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Research ArticleFuzzy Constrained Probabilistic Inventory ModelsDepending on Trapezoidal Fuzzy Numbers
Mona F. El-Wakeel1,2 and Kholood O. Al-yazidi1
1Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 22452,Riyadh 11495, Saudi Arabia2Higher Institute for Computers, Information and Management Technology, Tanta, Egypt
Correspondence should be addressed to Mona F. El-Wakeel; [email protected]
We discussed two different cases of the probabilistic continuous review mixture shortage inventory model with varying and con-strained expected order cost, when the lead time demand follows some different continuous distributions.The first case is when thetotal cost components are considered to be crisp values, and the other case iswhen the costs are considered as trapezoidal fuzzy num-ber. Also, some special cases are deduced. To investigate the proposedmodel in the crisp case and the fuzzy case, illustrative numer-ical example is added. From the numerical results we will conclude that Uniform distribution is the best distribution to get the exactsolutions, and the exact solutions for fuzzy models are considered more practical and close to the reality of life and get minimumexpected total cost less than the crisp models.
1. Introduction
Inventory system is one of the most diversified fields ofapplied sciences that are widely used in a variety of areasincluding operations research, applied probability, computersciences, management sciences, production system, andtelecommunications. More than fifty years ago, the analysisof inventory system has appeared in the reference books andsurvey papers. Hadley and Whitin [1] are considered oneof the first researchers who have discussed the analysis ofinventory systems, where they displayed a method for theanalysis of the mathematical model for inventory systems.Also, Balkhi and Benkherouf [2] have introduced productionlot size inventory model in which products deteriorate at aconstant rate and in which demand and production rates areallowed to vary with time. Inventory models may be eitherdeterministic or probabilistic, since the demand of commod-ity may be deterministic or probabilistic, respectively. Thesecases were dealt with by Hadley andWhitin [1], Abuo-El-Ataet al. [3], and Vijayan and Kumaran [4].
Some managers allow the shortage in inventory sys-tems; this shortage may be backorder case, lost sales case,
and mixture shortage case. Many authors are dealing withinventory problems with various shortage cases where thecost components are considered as crisp values which doesnot depict the real inventory system fully. For example,constrained probabilistic inventorymodel with varying orderand shortage costs using Lagrangian method has been inves-tigated by Fergany [5]. In addition, constrained probabilisticinventory model with continuous distributions and varyingholding cost was discussed by Fergany and El-Saadani [6].In 2006, several models of continuous distributions for con-strained probabilistic lost sales inventory models with vary-ing order cost under holding cost constraint using Lagrangianmethod by Fergany and El-Wakeel [7, 8] were discussed.Recently, El-Wakeel [9] deduced constrained backordersinventory system with varying order cost under holdingcost constraint: lead time demand uniformly distributedusing Lagrangian method. Also, El-Wakeel and Fergany[10] deduced constrained probabilistic continuous reviewinventory system with mixture shortage and stochastic leadtime demand.
Sometimes, the cost components are considered as fuzzyvalues, because, in real life, the various physical or chemical
Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2016, Article ID 3673267, 10 pageshttp://dx.doi.org/10.1155/2016/3673267
2 Advances in Fuzzy Systems
characteristics may cause an effect on the cost componentsand then precise values of cost characteristics become diffi-cult to measure as the exact amount of order, holding, andespecially shortage cost. Thus, in controlling the inventorysystem it may allow some flexibility in the cost parametervalues in order to treat the uncertainties which always fitthe real situations. Since we want to satisfy our requirementsfor such contradictions, the fuzzy set theory meets theserequirements to some extent. In 1965, Zadeh [11] first intro-duced the fuzzy set theory which studied the intention toaccommodate uncertainty in the nonstochastic sense ratherthan the presence of random variables. Syed and Aziz [12]have examined the fuzzy inventory model without shortagesusing signed distance method. Kazemi et al. [13] have treatedthe inventory model with backorders with fuzzy parametersand decision variables. Gawdt [14] presented a mixturecontinuous review inventory model under varying holdingcost constraint when the lead time demand follows Gammadistribution, where the costs were fuzzified as the trapezoidalfuzzy numbers.The continuous review inventory model withmixture shortage under constraint involving crashing costbased on probabilistic triangular fuzzy numbers by FerganyandGawdt [15] was discussed. A probabilistic periodic reviewinventory model using Lagrange technique and fuzzy adap-tive particle swarm optimization was presented by Ferganyet al. [16]. Fuzzy inventory model for deteriorating itemswith time dependent demand and partial backlogging isestablished by Kumar and Rajput [17]. Indrajitsingha et al.[18] give fuzzy inventory model with shortages under fullybacklogged using signed distance method. Recently, Patel etal. [19] introduced the continuous review inventory modelunder fuzzy environmentwithout backorder for deterioratingitems.
