. It follows from Definition 3 that the chance distribution of ξ is

(y) =

m  j=1

π j · M{τ j ≤ y} =

m 

π j ϒ j (y).

(10)

j=1

Since ϒ1 , ϒ2 , . . . , ϒm are all continuous, the chance distribution (y) is obviously a continuous function. As a result, by means of Theorem 2, the optimal order quantity y∗ is determined by the equation

π1 ϒ1 (y∗ ) + π2 ϒ2 (y∗ ) + · · · + πm ϒm (y∗ ) = ( p − w)/( p − s).

138

D. Wang et al. / Applied Mathematics and Computation 269 (2015) 133–145

Mixed uncertain random variable Let η1 , η2 , . . . , ηm be random variables and τ1 , τ2 , . . . , τl uncertain variables. Assume that f: m →  and g: l →  are both measurable functions. An uncertain random variable ξ is called mixed if it has the form ξ = f (η1 , . . . , ηm ) + g(τ1 , . . . , τl ). From Lemma 4, the chance distribution of ξ is

(y) =



+∞ −∞

ϒ (y − x)d (x)

(11)

where is the probability distribution of f (η1 , . . . , ηm ) and ϒ is the uncertainty distribution of g(τ1 , . . . , τl ). Applying Theorem 2 again, the optimal order quantity is determined by the equation



+∞

−∞

ϒ (y∗ − x)d (x) = ( p − w)/( p − s).

If m = l = 1 and f (x) = x, g(x) = x, then ξ = η + τ , i.e., the uncertain random variable ξ is the sum of a random variable and an uncertain variable, which is just the special case studied by Ding [2]. 4. Multi-item single-period inventory problem We next consider a multi-item case faced by the retailer without initial inventory. Assume that there are n items available. The unit ordering cost, unit selling price and unit salvage value are denoted by wi , pi and si , respectively, for the ith product. Similarly, we require that pi > wi > si > 0 for i = 1, 2, . . . , n. Denote by yi the quantity purchased for ith item at the beginning of the selling season and by ξ i the demand for ith item, which is considered as an uncertain random variable. The profit function for ith item is given by

hi (yi , ξi ) = ( pi − wi )yi − ( pi − si )(yi − ξi )+ .

 Then the total profit of the retailer is ni=1 hi (yi , ξi ). Assume that the available budget for the retailer is w, which implies that it is n required that a budget constraint i=1 wi yi ≤ w holds. Based on the criterion of expected value, we formulate an expected profit maximization model for the uncertain random multi-item single-period inventory problem,

⎧ ⎪ ⎪ ⎪ max ⎪ ⎪ ⎪ ⎨

E

n 

hi (yi , ξi )

i=1 n 

⎪ s.t. ⎪ ⎪ ⎪ ⎪ ⎪ ⎩





(12)

wi yi ≤ w,

i=1

yi ≥ 0,

i = 1, 2, . . . , n.

Mixture of random and uncertain demands We first consider a simple case in which some demands are random variables and the others are uncertain variables. It is true in the cases that some mature products have relatively regular demand patterns and more historical data on their sales, and the others are new products with volatile demand and almost no sales data. Without loss of generality, we assume that ξ1 , ξ2 , . . . , ξn1 are random variables and ξn1 +1 , . . . , ξn are uncertain variables. Then we have the following result. Theorem 3. Let ξ1 , ξ2 , . . . , ξn1 be random variables with probability distributions 1 , 2 , . . . , n1 , respectively, and ξn1 +1 , . . . , ξn independent uncertain variables with continuous uncertainty distributions ϒn1 +1 , . . . , ϒn , respectively. The Model (12) can be formulated as

⎧ ⎪ ⎪ ⎪max ⎨

n 

⎪ s.t. ⎪ ⎪ ⎩

w1 y1 + w2 y2 + · · · + wn yn ≤ w,

( pi − wi )yi −

i=1

n1 



( pi − si ) yi i (yi ) −

−∞

i=1

yi ≥ 0,

yi



n  xd i (x) − ( pi − si ) i=n1 +1

yi −∞

ϒi (x)dx (13)

i = 1, 2, . . . , n.

