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Probabilistic multiitem inventory model with varying mixture shortage cost under restrictions
 Hala A. Fergany^{1}Email author
 Received: 7 December 2015
 Accepted: 29 July 2016
 Published: 16 August 2016
Abstract
This paper proposed a new general probabilistic multiitem, singlesource 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
 Multiitem
 Varying mixture shortage
 Stochastic lead time demand
Backround
The multiitem, single source inventory system is the most general procurement system which may be described as follows; an inventory of nitems is maintained to meet the average demand rates designated \(\bar{D}_{1} ,\bar{D}_{2} ,\bar{D}_{3} , \ldots \ldots \bar{D}_{n}\). The 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 multiitem multi source concept and the probabilistic singleitem, single source (SISS) inventory system with zero leadtime, using the classical optimization. AbouElAta and Kotb (1996), AbouElAta et al. 2003) studied multiitem EOQ inventory modelswith varying costs under two restrictions. Moreover, Fergany and ElSaadani (2005, 2006; Fergany et al. 2014) treated constrained probabilistic inventory models with continuous distributions and varying costs.
The two basic questions that any continuous review \(\left\langle {{\text{Q}},\;{\text{r}}} \right\rangle\) 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 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 \(\left\langle {{\text{Q}},\;{\text{r}}} \right\rangle\) inventory models with mixture of backorders and lost were proposed by Ouyang et al. (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.
The following notations are adopted for developing the model

\(\left\langle {Q,\,r} \right\rangle\) = the continuous review inventory system

MISS = The Multiitem singlesource,

\(D_{i}\) = The demand rate of the ith item per period,

\(\bar{D}_{i}\) = The expected demand rate of the ith item per period,

\(Q_{i}\) = The order quantity of the ith item per period,

\(Q_{i}^{*}\) = The optimal order quantity of the ith item per period,

\(r_{i}\) = The reorder point of the ith item per period,

\(r_{i}^{*}\) = The optimal reorder point of the ith item per period,

\(\bar{n}_{i}\) = The expected number order of the ith item per period,

\(L_{i}\) = The leadtime between the placement of an order and its receipt of the ith item,

\(\bar{L}_{i}\) = The average value of the lead time \(L_{i}\),

\(x_{i}\) = The random variables represent the lead time demand of the ith item per period,

\(f\left( {x_{i} } \right)\) = The probability density function of the lead time demands,

\(E\left( {x_{i} } \right)\) = The expected value of \(x_{i}\),

\(r_{i}  x_{i}\) = The random variable represents the net inventory when the procurement quantity arrives if the leadtime demand x ≤ r,

\(\bar{H}_{i}\) = The average on hand inventory of the ith item per period

\(R\left( r \right) = p\left( {x_{i} > r} \right)\) = The probability of shortage = the reliability function,

\(\bar{S}\left( {r_{i} } \right)\) = The expected shortage quantity per period

\(c_{oi}\) = The order cost per unit of the ith item per period,

\(c_{hi}\) = The holding cost per unit of the ith item per period,

\(c_{si}\) = The shortage cost per unit of the ith item per period,

\(c_{bi}\) = The backorder cost per unit of the ith item per period,

\(c_{li}\) = The lost sales cost per unit of the ith item per period,

\(c_{si} (n)\) = The varying shortage cost of the ith item per period,

\(\Phi_{D} \left( t \right)\) = The characteristic function of demand,

\(\Phi_{x} \left( t \right)\) = The characteristic function of lead time demand x,

\(\beta\) = A constant real number selected to provide the best fit of estimated expected

cost function,

\(\gamma_{i}\) = The backorder fraction of the ith item, \(0 < \gamma_{i} < 1\),

E (OC) = The expected order (procurement) cost per period,

E (HC) = The expected holding (carrying) cost per period,

E (SC) = The expected shortage cost per period,

E (BC) = The expected backorder cost per period,

E (LC) = The expected lost sales cost per period,

E (TC) = The expected total cost function,

Min E (TC) = The minimum expected total cost function.

\(K_{bi}\) = The limitation on the expected annual varying backorder cost for

backorder model of the ith item,

\(K_{li}\) = The 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 leadtime L is constant and the distribution of the lead time demand (demand during the lead time) is known.
The optimal values \(Q_{i} \;{\text{and}}\;r_{i}\) can be calculated by setting each of the corresponding first partial derivatives of Eq. (3) equal to zero.
Clearly, there is no closed form solution of Eqs. (4), (5).
Mathematical derivation of the lead time demand
The 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.
We can deduce the corresponding distribution of the lead time demand X when the demand follows many continuous distributions. Consider X follows the normal distribution, the exponential distribution and the Chi square distribution.
The demand follows the normal distribution
The demand follows the exponential distribution
The demand follows the Chi square distribution
Special cases
Two special cases of the proposed model are deduced as follows;
Case 1
This 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).
Case 2
This is the unconstrained backorders continuous review inventory model with constant unit costs, which coincide with the result of Hadley and Whiten (1963).
Applications
The Maximum cost allowed (the limitations) for both backorder, lost sales and their fractions
Items  Costs  

K_{b}  K_{L}  γ  \(\left( {1  \gamma } \right)\)  
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 
Solution
Onesample Kolmogorov–Smirnov test of the demands
D1  D2  D3  

N  48  48  48 
Normal parameters^{a}  
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. (2tailed)  0.057  0.090  0.050 
The average units cost for each item 2004–2008
Items  Costs  

\(c_{o}\)  \(c_{h}\)  Shortage cost  
\(c_{b}\)  \(c_{l}\)  
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 
The optimal values \(Q^{*}\) and \(r^{*}\) for three items can be found by using (7) and (8) respectively. The iterative procedure will be used to solve the equations.

* Step 1: Assume that \(\bar{S}\, = \,0\) and \(r = E(x)\), then from Eq. (7) we have: \(Q_{0} = \sqrt {\frac{{2c_{oi} \bar{D}_{i} }}{{c_{hi} }}}\)

* Step 2: Substituting \(Q_{o}\) into Eq. (8) we obtain \(r_{0}\)

* Step 3: Substituting by \(r_{0}\) from step 2 into Eq. (7) we can deduce \(Q_{1}\)

* Step 4: the procedure is to change the values of \(\lambda_{i}\) in step 2 and step 3 until the smallest value of \(\lambda_{i} > 0\) is found such that the constraint varying shortage for the different values of β.
The optimal values of \(Q^{*} ,r^{*}\) and min E (TC) at different values of β
β  Item 1  Item 2  Item 3  

\(\lambda_{1}^{*}\)  \(\lambda_{2}^{*}\)  \(Q^{*}\)  \(r^{*}\)  min E (TC1)  \(\lambda_{1}^{*}\)  \(\lambda_{2}^{*}\)  \(Q^{*}\)  \(r^{*}\)  min E (TC2)  \(\lambda_{1}^{*}\)  \(\lambda_{2}^{*}\)  \(Q^{*}\)  \(r^{*}\)  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 
Conclusion
Upon studying the probabilistic multi item invetory model with varying mixture shortage cost under two restrictions using the Lagrange mulipliers technique, the optimal order quntity \(Q^{*}\) and the optimal reorder point \(r^{*}\) are introduced. Then, the minimum expected total cost min E(TC) for multi items are deduced. Three 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.
Declarations
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 EditorinChief 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.
Open AccessThis 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, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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