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Probabilistic multi‑item inventory
model withvarying mixture shortage cost
underrestrictions
Hala A. Fergany*
Backround
e multi-item, single source inventory system is the most general procurement system
which may be described as follows; an inventory of n-items is maintained to meet the
average demand rates designated
¯
D1,
¯
D2,
¯
D3,......
¯
Dn
. e objective is to decide when
to procure each item, how much of each item to procure, in the light of system and cost
parameters.
Hadley and Whiten (1963) treated the unconstrained probabilistic inventory models
with constant unit of costs. Fabrycky and Banks (1965) studied the multi-item multi
source concept and the probabilistic single-item, single source (SISS) inventory system
with zero lead-time, using the classical optimization. Abou-El-Ata and Kotb (1996),
Abou-El-Ata etal. 2003) studied multi-item EOQ inventory models-with varying costs
under two restrictions. Moreover, Fergany and El-Saadani (2005, 2006; Fergany etal.
2014) treated constrained probabilistic inventory models with continuous distributions
and varying costs.
e two basic questions that any continuous review
Q, r
inventory control system
has to answer are; when and how much to order. Over the years, hundreds of papers
and books have been published presenting models for doing this under a wide variety of
Abstract
This paper proposed a new general probabilistic multi-item, single-source inventory
model with varying mixture shortage cost under two restrictions. One of them is on
the expected varying backorder cost and the other is on the expected varying lost
sales cost. This model is formulated to analyze how the firm can deduce the optimal
order quantity and the optimal reorder point for each item to reach the main goal of
minimizing the expected total cost. The demand is a random variable and the lead
time is a constant. The demand during the lead time is a random variable that follows
any continuous distribution, for example; the normal distribution, the exponential
distribution and the Chi square distribution. An application with real data is analyzed
and the goal of minimization the expected total cost is achieved. Two special cases are
deduced.
Keywords: Probabilistic inventory model, Multi-item, Varying mixture shortage,
Stochastic lead time demand
Open Access
© 2016 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium,
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indicate if changes were made.
RESEARCH
Fergany SpringerPlus (2016) 5:1351
DOI 10.1186/s40064‑016‑2962‑2
*Correspondence:
halafergany@yahoo.com
Department of Mathematics,
Faculty of Science, Tanta
University, Tanta, Egypt
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Fergany SpringerPlus (2016) 5:1351
conditions and assumptions. Most authors have shown that the demand that cannot be
filled from stock then backordered or the lost sales model are used. Several
Q, r
inven-
tory models with mixture of backorders and lost were proposed by Ouyang etal. (1996),
Montgomery et al. (1973) and Park (1982). Also, Zipkin (2000) shows that demands
occurring during a stockout period are lost sales rather than backorders.
In this paper, we investigate a new probabilistic multi-item single-source (MISS)
inventory model with varying mixture shortage cost (backorder and lost sales) as shown
in Fig.1 under two restrictions. One of them is on the expected varying backorder cost
and the other one the expected varying lost sales cost. e optimal order quantity
Q∗
i
,
the optimal reorder point
r∗
i
and the minimum expected total cost [min E (TC)] are
obtained. Moreover, two special cases are deduced and an application with real data is
analyzed.