Proof. It is evident that the profit h1 (y1 , ξ1 ) + · · · + hn1 (yn1 , ξn1 ) is a random variable, and the profit hn1 +1 (yn1 +1 , ξn1 +1 ) + · · · + hn (yn , ξn ) is an uncertain variable. Assume that the former has a probability distribution and the latter has an uncertainty distribution ϒ . It follows from Lemma 4 that



E

n 



hi (yi , ξi )



=E

i=1

n1 

hi (yi , ξi ) +

i=1



=



n 

z + Eu



i=n1 +1 n 

i=n1 +1



hi (yi , ξi )



hi (yi , ξi )

=





Eu z +

d (z) = E p



n  i=n1 +1

n1  i=1



hi (yi , ξi ) d (z)



hi (yi , ξi ) + Eu



n  i=n1 +1

 hi (yi , ξi ) .

D. Wang et al. / Applied Mathematics and Computation 269 (2015) 133–145

139

Since ξn1 +1 , . . . , ξn are independent uncertain variables, hn1 +1 (yn1 +1 , ξn1 +1 ), . . . , hn (yn , ξn ) are also independent uncertain variables. By the linearity of expected value of independent uncertain variables, we have





E

n 

hi (yi , ξi )

i=1

=

n1 









n 

E p hi (yi , ξi ) +

Eu hi (yi , ξi ) .

i=n1 +1

i=1

For 1 ≤ i ≤ n1 , we have

E p [hi (yi , ξi )] = ( pi − wi )yi − ( pi − si )



yi −∞



(yi − x)d i (x) = ( pi − wi )yi − ( pi − si ) yi i (yi ) −

yi −∞

xd i (x) .

From the proof of Theorem 2, for n1 + 1 ≤ i ≤ n, we have

Eu [hi (yi , ξi )] = ( pi − wi )yi − ( pi − si )



yi −∞

ϒi (x)dx.

The theorem is completed.  Remark 2. The objective function of Model (13) may be converted into an analytical function of y1 , y2 , . . . , yn for some cases such as uniform, exponential and triangular probability distributions, and linear, zigzag and normal uncertainty distributions. The associated deterministic model can be easily solved by many algorithms or softwares. General uncertain random demands In this part, we assume that each demand ξ i is defined on the chance space ( , L, M) × ( , P, Pr) with a continuous chance  distribution i for i = 1, 2, . . . , n. Due to the complexity of this general case, we directly employ ni=1 E[hi (yi , ξi )] as the objective function. From the proof of Theorem 2, we have

E[hi (yi , ξi )] = ( pi − wi )yi − ( pi − si )



yi −∞

i (x)dx.

By summing we can obtain the converted objective function and the result is given in next theorem. Theorem 4. Assume that ξ1 , ξ2 , . . . , ξn are independent uncertain random variables with continuous chance distributions i . Then we can formulate the following multi-item single-period inventory model,

⎧ ⎪ ⎪ ⎨max ⎪ ⎪ ⎩s.t.

n 

( pi − wi )yi −

i=1

n 

( pi − si )

i=1



yi −∞

i (xi )dxi (14)

w 1 y1 + w 2 y2 + · · · + w n yn ≤ w yi ≥ 0, i = 1, 2, . . . , n.

Next we further discuss this model by providing more information of uncertain random demands. Corollary 1. Assume that each demand ξi (i = 1, 2, . . . , n) is a simple uncertain random variable given by

⎧ i τ1 , ⎪ ⎪ ⎨τ i , ξi = 2 ... ⎪ ⎪ ⎩ i τm i ,

with probability π1i with probability π2i with probability πmi i .

Then Model (14) is equivalent to the following form,

⎧ ⎪ ⎪ ⎪ ⎨max

n 

⎪ s.t. ⎪ ⎪ ⎩

w 1 y1 + w 2 y2 + · · · + w n yn ≤ w

( pi − wi )yi −

i=1

mi n  

( pi − si )π ji

i=1 j=1

yi ≥ 0,



yi −∞

ϒ ij (xi )dxi

i = 1, 2, . . . , n

where ϒ ij is the uncertainty distribution of uncertain variable τ ji for i = 1, 2, . . . , n and j = 1, 2, . . . , mi . Proof. It follows from Eq. (10) that the uncertainty distribution of ξ i is

i (xi ) =

mi  j=1

π ji ϒ ij (xi )

(15)

140

D. Wang et al. / Applied Mathematics and Computation 269 (2015) 133–145

which implies that n 

( pi − si )



yi 0

i=1

n 

i (xi )dxi =



( pi − si )

mi yi  0

i=1

π ji ϒ ij (xi )dxi =

mi n  

j=1

( pi − si )π ji

i=1 j=1



yi 0

ϒ ij (xi )dxi .