As we found earlier, many authors have studied theinventory models with different assumptions and conditions.These assumptions and conditions are represented in con-straints and costs (constant or varying).Therefore, due to theimportance of the inventory models we shall propose andstudy, in this paper, the mixture shortage inventory modelwith varying order cost under expected order cost constraintand the lead time demand follow Exponential, Laplace, andUniform distributions. Our goal of studying the inventorymodels is to minimize the total cost. The order quantity andthe reorder point are the policy variables for this model,which minimize the expected annual total cost. We evaluatedthe optimal order quantity and the reorder point in two cases:first case is when the cost components are considered as crispvalues, and the second case is when the cost componentsare fuzzified as a trapezoidal fuzzy numbers, which is calledthe fuzzy case. Finally this work is illustrated by numericalexample and we will make comparisons of all results andobtain conclusions.
2. Model Development
To develop any model of inventory models we need to putsome notations and assumptions represented in Notationssection.
2.1. Assumptions
(1) Consider that continuous review inventory modelunder order cost constraint and shortages are allowed.
(2) Demand is a continuous randomvariable with knownprobability.
(3) The lead time is constant and follows the knowndistributions.
(4) 𝛾 is a fraction of unsatisfied demand that will bebackordered while the remaining fraction (1 − 𝛾) iscompletely lost, where 0 ≤ 𝛾 ≤ 1.
(5) New order with size (𝑄) is placed when the inventorylevel drops to a certain level, called the reorder point(𝑟); assume that the system repeats itself in the sensethat the inventory position varies between 𝑟 and 𝑟+𝑄
during each cycle.
3. Model (I): The Mixture ShortageModel Where the Cost ComponentsAre Considered as Crisp Values
In this section, we consider that the continuous reviewinventory model with shortage is allowed. Some customersare willing to wait for the new replenishment and the othershave no patience; this case is called mixture shortage orpartial backorders.
The expected annual total cost consisted of the sum ofthree components:
and we assume the varying order cost function, where theorder cost is a decreasing function of the order quantity 𝑄.Then, the expected order cost is given by
𝐸 (OC) = 𝑐𝑜(𝑄)
𝐷
𝑄= 𝑐𝑜𝑄−𝛽
𝐷
𝑄= 𝑐𝑜𝐷𝑄−𝛽−1
,
𝐸 (HC) = 𝑐ℎ𝐻 = 𝑐
ℎ[𝑄
2+ 𝑟 − 𝐸 (𝑥) + (1 − 𝛾) 𝑆 (𝑟)] ,
𝐸 (BC) =𝑐𝑏𝛾𝐷
𝑄𝑆 (𝑟) ,
𝐸 (LC) =𝑐𝑙(1 − 𝛾)𝐷
𝑄𝑆 (𝑟) .
(3)
Our objective is to minimize the expected total costs[min𝐸(TC(𝑄, 𝑟))] with varying order cost under theexpected order cost constraint which needs to find the
Advances in Fuzzy Systems 3
optimal values of order quantity 𝑄 and reorder point 𝑟. Tosolve this primal function, let us write it as follows:
𝐸 (TC (𝑄, 𝑟)) = 𝑐𝑜𝐷𝑄−𝛽−1
+ 𝑐ℎ[𝑄
2+ 𝑟 − 𝐸 (𝑥) + (1 − 𝛾) 𝑆 (𝑟)]
+𝑐𝑏𝛾𝐷
𝑄𝑆 (𝑟) +
𝑐𝑙(1 − 𝛾)𝐷
𝑄𝑆 (𝑟)
= 𝑐𝑜𝐷𝑄−𝛽−1
+ 𝑐ℎ(𝑄
2+ 𝑟 − 𝐸 (𝑥))
+𝑐𝑏𝛾𝐷
𝑄𝑆 (𝑟)
+ (𝑐ℎ+
𝑐𝑙𝐷
𝑄) (1 − 𝛾) 𝑆 (𝑟)
(4)
Subject to: 𝑐𝑜𝐷𝑄−𝛽−1
≤ 𝐾. (5)We use the Lagrange multiplier technique to get the optimalvalues 𝑄∗ and 𝑟
∗ which minimize (4) under constraint (5) asfollows:
𝐺 (𝑄, 𝑟, 𝜆) = 𝑐𝑜𝐷𝑄−𝛽−1
+ 𝑐ℎ(𝑄
2+ 𝑟 − 𝐸 (𝑥))
+𝑐𝑏𝛾𝐷
𝑄𝑆 (𝑟) + (𝑐
ℎ+
𝑐𝑙𝐷
𝑄) (1 − 𝛾) 𝑆 (𝑟)
+ 𝜆 (𝑐𝑜𝐷𝑄−𝛽−1
− 𝐾) .