The followingnotations are adopted fordeveloping the model
Q,r
=the continuous review inventory system
MISS=e Multi-item single-source,
Di
=e demand rate of the ith item per period,
¯
Di
=e expected demand rate of the ith item per period,
Qi
=e order quantity of the ith item per period,
Q∗
i
=e optimal order quantity of the ith item per period,
ri
=e reorder point of the ith item per period,
r∗
i
=e optimal reorder point of the ith item per period,
¯ni
=e expected number order of the ith item per period,
Li
=e lead-time between the placement of an order and its receipt of the ith item,
¯
Li
=e average value of the lead time
Li
,
xi
=e random variables represent the lead time demand of the ith item per period,
f(xi)
=e probability density function of the lead time demands,
E(xi)
=e expected value of
xi
,
ri−xi
= e random variable represents the net inventory when the procurement
quantity arrives if the lead-time demand x≤r,
¯
Hi
=e average on hand inventory of the ith item per period
R(r)=p(xi>r)
=e probability of shortage=the reliability function,
¯
S(ri)
=e expected shortage quantity per period
Fig. 1 The inventory model
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Fergany SpringerPlus (2016) 5:1351
coi
=e order cost per unit of the ith item per period,
chi
=e holding cost per unit of the ith item per period,
csi
=e shortage cost per unit of the ith item per period,
cbi
=e backorder cost per unit of the ith item per period,
cli
=e lost sales cost per unit of the ith item per period,
csi(n)
=e varying shortage cost of the ith item per period,
�D(t)
=e characteristic function of demand,
�x(t)
=e characteristic function of lead time demand x,
β
= A constant real number selected to provide the best fit of estimated expected
cost function,
γi
=e backorder fraction of the ith item,
0<γ
i<1
,
E (OC)=e expected order (procurement) cost per period,
E (HC)=e expected holding (carrying) cost per period,
E (SC)=e expected shortage cost per period,
E (BC)=e expected backorder cost per period,
E (LC)=e expected lost sales cost per period,
E (TC)=e expected total cost function,
Min E (TC)=e minimum expected total cost function.
Kbi
=e limitation on the expected annual varying backorder cost for
backorder model of the ith item,
Kli
=e limitation on the expected annual varying lost sales cost for
lost sales model of the ith item.
Mathematical model
We will study the proposed model with varying mixture shortage cost constraint when
the demand D is a continuous random variable, the lead-time L is constant and the dis-
tribution of the lead time demand (demand during the lead time) is known.
It is possible to develop the expected annual total cost as follows:
i.e.
where;
∞
r
(x
i−
r
i
)f(x
i
)dx
i=¯
S(r
i)
e objective is to minimize the expected annual total cost E [TC (Q, r)] under two
constraints:
E
(Total Cost)=
m
i=1
[E(Order Cost)+E(Holding Cost)+E(Shortage Cost)
]
E
[TC(Q,r)]=
m
i=1
coi¯
Di
Qi+chi Qi
2+ri−E(xi)+cbiγ¯
Di
Qiβ+
1
∞
r
(xi−ri)f(xi)dxi
+
cli¯
Di
Qiβ+1
+chi
(1−γi)
∞
r
(xi−ri)f(xi)dxi
c
biγi
¯
Di
Qi
β+1
¯
S(ri)−Kbi ≤
0
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To solve this primal function which is a convex programming problem, let us write the
previews equations in the following form:
Subject to:
To find the optimal values
Q∗and r∗
which minimize Eq.(1) under the constraints (2),
the Lagrange multiplier technique is used as follows:
where
1i,2i
are the Lagrange multipliers.
e optimal values
Qiand ri
can be calculated by setting each of the corresponding
first partial derivatives of Eq.(3) equal to zero.
i.e.
then we obtain:
c
li (1−γi)
¯
Di
Qi
β+1
¯
S(ri)−Kli ≤
0
(1)
E[TC(Q,r)]=
m
i=1
coi
¯
Di
Qi
+chiQi
2+ri−E(xi)
+
cbiγ¯
Di
Qiβ+1
¯
S(ri)
+
cli¯
Di
Qiβ+1
+chi
(1−γi)¯
S(ri)
(2)
cbi γi¯
Di
Qi
β+1
¯
S(ri)−Kbi ≤0
c
li (1−γi)
¯
Di
Qi
β+1
¯
S(ri)−Kli ≤0
(3)
L
(Qi,ri,i1i2)=
m
i=1
¯
Di
Qi
+chiQi
2+ri−E(xi)+cbiγi¯
Di
Qiβ+1
¯
S(ri)
+
cli¯
Di
Qiβ+1
+chi
(1−γi)¯
S(ri)+1i
cbiγi
¯
Di
Qiβ+1
¯
S(ri)−kbi
+2i
Cli(1−γi)¯
Di
Qiβ+1
¯
S(ri)−kli
,
∂L
∂Qi
=
0
∂L
∂ri
=0,
(4)
Cbi
Q
∗β+2
i−
2C
oi
Q
∗β
i−
2A(β
+
1)
¯
S(r
i
)
=0,
(5)
R
r∗
i
=
ChiQ∗B+1
i
A
+
C
hi
(1
−
γ
i
)Q∗β+1
i
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Fergany SpringerPlus (2016) 5:1351
where A
=¯
D
β+1
i
[γ
i
C
hi
(1
+
λ1
i
)
+
(1
−
γ
i
)C
li
(1
+
λ2
i
)
]
Clearly, there is no closed form solution of Eqs.(4), (5).