This proves the corollary.  Remark 3. If τ ji = L(aij , bij ) is a linear uncertain variable for i = 1, 2, . . . , n and j = 1, 2, . . . , mi , then the uncertainty distribution function is ϒ ij (xi ) = n 

xi −aij

I

i i bij −aij {a j ≤xi
+ I{x ≥bi } and the objective function of Model (15) is equal to i

j



(yi − aij )2

n mi 1  π ji ( pi − si ) 2

( pi − wi )yi −

i=1

bij − aij

i=1 j=1

 I{ai ≤yi
j

j

in which I{·} is the indicator function of the set {·}. If τ ji = Z (aij , bij , cij ) is a zigzag uncertain variable for i = 1, 2, . . . , n and xi −aij

j = 1, 2, . . . , mi , then the uncertainty distribution function is ϒ ij (xi ) = objective function of Model (15) is equal to n 

( pi − wi )yi −

i=1



n mi 1  π ji ( pi − si ) 4

(yi − aij )2 −

bij

i=1 j=1

aij

I{ai ≤yi
2(bij −aij )

I{ai ≤x
j

j

(yi + cij − 2bij )2 cij − bij

j



xi +cij −2bij

+ I{x ≥ci } and the

I

i i 2(cij −bij ) {b j ≤xi
i

j

I{bi ≤yi
j

+ 4(yi − bij )I{yi ≥ci } − (aij − 2bij + cij )I{yi ≥bi } . j

j

If τ ji = N (eij , σ ji ) is a normal uncertain variable for i = 1, 2, . . . , n and j = 1, 2, . . . , mi , then the uncertainty distribution function is ϒ ij (xi ) = (1 + exp ( n 

π (eij −xi ) √

3σ ji

))−1 and the objective function of Model (15) is equal to

√ n mi 3 

( pi − wi )yi −

π

i=1

 



π ji σ ji ( pi − si ) ln 1 + exp

i=1 j=1

π (eij − yi )

 −

√ 3σ ji

π (eij − yi ) √ 3σ ji

 .

Corollary 2. Let ηi be a random variable with probability distribution i and τ i an uncertain variable with uncertainty distribution Yi for i = 1, 2, . . . , n. Assume that each demand ξ i is a mixed uncertain random variable defined as ξi = ηi + τi . Then Model (14) is equivalent to the following form,

⎧ ⎪ ⎪ ⎨max

n 

⎪ s.t. ⎪ ⎩

w 1 y1 + w 2 y2 + · · · + w n yn ≤ w

( pi − wi )yi −

i=1

n 

( pi − si )



−∞

i=1

yi ≥ 0,



yi

+∞

−∞

ϒi (xi − r)d i (r)dxi (16)

i = 1, 2, . . . , n.

Proof. Similar to the proof of Corollary 1 by using Eq. (11).  Example 1. When the probability distribution is exponential and the uncertainty distribution is linear, we can analytically calculate the objective function. If τi = L(ai , bi ) is a linear uncertain variable for i = 1, 2, . . . , n, then we have



yi −∞



+∞

−∞

ϒi (xi − r)d i (r)dxi =



yi



−∞

xi −ai xi −bi

xi − r − ai d i (r) + bi − ai



xi −bi −∞

 d i (r) dxi .

By using the integral by parts formula and noting that limr→−∞ i (r) = 0, we get



yi −∞



+∞

−∞

ϒi (xi − r)d i (r)dxi =

1 bi − ai





yi −∞

xi −ai xi −bi

i (r)drdxi .

Let i (r) = 1 − exp (−r/λi ) be an exponential distribution. Then we have



yi −∞



xi −ai

xi −bi

i (r)drdxi = I{ai ≤yi ≤bi }

= I{ai ≤yi ≤bi }

+ I{yi >bi }

yi ai yi

bi

 



yi



ai

xi − ai − λi + λi exp

xi −ai 0

i (r)drdxi + I{yi >bi }

ai − xi

!"

dxi + I{yi >bi }





bi ai

bi





xi −ai 0

i (r)drdxi +

xi − ai − λi + λi exp

λi a  i ! ai − xi b i − xi bi − ai + λi exp − λi exp dxi λi λi



yi

bi



xi −bi

ai − xi

λi

xi −ai

i (r)drdxi

!" dxi

D. Wang et al. / Applied Mathematics and Computation 269 (2015) 133–145

141

 !" I{ai ≤yi ≤bi }  bi + ai ai − yi 2 2 2 (yi − ai − λi ) + λi − 2λi exp = + I{yi >bi } (bi − ai ) yi − λi − 2 λi 2   ! b i − yi ai − yi + I{yi >bi } λ2i exp − exp . λi λi Then the objective function of Model (16) is equal to n 