(6)
Putting each of the corresponding first partial derivatives of(6) equal to zero at 𝑄 = 𝑄
∗ and 𝑟 = 𝑟∗, respectively, we get
𝑐ℎ𝑄∗2
+ 𝐵𝐷𝑄∗−𝛽
− 2𝐴𝑆 (𝑟∗
) = 0,
𝑅 (𝑟∗
) =𝑐ℎ𝑄∗
𝑐ℎ(1 − 𝛾)𝑄∗ + 𝐴
,
(7)
where𝐴 = 𝐷 [𝑐
𝑏𝛾 + 𝑐𝑙(1 − 𝛾)] ,
𝐵 = 2𝑐𝑜(−𝛽 − 1) [1 + 𝜆] .
(8)
Clearly, it is difficult to find an exact solution of 𝑄∗ and 𝑟∗
of (7), so we can suppose that the lead time demand followssome distributions.
3.1. Lead Time Demand Follows Exponential Distribution.Supposing that the lead time demand follows the Exponentialdistribution with parameters ], then its probability densityfunction is given by
𝑓 (𝑥) = ]𝑒−]𝑥; 𝑥 ≥ 0, ] > 0
with 𝐸 (𝑥) =1
],
𝑅 (𝑟) = 𝑒−]𝑟
,
𝑆 (𝑟) =1
]𝑒−]𝑟
.
(9)
The optimal order quantity and the optimal reorder levelwhich minimize the expected relevant annual total costcan be obtained by substituting (9) into (7). Solving themsimultaneously we get
]𝑐2ℎ(1 − 𝛾)𝑄
∗3
+ ]𝑐ℎ𝐴𝑄∗2
− 2𝑐ℎ𝐴𝑄∗
+ ]𝑐ℎ(1 − 𝛾) 𝐵𝐷𝑄
∗1−𝛽
+ ]𝐴𝐵𝐷𝑄∗−𝛽
= 0,
𝑟∗
= −1
]ln[
𝑐ℎ𝑄∗
𝑐ℎ(1 − 𝛾)𝑄∗ + 𝐴
] ,
(10)
which give exact solutions for model (I).
3.2. Lead Time Demand Follows Laplace Distribution. If thelead time demand follows the Laplace distribution withparameters 𝜇, 𝜃, the probability density function will be
𝑓 (𝑥) =1
2𝜃𝑒−|𝑥−𝜇|/𝜃
; − ∞ < 𝑥 < ∞, 𝜃 > 0
with 𝐸 (𝑥) = 𝜇,
𝑅 (𝑟) =1
2𝑒
−((𝑟−𝜇)/𝜃)
,
𝑆 (𝑟) =𝜃
2𝑒−((𝑟−𝜇)/𝜃)
.
(11)
The optimal order quantity and the optimal reorder levelwhich minimize the expected relevant annual total cost canbe obtained by substituting (11) into (7), and, solving themsimultaneously, we obtain
𝑐2
ℎ(1 − 𝛾)𝑄
∗3
+ 𝑐ℎ𝐴𝑄∗2
− 2𝑐ℎ𝜃𝐴𝑄∗
+ 𝑐ℎ(1 − 𝛾) 𝐵𝐷𝑄
∗1−𝛽
+ 𝐴𝐵𝐷𝑄∗−𝛽
= 0,
𝑟∗
= 𝜇 − 𝜃 ln[2𝑐ℎ𝑄∗
𝑐ℎ(1 − 𝛾)𝑄∗ + 𝐴
] ,
(12)
which give exact solutions for model (I).
