3. SINGLE PERIOD NEWSBOY PROBLEM WITH STOCHASTIC DEMAND
AND PARTIAL BACKLOGGING
3.1 INTRODUCTION
In today’s highly competitive business environment and inventory
management, the ability to plan and control inventories to meet the
competitive priorities is becoming increasingly important in many types of
organizations. Depending on this, the type of inventory problem frequently
encountered with seasonal or customized products is the newsboy problem,
also called the newsvendor problem or single period stochastic inventory
problem because only a single procurement is made. The typical examples
are the dilemmas of making a one-period decision on the quantity of
newspapers that a newsboy should buy on a given day or the quantity of
seasonal goods that a retailer should purchase for the current year or goods
that cannot be sold the next year because of style changes.
The single period inventory model has wide application in the real world in
assisting the decision maker to determine the optimal quantity to order. This
is one type of inventory problem frequently discussed in the literature. A wide
variety of real world problems including the stocking of spare parts,
perishable items, style goods and special season items offer practical
example of this sort of situation. These types of problems are referred to as
the newsboy problem. Since it can be phrased as a problem of deciding how
many newspapers a boy should buy on a given day for his corner news
stand.
In some real life situation there is a part of the demand which cannot
be satisfied from the inventory and it leaves the system stock-out. If the order
quantity is larger than the realised demand, the items which are left over at
the end of period are sold at a salvage value or disposed off. Hence, both the
factors such as salvage and stock-out situations are equally important. The
basic Newsboy inventory model has been discussed in Hanssman F [33]. If
the demand is uncertain then it must be predicted and the continuous
43
sources of uncertainty or stochastic demand, has a different impact on
optimal inventory settings and prevents optimal solutions from being found in
closed form. Notably, there are cases in which the probability distribution of
the demand for new products is typically unknown because of a lack of
historical information, and the use of linguistic expressions by experts for
demand forecasting is often employed. The Assorted level of demand is
viewed in form of a special class of inventory evolution known as finite
inventory process.
In this chapter, a variation of the finite Inventory process model i.e.,
the classical Newsboy problem is attempted. This variation of the Newsboy
problem discussed has not been investigated earlier in the literature,
although compound demand processes have been studied for a long time. A
closed form is introduced and there are many benefits of having a closedform approximate solution. The objective is to obtain the optimal solution in
which the demand 𝑄0 is varied according to the SCBZ property. Appropriate
Numerical illustrations provide a justification for its unique existence.
3.2 ASSUMPTIONS AND NOTATIONS
ℎ, 𝐶1
- The cost of each unit produced but not sold called holding
cost.
𝑑, 𝐶2
- The shortage cost arising due to each unit of unsatisfied
demand.
𝑅
- Random variable denoting the demand.
𝑥0
- The truncation point.
𝐶
- Total cost per unit time.
𝑆
- Supply level 𝑆 and 𝑆̂ is the optimal value.
𝑅1 , 𝑅2
- A random variables denoting the demand for both the case
44
when demand is less and more than the production, its PDF
is given by (𝑅1 ) , 𝑓(𝑅2 ).
𝑓(𝑅1 , 𝜃1 )
- The probability distribution function when 𝑅1 ≤ 𝑥0 .
𝑓(𝑅2 , 𝜃2 )
- The probability distribution function when 𝑅2 > 𝑥0 .
𝐶1 (𝑄𝑖 )
- The cost of each unit of Newspapers purchased for several
individual demand but not sold called salvage loss.
𝐶2 (𝑄𝑖 )
- The shortage cost arising due to each unit of unsatisfied
individual demand of Newspapers.
𝑆𝑖
- Optimal supply size.
𝑓1 (𝑄𝑖 , 𝜃1 )
- The probability distribution function when 𝑄𝑖 ≤ 𝑄0 .
𝑓2 (𝑄𝑖 , 𝜃1 )
- The probability distribution function when 𝑄𝑖 > 𝑄0 .
𝜓(𝑍), 𝐸(𝐶) - The expected total cost
𝜓(𝑍̂), 𝐸(𝐶̂ ) - The optimal expected cost
𝜓1 (𝑍)
- Total holding cost
𝜓2 (𝑍)
- Total shortage cost
𝑄𝑖
- Random variable denoting the several individual demands at
the 𝑖 th location where 𝑖 = 1,2, … , 𝑛
𝐸(𝑄𝑖 )
- The
expected
several
individual
1
demand
before
the
truncation point 𝑥0 , 𝐸(𝑄𝑖 ) = 𝜃 and after the truncation point
1
1
is 𝐸(𝑄𝑖 ) = 𝜃
2
45
3.3 BASIC MODEL
Many researchers have suggested that the probability of achieving a
target profit level is a realistic managerial objective in the Newsboy problem.