( pi − wi )yi −

i=1

−

n  ( pi − si )I{ai ≤yi ≤bi }  i=1

n  i=1



(yi − ai − λi )2 + λ2i − 2λ2i exp

2(bi − ai )

b + ai ( pi − si )I{yi >bi } yi − λi − i 2

−

n 

λ ( pi − si )I{yi >bi } 2 i

 exp

bi − ai

i=1

ai − yi

!"

λi 

b i − yi



λi

− exp

ai − yi

!

λi

which is an analytical function of the decision variables y1 , y2 , . . . , yn . Remark 4. For most distributions, we cannot obtain the analytical expression of the objective function. Thus, we have to use numerical integration techniques to calculate the objective. 5. Numerical examples In this section, we present several numerical examples to illustrate the proposed modelling approach and to provide intuitive understanding. These examples are based on the procurement decision of pharmaceutical reference standard materials. Suppose that a decision maker needs to purchase these materials before a selling season. The decision maker first need to estimate the future demand for each kind of pharmaceutical reference standard materials. Based on different types of the estimated demands, the decision maker may choose appropriate model to determine the associated optimal order quantities. Example 2. We consider a single-item situation and the demand is described by a simple uncertain random variable with two normal uncertainty distributions

  −1   −1 π (e1 − y) π (e2 − y) ϒ1 (y) = 1 + exp and ϒ2 (y) = 1 + exp . √ √ 3σ1 3σ2 Without loss of generality, we let π1 = 0.4, π2 = 0.6, e1 = 200, σ1 = 10, e2 = 300 and σ2 = 20. Then the optimal order quantity y∗ is the solution of the nonlinear equation with one unknown quantity



2 1 + exp 5



π (200 − y) √ 10 3

−1



+

3 1 + exp 5



π (300 − y)

−1 −

√ 20 3

p−w = 0. p−s

(17)

The Matlab function “fzero” is used to find the root of Eq. (17) when p, w and s are given. Fig. 1 provides the values of optimal order quantity y∗ with respect to one parameter when other two parameters are fixed. For example, the first figure shows the sensitivity of y∗ to the change of unit selling price p when the unit salvage value s is 1 and the unit ordering cost w is 11. Example 2 implies that the optimal order quantity is easy to obtain when the demand is described by a simple uncertain random variable. Actually, even if ϒ1 , ϒ2 , . . . , ϒm are of different forms, the optimal order quantity can also be numerically obtained by using fzero according to Theorem 2. Thus, the key is to accurately estimate the uncertainty distributions and their associated occurrence probability when applying the proposed model in practice. The related estimate methods may consult uncertain statistics in Liu [9]. Example 3. We continue considering a single-item issue but the demand is described by a mixed uncertain random variable

ξ = η + τ . Suppose that η ∼ N (μ, σ 2 ) is a normal random variable and τ = L(a, b) is a linear uncertain variable. From Eq. (11), the chance distribution of ξ is

(y) = √

1

2πσ



+∞ −∞

 (x − μ)2 ϒ (y − x) · exp − dx 2σ 2

where ϒ is the uncertainty distribution of τ . Substituting the expression of ϒ into the above equation, we can obtain that



!





y−x−a (x − μ)2 I{y−b≤x≤y−a} + I{x≤y−b} · exp − dx b−a 2σ 2 2πσ −∞  

y−a

y−b 1 1 (x − μ)2 (x − μ)2 = √ (y − x − a) exp − dx + √ exp − dx. 2σ 2 2σ 2 2πσ (b − a) y−b 2πσ −∞

(y) = √

1

+∞

142

D. Wang et al. / Applied Mathematics and Computation 269 (2015) 133–145 320

340

300

320 p=30, s=1 300 Optimal order quantity, y*

Optimal order quantity, y*

280 s=1,w=11 260 240 220 200

280 260 240 220 200 180

180

160 160 10

15

20

25 30 Unit selling price p

35

140

40

5

10

15 20 Unit ordering cost w

25

30

380 370

p=30, w=15

Optimal order quantity, y*

360 350 340 330 320 310 300 290 280 0

5

10 Unit salvage value s

15

Fig. 1. Sensitivity of the optimal order quantity with respect to p, w and s, respectively.