3.3. Lead Time Demand Follows Uniform Distribution. Simi-larly, suppose that the lead time demand follows the Uniformdistribution over the range from 0 to 𝑏, that is, 𝑥 ∼
Uniform(0, 𝑏); then its probability density function is givenby
𝑓 (𝑥) =1
𝑏; 0 ≤ 𝑥 ≤ 𝑏
with 𝐸 (𝑥) =𝑏
2,
𝑅 (𝑟) = 1 −𝑟
𝑏,
𝑆 (𝑟) =1
2𝑏(𝑟 − 𝑏)
2
.
(13)
The optimal order quantity and the optimal reorder levelwhich minimize the expected relevant annual total cost
4 Advances in Fuzzy Systems
can be obtained by substituting (13) into (7). Solving themsimultaneously, we find
𝑐3
ℎ(1 − 𝛾)
2
𝑄∗4
+ 2𝑐2
ℎ(1 − 𝛾)𝐴𝑄
∗3
+ 𝑐ℎ𝐴 [𝐴 − 𝑏𝑐
ℎ] 𝑄∗2
+ 𝑐2
ℎ(1 − 𝛾)
2
𝐵𝐷𝑄∗2−𝛽
+ 2𝑐ℎ(1 − 𝛾)𝐴𝐵𝐷𝑄
∗1−𝛽
+ 𝐴2
𝐵𝐷𝑄∗−𝛽
= 0,
𝑟∗
= 𝑏 [1 −𝑐ℎ𝑄∗
𝑐ℎ(1 − 𝛾)𝑄∗ + 𝐴
] ,
(14)
which give exact solutions for model (I).Thus, the exact solution for constrained continuous
review inventory model with mixture shortage and varyingorder cost can obtained by solving previous equations foreach distribution separately at different values of 𝛽 andvarying 𝜆 until the smallest positive value is found such thatthe constraint holds.
4. Model (If): The Mixture ShortageModel Where the Cost Components AreConsidered as Fuzzy Numbers
Consider continuous review inventory model similar tomodel (I), but assuming that the cost components 𝑐
𝑜, 𝑐ℎ, 𝑐𝑏,
and 𝑐𝑙are all fuzzy numbers, to control various uncertainties
from various physical or chemical characteristics where theremay be an effect on the cost components.
We represent these costs by trapezoidal fuzzy numbers asgiven below:
��𝑜= (𝑐𝑜− 𝛿1, 𝑐𝑜− 𝛿2, 𝑐𝑜+ 𝛿3, 𝑐𝑜+ 𝛿4) ,
��ℎ= (𝑐ℎ− 𝛿5, 𝑐ℎ− 𝛿6, 𝑐ℎ+ 𝛿7, 𝑐ℎ+ 𝛿8) ,
��𝑏= (𝑐𝑏− 𝜃1, 𝑐𝑏− 𝜃2, 𝑐𝑏+ 𝜃3, 𝑐𝑏+ 𝜃4) ,
��𝑙= (𝑐𝑙− 𝜃5, 𝑐𝑙− 𝜃6, 𝑐𝑙+ 𝜃7, 𝑐𝑙+ 𝜃8) ,
(15)
where 𝛿𝑖and 𝜃𝑖, 𝑖 = 1, 2, . . . , 8 are arbitrary positive numbers
and should satisfy the following constraints:
𝑐𝑜> 𝛿1> 𝛿2,
𝛿3< 𝛿4,
𝑐ℎ> 𝛿5> 𝛿6,
𝛿7< 𝛿8.
(16)
Similarly,
𝐶𝑏> 𝜃1> 𝜃2,
𝜃3< 𝜃4,
𝑐𝑙> 𝜃5> 𝜃6,
𝜃7< 𝜃8.
(17)
We can represent the order cost as a trapezoidal fuzzy numberas shown in Figure 1 and similarly for the remaining costs.
0
1
𝜇c𝑜(x)
(co − 𝛿2, 1)
(co − 𝛿1, 0)
(co + 𝛿3, 1)
(co + 𝛿4, 0)
𝛼 − cut
x
Figure 1: Order cost as a trapezoidal fuzzy number.