Mathematical derivation ofthe lead time demand
e lead time demand
X
is the total demand D which accrue during the lead time L.
Consider that the lead time is a constant number of periods and demand is random
variable.
en,
To determine the distribution of the lead time demand X: consider the characteristic
function of
X
and D are related as:
We can deduce the corresponding distribution of the lead time demand X when the
demand follows many continuous distributions. Consider X follows the normal distribu-
tion, the exponential distribution and the Chi square distribution.
The demand follows the normal distribution
If the demand D have the normal distribution with parameters
µ,σ
,
en the lead time demand follows the normal distribution with parameters
µL,Lσ2
Also:
R
(r)
=∞
r
f(x)d(x
)
i.e.
and
where
X
=
L
i=1
Di,i=1, 2, ......,
L
�
x(t)=
L
i=1
�D(t)=[�D(t)]
L
f(D)
=
1
σ√2π
e−1
2D−µ
σ
2
,
−∞
<D<
∞
,
−∞
<µ<
∞
,σ>
0
f(x)
=
1
σ√2πL
e−1
2Lx−µL
σ
2
,
−∞
<x<
∞
,
−∞
<µL<
∞
,σL>
0
R
(r)
=
1
−
φ
r−µL
σ√L=
ϕ
r−µL
σ√L
(6)
¯
S
(r)
=
σ
√
L�
r
−µ
L
σ√L+
(µL
−
r)ϕ
r
−µ
L
σ√L
�
r−µL
σ√L=
1
√2π
∞
r−µL
σ√L
ye
−1
2y2
dy
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Hence, the expected annual total cost can be minimized mathematically by substitut-
ing from Eq.(6) into (4), (5) we get (7), (8)
and
The demand follows the exponential distribution
If the demand D have the exponential distribution with parameter
α
,
en, lead time demand follows the Gamma distribution with parameters
L,α
also
R
(r)
=
α
L
Ŵ(L)∞
r
xL−1e−αx
dx
then,
R
(r)
=
L−1
i=0
(αr)ie−αr
i
!
,
Hence, the expected annual total cost can be minimized mathematically by substitut-
ing from Eq.(9) into (4), (5) we get (10), (11)
and
The demand follows the Chi square distribution
If the demand D follows Chi-squire distribution with parameter
η
2
(7)
C
hiQ∗β+2
i
−
2CoiQ∗β
r
−
2A(β
+
1)
σ
√
L�
r−µL
σ√L+
(µL
−
r)ϕ
r−µL
σ√L,
(8)
φ
r−µL
σ√L
=
ChiQ
∗β+1
i
Chi
(1
−
γ)
Q
∗β+1
i+A
f(x)=αe−αD
,0
<D<∞
,
α>0
f(x)
=
α
L
Ŵ(L)
xL−1e−αx,0<x<
∞
,L>0, α>
0,
¯
S
(r)=
∞
r
(x−r)f(x)dx =αL
Ŵ(L)
∞
r
(x−r)xL−1e−αrdx =αL
Ŵ(L)
∞
r
xLe−αrdx −rR(r
)
(9)
¯
S
(r)=
L
α
L
i=0
(αr)ie−αr
i
!
−r
L−1
i=0
(αr)ie−αr
i
!
(10)
C
hi Q∗β+2
i−2Coi Q∗β
i−2A(β+1)
L
α
L
i=0
(αr)ie−αr
i!