Setting a = 100, b = 200, μ = 150 and σ = 10, then the above equation is simplified as

(y) =

1 √ 1000 2π



y−100 y−200

 (x − 150)2 (y − x − 100) exp − dx + 200

1 √ 10 2π



y−200

−∞

 exp −

(x − 150)2 200

dx.

The Matlab function “fzero” is used to find the solution of the equation (y) = ( p − w)/( p − s), in which the integrals are calculated by the function “quadl”. For example, letting p = 25, w = 11 and s = 1, the optimal order quantity is y∗ = 308.33. Example 4. We consider a two-item single-period inventory problem in which the demands are described by an exponential distribution and a normal uncertainty distribution, respectively. That is,

(x) = 1 − exp −

x

λ

!

 , x ≥ 0 and ϒ (x) =

Then Model (13) is converted into the following model,

⎧ ⎪ max ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ s.t. ⎪ ⎪ ⎩

 1 + exp



π (e − x) √ 3σ

−1

!"

y1 −(w1 − s1 )y1 − (w2 − s2 )y2 + λ( p1 − s1 ) 1 − exp − λ   √ 3σ ( p2 − s2 ) π (e − y2 ) +( p2 − w2 )e − ln 1 + exp √ π 3σ w 1 y1 + w 2 y2 ≤ w y1 ≥ 0, y2 ≥ 0.

,

x ∈ .

D. Wang et al. / Applied Mathematics and Computation 269 (2015) 133–145 90

95

y*1

90

y*2

The optimal order quantities y*1 and y2*

100

The optimal order quantities y*1 and y*2

143

85 80 75 70 65

88

y*1

86

y*2

84 82 80 78 76 74

60 55 90

95

100 105 110 115 120 125 130 The expected value of random demand, i.e., λ

135

72 90

140

95

100 105 110 115 120 125 The expected value of uncertain demand, i.e., e

130

Fig. 2. Optimal order quantities y∗1 and y∗2 when λ and e changes, respectively.

Table 1 Unit selling prices, unit ordering costs and unit salvage values. Materials No.

Unit selling price (pi )

Unit ordering cost (wi )

Unit salvage value (si )

1 2 3 4 5 6

20 25 30 35 40 45

10 12 14 16 18 20

1 2 3 4 5 6

√ Let p1 = 23, w1 = 11, s1 = 1, p2 = 29, w2 = 14, s2 = 2, σ = 50 3π and the available capital is w = 2000. We employ the Matlab function “fmincon” to search for the optimal order quantities for two items. To show the impact of expected demands to the optimal order quantities, we first set e = 100 and fix e, and then solve the above model by changing the value of λ. The associated results are shown in the left side of Fig. 2, which implies that y∗1 is almost a linearly increasing function of λ, but y∗2 is almost a linearly decreasing function of λ. A similar observation can be found when changing the value of e. The right side of Fig. 2 shows the sensitivity of optimal order quantities with e when setting λ = 140 and fixing λ.

Example 5. We consider a relatively complex case and solve the proposed multi-product single-period inventory model. A decision maker needs to purchase six types of pharmaceutical reference standard materials before the selling season. The associated parameters are listed in Table 1. In this numerical experiment, the demands are assumed to be simple uncertain random variables given by

⎧ ⎧ ⎧ ⎨Z (200, 250, 300), π1 = 1/6 ⎨Z (220, 260, 300), π1 = 1/6 ⎨Z (180, 210, 240), π1 = 1/6 ξ1 = Z (250, 300, 350), π2 = 1/2 ξ2 = Z (260, 300, 340), π2 = 1/2 ξ3 = Z (210, 240, 270), π2 = 1/2 ⎩ ⎩ ⎩ Z (300, 350, 400), π3 = 1/3, Z (300, 340, 380), π3 = 1/3, Z (240, 270, 300), π3 = 1/3, ⎧ ⎧ ⎧ ⎨Z (170, 200, 230), π1 = 1/6 ⎨Z (160, 200, 240), π1 = 1/6 ⎨Z (100, 150, 200), π1 = 1/6 ξ4 = Z (200, 230, 260), π2 = 1/2 ξ5 = Z (200, 240, 280), π2 = 1/2 ξ6 = Z (150, 200, 250), π2 = 1/2 ⎩ ⎩ ⎩ Z (230, 260, 290), π3 = 1/3, Z (240, 280, 320), π3 = 1/3, Z (200, 250, 300), π3 = 1/3. In this case, we actually assume that there are three scenarios in the future and their appearance probabilities are 1/6, 1/2 and 1/3, respectively. In each scenario, the demand is characterized by a zigzag uncertain variable. For example, the demand of the first scenario is Z (200, 250, 300) which represents the demand is about 250 and the minimum possible quantity is 200, the maximum possible quantity is 300.