Note that themembership function of ��𝑜is 1 at points 𝑐
𝑜−
𝛿2and 𝑐𝑜+𝛿3, decreases as the point deviates from 𝑐
𝑜−𝛿2and
𝑐𝑜+ 𝛿3, and reaches zero at the endpoints 𝑐
𝑜− 𝛿1and 𝑐𝑜+ 𝛿4.
The left and right limits of 𝛼 – cut of 𝑐𝑜, 𝑐ℎ, 𝑐𝑏, and 𝑐
𝑙are
given as follows:
��𝑜V (𝛼) = 𝑐
𝑜− 𝛿1+ (𝛿1− 𝛿2) 𝛼,
��𝑜𝑢
(𝛼) = 𝑐𝑜+ 𝛿4− (𝛿4− 𝛿3) 𝛼,
��ℎV (𝛼) = 𝑐
ℎ− 𝛿5+ (𝛿5− 𝛿6) 𝛼,
��ℎ𝑢
(𝛼) = 𝑐ℎ+ 𝛿8− (𝛿8− 𝛿7) 𝛼,
��𝑏V (𝛼) = 𝑐
𝑏− 𝜃1+ (𝜃1− 𝜃2) 𝛼,
��𝑏𝑢
(𝛼) = 𝑐𝑏+ 𝜃4− (𝜃4− 𝜃3) 𝛼,
��𝑙V (𝛼) = 𝑐
𝑙− 𝜃5+ (𝜃5− 𝜃6) 𝛼,
��𝑙𝑢(𝛼) = 𝑐
𝑙+ 𝜃8− (𝜃8− 𝜃7) 𝛼.
(18)
The expected annual total cost for this case under theexpected order cost constraint and when all cost componentsare fuzzy can be expressed as follows:
�� (��𝑜, ��ℎ, ��𝑏, ��𝑙) = ��𝑜𝐷𝑄−𝛽−1
+ ��ℎ[𝑄
2+ 𝑟 − 𝐸 (𝑥) + (1 − 𝛾) 𝑆 (𝑟)]
+��𝑏𝛾𝐷
𝑄𝑆 (𝑟) +
��𝑙(1 − 𝛾)𝐷
𝑄𝑆 (𝑟)
= ��𝑜𝐷𝑄−𝛽−1
+ ��ℎ(𝑄
2+ 𝑟 − 𝐸 (𝑥))
+��𝑏𝛾𝐷
𝑄𝑆 (𝑟)
+ (��ℎ+
��𝑙𝐷
𝑄) (1 − 𝛾) 𝑆 (𝑟)
(19)
Subject to: ��𝑜𝐷𝑄−𝛽−1
≤ 𝐾. (20)
Advances in Fuzzy Systems 5
We use the Lagrange multiplier technique to find the optimalvalues 𝑄∗ and 𝑟
∗ which minimize (19) under constraint (20)as follows:
�� (��𝑜, ��ℎ, ��𝑏, ��𝑙) = ��𝑜𝐷𝑄−𝛽−1
+ ��ℎ(𝑄
2+ 𝑟 − 𝐸 (𝑥))
+��𝑏𝛾𝐷
𝑄𝑆 (𝑟)
+ (��ℎ+
��𝑙𝐷
𝑄) (1 − 𝛾) 𝑆 (𝑟)
+ 𝜆 (��𝑜𝐷𝑄−𝛽−1
− 𝐾) .
(21)
We can obtain the formof left and right𝛼– cut of the fuzzifiedcost function (21), respectively, as follows:
�� (��𝑜, ��ℎ, ��𝑏, ��𝑙)V (𝛼) = ��
𝑜V𝐷𝑄−𝛽−1
+ ��ℎV (
𝑄
2+ 𝑟 − 𝐸 (𝑥))
+��𝑏V𝛾𝐷
𝑄𝑆 (𝑟)
+ (��ℎV +
��𝑙V𝐷
𝑄) (1 − 𝛾) 𝑆 (𝑟)
+ 𝜆 (��𝑜V𝐷𝑄−𝛽−1
− 𝐾) ,
�� (��𝑜, ��ℎ, ��𝑏, ��𝑙)𝑢(𝛼) = ��
𝑜𝑢𝐷𝑄−𝛽−1
+ ��ℎ𝑢
(𝑄
2+ 𝑟 − 𝐸 (𝑥))
+��𝑏𝑢𝛾𝐷
𝑄𝑆 (𝑟)
+ (��ℎ𝑢
+��𝑙𝑢𝐷
𝑄) (1 − 𝛾) 𝑆 (𝑟)
+ 𝜆 (��𝑜𝑢𝐷𝑄−𝛽−1
− 𝐾) .