−r
L−1
i=0
(αr)ie−αr
i!
,
(11)
ϕ
r−µL
σ√L
=
Chi Q
∗β+1
i
C
hi
(1
−
γ)Q∗β+1
i+
A
=
L−1
i=0
(αr)ie−αr
i
!
f(D)=
1
2η
2Ŵ
η
2
Dη
2−1,0<D<∞,
η
2>
0
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en lead time demand X follows the Chi-squire distribution with parameters
Lη
2
also
and
Hence, the expected annual total cost can be minimized mathematically by substitut-
ing from Eq.(12) into (4), (5) we get (13), (14):
and
Special cases
Two special cases of the proposed model are deduced as follows;
Case 1
Let γi=
0,
β=
0
and Kbi →∞ ⇒cs(¯n)β=csand λi=0.
us Eqs.(4) and
(5) become:
is is the unconstrained lost sales continuous review inventory model with constant
units of cost, which are the same results as in Hadley and Whiten (1963).
f(x)=
1
2Lη
2Ŵ
Lη
2
xLη
2−10<x<∞,
Lη
2>
0,
R
(r)=
Lη
2−1
i=0
r
2
ie−r
2
i
!,
(12)
¯
S
(r)=Lη
Lη
2
i
=
0r
2ie−r
2
i!
−r
Lη
2−1
i
=
0r
2ie−r
2
i!
(13)
C
hi Q∗β+2
i−2CoiQ∗β
i−2A(β+1)
Lη
Lη
2
i
=
0r
2ie−r
2
i!−r
Lη
2
i
=
0r
2ie−r
2
i!
,
(14)
ϕ
r−µL
σ√L
=
Chi Q∗β+1
i
C
hi
(1
−
γ)Q∗β+1
i+
A
=
Lη
2−1
i=0
r
2
ie−r
2
i
!
Q
∗=
2¯
D
co+cl¯
S(r)
c
h
and R
r∗
=chQ∗
chQ∗+cl¯
D
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Case 2
us Eqs.(4) and (5) become:
is is the unconstrained backorders continuous review inventory model with con-
stant unit costs, which coincide with the result of Hadley and Whiten (1963).
Applications
A company for ready clothes produces three Items [Trousers: I, Shirt: II, and Jacket: III]
of seasonal products (production takes two cycles and each cycle lasts for 6months).
Table5 in Appendix shows the order quantity and the demand rate during the interval
2004–2008. But for some un expected reasons in some cycles, the company faces short-
age and it has to pay penalty at least 1% for month for backorder and 3% for lost sale.
Table1 shows the maximum cost allowed for backorder
Kb
, lost sales
KL
and their frac-
tions. Hence, the company wishes to put an optimal policy for production to minimize
the expected total cost.
Solution
By using SPSS program, One-Sample Kolmogorov–Smirnov Test, the demand for the
three Items is fitted to normal distribution, where Table2 shows the K-S statistic with
their P values. Table3 shows the average units cost for each item 2004–2008
e optimal values
Q∗
and
r∗
for three items can be found by using (7) and (8) respec-
tively. e iterative procedure will be used to solve the equations.
Use the following numerical procedure:
* Step 1: Assume that
¯
S=0
and
r=E(x)
, then from Eq.(7) we have:
Q
0
=
2coi ¯
Di
c
hi
* Step 2: Substituting
Qo
into Eq.(8) we obtain
r0
* Step 3: Substituting by
r0
from step 2 into Eq.(7) we can deduce
Q1
Let γi=1β=0and Kli →∞ ⇒cs(¯n)β=csand λi=0.