144

D. Wang et al. / Applied Mathematics and Computation 269 (2015) 133–145

Based on these given parameters, Model (15) is reformulated as

⎧ max ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪s.t. ⎩

24(10y1 + 13y2 + 16y3 + 19y4 + 22y5 + 25y6 )

    +23 1040I{y2 ≥260} − 4(y2 − 1160)I{y2 ≥300} − 4(3y2 − 905)I{y2 ≥340} − 8(y2 − 340)I{y2 ≥380}   +27 840I{y3 ≥210} − 4(y3 − 930)I{y3 ≥240} − 12(y3 − 420)I{y3 ≥270} − 8(y3 − 270)I{y3 ≥300}   +31 800I{y4 ≥200} − 4(y4 − 890)I{y4 ≥230} − 4(3y4 − 1210)I{y4 ≥260} − 8(y4 − 260)I{y4 ≥290}   +35 800I{y5 ≥200} − 4(y5 − 920)I{y5 ≥240} − 4(3y5 − 1280)I{y5 ≥280} − 8(y5 − 280)I{y5 ≥320}   +39 600I{y6 ≥150} − 4(y6 − 750)I{y6 ≥200} − 4(3y6 − 1100)I{y6 ≥250} − 8(y6 − 250)I{y6 ≥300}  19  − (y1 − 200)2 I{200≤y1 <300} + 3(y1 − 250)2 I{250≤y1 <350} + 2(y1 − 300)2 I{300≤y1 <400} 50  23  − (y2 − 220)2 I{220≤y2 <300} + 3(y2 − 260)2 I{260≤y2 <340} + 2(y2 − 300)2 I{300≤y2 <380} 40  27  − (y3 − 180)2 I{180≤y3 <240} + 3(y3 − 210)2 I{210≤y3 <270} + 2(y3 − 240)2 I{240≤y3 <300} 30  31  − (y4 − 170)2 I{170≤y4 <230} + 3(y4 − 200)2 I{200≤y4 <260} + 2(y4 − 230)2 I{230≤y4 <290} 30  35  − (y5 − 160)2 I{160≤y5 <240} + 3(y5 − 200)2 I{200≤y5 <280} + 2(y5 − 240)2 I{240≤y5 <320} 40  39  − (y6 − 100)2 I{100≤y6 <200} + 3(y6 − 150)2 I{150≤y6 <250} + 2(y6 − 200)2 I{200≤y6 <300} +19 1000I{y1 ≥250} − 4(y1 − 1150)I{y1 ≥300} − 4(3y1 − 925)I{y1 ≥350} − 8(y1 − 350)I{y1 ≥400}

(18)

50 10y1 + 12y2 + 14y3 + 16y4 + 18y5 + 20y6 ≤ w y1 , y2 , . . . , y6 ≥ 0.

Although the objective function is complex, it is actually a deterministic nonlinear programming. Assume that the given budget w is 15000. It is easily to be solved by using the Matlab function “fmincon”. Since fmincon relies on an initial guess, we need to try different initial values to make the maximization guaranteed. The optimal order quantities are (y∗1 , y∗2 , . . . , y∗6 ) = (40.6, 24.4, 296.4, 279.9, 280, 31.7) and the associated maximum profit is 36195.4. 6. Conclusions The fundamental difference of this paper with the existing literature is to consider a complex situation in which the demand is originated from subjective evaluations and/or small amount of data, not only separate indeterministic environment such as stochastic or fuzzy/uncertain case. The total demands were assumed to be uncertain random variables such as the sum of random variable and uncertain variable and so on. In this paper we first derived the optimality condition for optimal order quantity for single-item case and then considered a multi-item situation with a budget constraint. The equivalent deterministic forms were given in some special cases. Numerical examples were provided to illustrate the application of the proposed method. The proposed single-period inventory models not only cover random case but also uncertain case. Thus, they are more applicable for solving complex practical problems. Acknowledgments This work was supported in part by National Natural Science Foundation of China (Nos. 71371019 and 71332003), and in part by the Program for New Century Excellent Talents in University (No. NCET-12-0026). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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