(22)
Since ��V(𝛼) and ��𝑢(𝛼) exist and are integrable for 𝛼 ∈ [0, 1],
as in Yao and Wu [20], we have
𝑑 (��, 0) =1
2∫
1
0
(��V (𝛼) + ��𝑢(𝛼)) 𝑑𝛼. (23)
We get the defuzzified value of ��(��𝑜, ��ℎ, ��𝑏, ��𝑙)(𝛼) by using
(23) for (22) as follows:
𝑑 (��, 0) = 𝐴1𝐷𝑄−𝛽−1
+ 𝐴2(𝑄
2+ 𝑟 − 𝐸 (𝑥))
+𝐴3𝛾𝐷
𝑄𝑆 (𝑟)
+ (𝐴2+
𝐴4𝐷
𝑄) (1 − 𝛾) 𝑆 (𝑟)
+ 𝜆 (𝐴1𝐷𝑄−𝛽−1
− 𝐾) ,
(24)
where
𝐴1=
(4𝑐𝑜− 𝛿1− 𝛿2+ 𝛿3+ 𝛿4)
4,
𝐴2=
(4𝑐ℎ− 𝛿5− 𝛿6+ 𝛿7+ 𝛿8)
4,
𝐴3=
(4𝑐𝑏− 𝜃1− 𝜃2+ 𝜃3+ 𝜃4)
4,
𝐴4=
(4𝑐𝑙− 𝜃5− 𝜃6+ 𝜃7+ 𝜃8)
4.
(25)
Similarly, as in model (I), to get the optimal values𝑄∗ and 𝑟∗
put each of the corresponding first partial derivatives of (24)equal to zero at 𝑄 = 𝑄
∗ and 𝑟 = 𝑟∗, respectively; we obtain
(2𝐴1(−𝛽 − 1)𝐷𝑄
∗−𝛽
) (1 + 𝜆) + 𝐴2𝑄∗2
− 2𝐴3𝛾𝐷𝑆 (𝑟) − 2𝐴
4(1 − 𝛾)𝐷𝑆 (𝑟) = 0
(26)
and the probability of the shortage is
𝑅 (𝑟∗
) =𝐴2𝑄∗
𝐴2(1 − 𝛾)𝑄∗ + 𝐴
3𝛾𝐷 + 𝐴
4(1 − 𝛾)𝐷
. (27)
Clearly, there is no closed form solution of (26) and (27). Wecan solve these equations by using the same manner as inmodel (I).
5. Special Cases
(1) Letting 𝛾 = 0, 𝛽 = 0 and 𝐾 → ∞ ⇒ 𝐶𝑜(𝑄) = 𝑐
𝑜and
𝜆 = 0, thus 𝐴 = 𝑐𝑙𝐷, 𝐵 = −2𝑐
𝑜and hence (7) reduces
to
𝑄∗
= √2𝐷 (𝑐𝑜+ 𝑐𝑙𝑠 (𝑟∗
))
𝑐ℎ
,
𝑅 (𝑟∗
) =𝑐ℎ𝑄∗
𝑐ℎ𝑄∗ + 𝑐
𝑙𝐷
.
(28)
This is an unconstrained lost sales continuous reviewinventory model with constant units of costs, whichare the same results as in Hadley and Whitin [1].
(2) Letting 𝛾 = 1, 𝛽 = 0 and 𝐾 → ∞ ⇒ 𝐶𝑜(𝑄) = 𝑐
𝑜
and 𝜆 = 0, thus 𝐴 = 𝑐𝑏𝐷, 𝐵 = −2𝑐
𝑜; thus (7) reduces
to
𝑄∗
= √2𝐷 (𝑐𝑜+ 𝑐𝑏𝑠 (𝑟∗
))
𝑐ℎ
,
𝑅 (𝑟∗
) =𝑐ℎ𝑄∗
𝑐𝑏𝐷
.
(29)
This is an unconstrained backorders continuousreview inventory model with constant units of costs,which are the same results as in Hadley and Whitin[1].