Q
∗=
2¯
D
co+cb¯
S(r)
c
h
and R
r∗
=ch
cb¯
D
Q
,
Table 1 The Maximum cost allowed (the limitations) for both backorder, lost sales
andtheir fractions
Items Costs
KbKLγ
(1
−
γ)
Item (I) 1680 13,720 0.56 0.44
Item (II) 1800 9300 0.70 0.30
Item (III) 1052 10,820 0.67 0.33
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* Step 4: the procedure is to change the values of
λi
in step 2 and step 3 until the small-
est value of
λi>0
is found such that the constraint varying shortage for the different
values of β.
e numerical computation are done by using mathematica program for three items
at different values of β, Table4 shows the optimal values
Q∗,r∗
E(TC) and min E(TC) at
different values of
β
. Hence we can draw the optimal routes of
Q∗,r∗
and E (TC) against
β for all three items as shown in Figs.2, 3 and 4. It is evident that the min E(TC) is
achieved at minimum value for β.
Conclusion
Upon studying the probabilistic multi item invetory model with varying mixture short-
age cost under two restrictions using the Lagrange mulipliers technique, the optimal
order quntity
Q∗
and the optimal reorder point
r∗
are introduced. en, the minimum
Table 2 One-sample Kolmogorov–Smirnov test ofthe demands
a Test distribution is normal
D1 D2 D3
N 48 48 48
Normal parametersa
Mean 1.07E4 1.12E4 6109.38
SD 2.300E3 2.258E3 3.603E3
Most extreme differences
Absolute 0.193 0.180 0.196
Positive 0.091 0.109 0.176
Negative −0.193 −0.180 −0.196
Kolmogorov–Smirnov Z 1.335 1.245 1.359
Asymp. Sig. (2-tailed) 0.057 0.090 0.050
Table 3 The average units cost foreach item 2004–2008
Items Costs
co
ch
Shortage cost
cb
cl
Item (I) 2.23 7.898 0.90 9.350
Item (II) 2.14 7.567 1.10 13.254
Item (III) 9.77 34.542 3.28 68.460
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Table 4 The optimal values of
Q∗,r∗
andmin E (TC) atdierent values ofβ
βItem 1 Item 2 Item 3
∗
1
∗
2
Q∗
r∗
min E (TC1)
∗
1
∗
2
Q∗
r∗
min E (TC2)
∗
1
∗
2
Q∗
r∗
min E
(TC3)
0.1 0.02 0.021 3635.43 10,543 39,538 0.001 0.012 3758 1161 36,431 0.14 0.012 3322 9282 159,060
0.2 0.024 0.025 3786.93 10,635 40,586 0.001 0.021 3926 11,699 37,443 0.13 0.18 3430 9323 161,426
0.3 0.025 0.027 3931.32 10,727 41,582 0.002 0.022 4071 11,789 38,400 0.13 0.19 3584 9364 164,554
0.4 0.032 0.034 4083.49 10,819 42,467 0.002 0.027 4210 11,879 39,256 0.13 0.19 3717 9384 166,737
0.5 0.039 0.040 4246 10,888 43,302 0.004 0.042 4404 11,857 39,603 0.12 0.19 3852 9384 168,639
0.6 0.042 0.043 4413.04 10,934 44,124 0.005 0.052 4554 11,992 40,902 0.12 0.19 3990 9384 170,104
0.7 0.043 0.044 4554.17 11,003 44,886 0.008 0.063 4719 12,069 41,634 0.12 0.19 4124 9405 172,461
0.8 0.048 0.046 4730.91 11,026 45,598 0.01 0.068 4881 12,104 42,323 0.12 0.19 4261 9405 174,186
0.9 0.049 0.048 4876.67 11,026 45,865 0.01 0.071 5056 12,149 43,008 0.13 0.19 4455 9364 175,779
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expected total cost min E(TC) for multi items are deduced. ree curves
Q∗
,
r∗
and min
E(TC) are displayed to illustate them for multi items against the different values of β.
Finally, the min E(TC) is achieved at minimum value for β.
Acknowledgements
I would like to greatly appreciate the anonymous referees for their very valuable and helpful suggestions. I take this
favorable chance to express my indebtedness to the Honorable Editor-in-Chief and his Editorial Board for their helpful
support. I am also grateful to Department of Math & Stat, Faculty of Science, Tanta University for infrastructural assistance
to carry out the research.
Competing interests
The author declare that he have no competing interests.