6 Advances in Fuzzy Systems
(i) Equations (10) give unconstrained backorders con-tinuous review of inventory model with constantunits of cost and the lead time demand follows theExponential distribution, which are the same resultsas in Hillier and Lieberman [21].
(ii) Equations (12) give unconstrained backorders contin-uous review inventory model with constant units ofcost and the lead time demand follows the Laplacedistribution, which agree with results of Nahmias[22].
(iii) Equations (14) give unconstrained backorders contin-uous review inventory model with constant units ofcost and the lead time demand follows the Uniformdistribution, which are the same results as in Fabryckyand Banks [23].
6. Numerical Example
Consider an inventory system with the following data:
𝐷 = 1050 units per year,
𝑐𝑜= 70 SR per unit ordered,
𝑐ℎ= 25 SR per unit per year,
𝑐𝑏= 7 SR per unit backorder,
𝑐𝑙= 15 SR per unit lost,
the backorder fraction has the values 𝛾 = 0.1, 𝛾 = 0.3,and 𝛾 = 0.7,
let 𝐾 = 140 SR,
and take
𝛿1= 60,
𝛿2= 48,
𝛿3= 10,
𝛿4= 50,
𝛿5= 19,
𝛿6= 10,
𝛿7= 1,
𝛿8= 2,
𝜃1= 6,
𝜃2= 4,
𝜃3= 2,
𝜃4= 4,
𝜃5= 12,
𝜃6= 7,
0
1000
2000
3000
4000
0.2 0.4 0.6 0.80.0𝛽
min E(TC) in the fuzzy casemin E(TC) in the crisp case
min
E(T
C)
Figure 2: The comparison between the crisp and fuzzy cases forExponential at 𝛾 = 0.7.
𝜃7= 1,
𝜃8= 2.
(30)
Determine 𝑄∗ and 𝑟
∗ for both cases of the previousmodel, when the lead time demand has the following distri-butions:
(i) Exponential distribution with ] = 0.077 units.(ii) Laplace distribution with 𝜇 = 13 and 𝜃 = 10 units.(iii) Uniform distribution with 𝑏 = 26 units.
Depending on the above data, we can obtain all results bysolving the previous deduced equations at different values of𝛽, 𝜆, and 𝛾 as shown in the Tables 1, 2, and 3 which givethe optimal values of 𝑄∗ and 𝑟
∗ that minimize the expectedtotal cost, when the lead time demand follows Exponential,Laplace, and Uniform distribution, respectively, for model (I)and model (If ).
From Table 1 we have that
at 𝛾 = 0.1, we will make backorders by 10% of neworders quantity;at 𝛾 = 0.3, we will make backorders by 30% of neworders quantity;at 𝛾 = 0.7, we will make backorders by 70% of neworders quantity.
After comparison of the crisp case and fuzzy case for Expo-nential distribution, we can deduce that the least min𝐸(TC)
was obtained at 𝛾 = 0.7. We can draw the minimum expectedtotal cost for model (I) and model (If ) against 𝛽 for theExponential distribution at 𝛾 = 0.7 as shown in Figure 2.
From Table 2 we have that
at 𝛾 = 0.1, we will make backorders by 10% of neworders quantity;at 𝛾 = 0.3, we will make backorders by 30% of neworders quantity;
Advances in Fuzzy Systems 7
Table 1: The exact solutions and min𝐸(TC) for model (I) and model (If ) at Exponential distribution.
at 𝛾 = 0.7, we will make backorders by 70% of neworders quantity.
After comparison of the crisp case and fuzzy case forLaplace distribution, we can deduce that the least min𝐸(TC)
was obtained at 𝛾 = 0.7. We can draw the minimum expectedtotal cost for model (I) and model (If ) against 𝛽 for theLaplace distribution at 𝛾 = 0.7 as shown in Figure 3.
From Table 3 we have that
at 𝛾 = 0.1, we will make backorders by 10% of neworders quantity;at 𝛾 = 0.3, we will make backorders by 30% of neworders quantity;at 𝛾 = 0.7, we will make backorders by 70% of neworders quantity.
After comparison of the crisp case and fuzzy case forUniformdistribution, we can deduce that the leastmin𝐸(TC)
was obtained at 𝛾 = 0.7. We can draw the minimum expectedtotal cost for model (I) and model (If ) against 𝛽 for theUniform distribution at 𝛾 = 0.7 as shown in Figure 4.
7. Conclusion
In this study we discussed two cases for mixture shortageinventory model under varying order cost constraint when
0
1000
2000
3000
4000
0.2 0.4 0.6 0.80.0𝛽
min E(TC) in the fuzzy casemin E(TC) in the crisp case
min
E(T
C)
Figure 3: The comparison between the crisp and fuzzy cases forLaplace at 𝛾 = 0.7.
lead time demand follows Exponential, Laplace, andUniformdistributions.Wehave evaluated the exact solutions of𝑄∗ and𝑟∗ for each value of 𝛽 and 𝜆
∗ which yields our expected ordercost constraint and then obtain the minimum expected totalcost by using Lagrangian multiplier technique.
By comparing between the minimum expected total costfor model (I) and model (If ) at each distribution, we can
8 Advances in Fuzzy Systems
Table 2: The exact solutions and min𝐸(TC) for model (I) and model (If ) at Laplace distribution.
min E(TC) in the fuzzy casemin E(TC) in the crisp case
Figure 4: The comparison between the crisp and fuzzy cases forUniform at 𝛾 = 0.7.
deduce that the least min𝐸(TC) was obtained when thelead time demand follows Uniform distribution and equals844.584 SR with order quantity 𝑄
∗
= 32.4596 and reorderpoint 𝑟
∗
= 23.9138 for model (I), while the minimumexpected annual total cost for model (If ) is 634.709 SR with
order quantity 𝑄∗
= 29.3328 and reorder point 𝑟∗ = 24.2447
as shown in Table 3.Thismeans that we can conclude that theminimum expected total cost in fuzzy case is less than in thecrisp case, which indicates that the fuzziness is very close tothe actuality of life and gets minimum expected total cost lessthan the crisp case.
For the results of the numerical example, we note thatwhen 𝛽 increases, 𝑟∗ increases, and thus 𝑄∗ decreases whichindicate that the min𝐸(TC) decreases.
Also, the different values of 𝛽 lead to changes of 𝑄∗ ineach distribution separately. But in all distributions we notethat values of 𝑄∗ are almost fixed, due to the constraint onthe varying order cost. Also, we note that when 𝛾 increases,min𝐸(TC) decreases; this indicates that 70% of the shortagescan be met at the lowest possible cost.
Finally, our study in particular provides the ample scopefor further research and exploration. For instance, we haveconsidered probabilistic mixture shortage inventory modelunder varying order cost constraint. This work can befurther developed by considering an ample range of differentassumptions and conditions represented in constraints andcosts (constant or varying), such as varying two costs undertwo constraints or varying two costs under constraint orvarying one cost under two constraints. Also, we can studysome of the inventory models with the system multiechelon-multisource.
Advances in Fuzzy Systems 9
Table 3: The exact solutions and min𝐸(TC) for model (I) and model (If ) at Uniform distribution.
𝐷: A random variable denoting the demandrate per period
𝑄: A decision variable representing the orderquantity per cycle
𝑟: A decision variable representing thereorder point
𝐿: The lead time between the placement of anorder and its receipt
𝑥: The continuous random variablerepresenting the demand during 𝐿
𝑓(𝑥): The probability density function of thelead time demand and (𝑥) is itsdistribution function
𝑅(𝑟): The probability of the shortage= 1 − 𝐹(𝑟) = ∫
∞
𝑟
𝑓(𝑥) 𝑑𝑥
𝑆(𝑟): The expected value of shortages per cycle= ∫∞
𝑟
(𝑥 − 𝑟)𝑓(𝑥) 𝑑𝑥
𝑐𝑜: The order cost per unit
𝐶𝑜(𝑄) = 𝑐
𝑜𝑄−𝛽: The varying order cost per cycle
𝛽: A constant real number selected toprovide the best fit of estimated expectedcost function
𝑐ℎ: The holding cost per unit per period
𝑐𝑠: The shortage cost per unit
𝑐𝑏: The backorders cost per unit
𝑐𝑙: The lost sales cost per unit
𝐾: The limitation on the expected annualorder cost
𝜆: The Lagrangian multiplier.
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper.
Acknowledgments
This research project was supported by a grant from the“Research Center of the Female Scientific and Medical Col-leges,”Deanship of ScientificResearch, King SaudUniversity.
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10 Advances in Fuzzy Systems
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