Appendix
See Table5.
0
1000
2000
3000
4000
5000
6000
0.
1
0.
3
0.
5
0.
7
0.
9
B
Q*
Item (I)
Item(II)
Item (III)
Fig. 2 The optimal values of Q* against β
0
2000
4000
6000
8000
10000
12000
14000
0.
1
0.
3
0.
5
0.
7
0.
9
B
r*
ItemI
Item II
Item III
Fig. 3 The optimal values of r* against β
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
200000
0
.1
0.
3
0.
5
0.
7
0
.9
B
E(TC)
Item (I)
Item (II)
Item (III)
Fig. 4 The optimal values of E(TC) against β
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Table 5 The actual inventory quantity anddemand rate, fromMay 2004 toApril 2008
Year No. ofcycle Month Item 1 Item 2 Item 3
Q1 D1 Q2 D2 Q3 D3
2004 1 May 5800 6000 10,500 10,500 8000 900
June 9000 8000 9000 10,000 5500 500
July 11,800 12,000 12,000 12,000 8000 900
Aug 11,800 12,000 12,000 12,500 6000 500
Sept. 8000 8500 10,000 9000 4000 400
Oct. 7200 7000 7500 7000 3000 400
2Nov. 10,000 10,000 10,000 10,500 5500 500
Dec. 11,000 12,000 9000 9000 5500 500
2005 Jan. 12,800 12,800 11,000 11,000 5000 550
Feb. 11,000 10,000 7500 7500 4000 500
March 6000 6500 12,500 12,500 5000 500
April 9500 8500 13,000 12,500 7000 600
3May 12,000 12,000 11,000 12,000 9500 10,000
June 12,000 12,500 10,000 9000 6500 6000
July 8500 9000 12,500 12,800 9000 10,000
Aug. 7000 7500 17,000 16,000 7000 6000
Sept. 11,000 12,000 9000 10,000 5000 5000
Oct. 13,400 11,000 7800 8000 4000 5000
4Nov. 12,850 13,500 12,500 12,000 6500 6000
Dec. 12,830 13,000 11,000 12,000 6500
2006 Jan. 12,850 12,500 11,850 10,500 7000 7500
Feb. 12,830 11,850 6830 8000 6000 7000
March 12,820 12,000 11,820 12,500 7000 7000
April 10,730 11,030 12,730 12,230 9000 8000
5May 6500 7000 11,500 12,000 10,000 11,000
June 9800 8500 10,000 9500 7500 7000
July 12,500 13,000 12,800 12,950 10,000 11,000
Aug. 12,200 13,000 17,000 16,000 8500 7000
Sept. 9000 8600 9000 9500 6000 6000
Oct. 7000 7300 8500 8750 5000 6000
6Nov. 10,000 12,000 13,000 12,000 7500 7000
Dec. 12,000 10,500 11,500 12,500 7500 7000
Jan. 13,000 14,000 12,000 11,000 8000 8500
Feb. 13,000 13,000 7000 8000 7000 8000
March 13,000 12,000 12,000 13,000 8000 8000
April 11,000 10,000 13,000 13,000 10,000 9000
May 7000 7000 12,000 13,000 11,500 12,000
June 10,000 11,000 10,000 9000 8500 8000
July 13,000 13,000 13,000 14,000 11,000 12,000
Aug. 12,000 13,000 17,000 16,000 9000 8000
Sept. 9000 9000 11,000 9000 7000 7000
Oct. 10,000 8000 8000 9000 7000 7000
8Nov. 10,000 12,000 13,000 12,000 8500 8000
Dec. 12,000 10,000 11,500 12,000 8500 8000
2008 Jan. 14,000 14,500 12,500 12,000 9000 9500
Feb. 13,000 13,200 8000 7500 8000 9000
March 13,000 13,000 13,000 13,000 9000 9000
April 11,000 10,000 14,000 14,000 11,000 10,000
Page 13 of 13
Fergany SpringerPlus (2016) 5:1351
Received: 7 December 2015 Accepted: 29 July 2016
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