However Hanssman F [33] has given a different perspective of the Newsboy
model. In this model, somewhat different problem arises when the salvage
loss for the left-over units is negligible but a significant holding cost ℎ per unit
time is incurred. It is assumed that the demand 𝑅 materializes in a linear
fashion during a given planning interval 𝑇. The shortage cost is assumed
proportional to the area under the negative part of the inventory curve. If the
total cost per unit time is 𝐶, then the cost incurred during the interval 𝑇 is
given as
𝑅
ℎ (𝑆 − 2 ) 𝑇
𝐶𝑇 = { ℎ
𝑑
𝑆𝑡1 + 2 (𝑅 − 𝑆)𝑡2
2
, 𝑅≤𝑆
(3.1)
, 𝑅>𝑆
The expected total cost given in Hanssman.F [33] is as follows
𝑆
∞
∞
(𝑅 − 𝑆)2
𝑅
𝑆2
𝐸(𝐶) = ℎ ∫ (𝑆 − ) 𝑓(𝑅)𝑑𝑅 + ℎ ∫
𝑓(𝑅)𝑑𝑅 + 𝑑 ∫
𝑓(𝑅)𝑑𝑅
2
2𝑅
2𝑅
0
𝑆
𝑆
(3.2)
Motivated from Newsboy model discussed above and the concept of
the setting the clock back to zero property, this chapter follows the different
variation of the probability distribution function. A brief discussion on the
SCBZ property can be followed from chapter 1 and in present chapter this
property is slightly modified with respect to the model and it is defined as a
random variable 𝑅𝑖 is said to satisfy the SCBZ property if
𝑆(𝑅𝑖 +𝑥0 ,𝜃1 ,𝜃2 )
𝑆(𝑥0 ,𝜃1 )
= 𝑆(𝑅𝑖 , 𝜃2 ), ∀ 𝑖 = 1, 2 where 𝑆(𝑅𝑖 , 𝜃) denotes the survivor function
which is 𝑆(𝑅𝑖 , 𝜃) = 1 − 𝐹(𝑅𝑖 , 𝜃). Here 𝑥0 is called the truncation point of the
random variable 𝑅𝑖 . SCBZ property means that the probability distribution of
the random variable undergoes a parametric change after the truncation
46
point 𝑥0 . Similar definition of SCBZ property was discussed in Raja Rao et.al
[53]. Accordingly SCBZ property is defined by the pdf as
𝜃1 𝑒 −𝜃1 𝑅1
; 𝑅1 ≤ 𝑥0
𝑓(𝑅) = { 𝑥 (𝜃 −𝜃 )
−𝜃2 𝑅2
0
2
1
𝑒
𝜃2 𝑒
; 𝑅2 > 𝑥0
(3.3)
where 𝑥0 is constant denoting truncation point. The probability distribution
function is denoted as 𝑓(𝑅1 , 𝜃1 ) if 𝑅1 ≤ 𝑥0 and 𝑓(𝑅2 , 𝜃2 ) if 𝑅2 > 𝑥0
(3.4)
Similar model is discussed by Sathiyamoorthy R et.al [63]. Use of this
property is throughout the thesis.
3.4 FINITE PROCESS INVENTORY MODEL USING SCBZ PROPERTY
During certain situations the inventory process gets terminated after a
finite duration. In order to control this, an inventory model is studied. The
Newsboy problem is under the category of finite inventory process. There is a
onetime supply of the items per day and the demand is probabilistic. The
classical model assumes that if the order quantity is larger than the realized
demand, the items which are left over at the end of period are sold at a
salvage value or are disposed of. Further in cases of stock-out unsatisfied
demand is lost. From Hanssman F [33], the total cost incurred during the
interval 𝑇 is modified as given
𝐶1 (𝑆 − 𝑅1 )𝑇
𝐶𝑇 = {𝐶1
𝐶
𝑆𝑡1 + 2 (𝑅2 − 𝑆)𝑡2
2
2
, 𝑅1 ≤ 𝑆
, 𝑅2 > 𝑆
(3.5)
Where 𝑡1 and 𝑡2 are defined as in Figure 3.1. Hence the expected total cost
function per unit time is given in the form
𝑆
∞
𝐸(𝐶) = 𝐶1 ∫(𝑆 − 𝑅1 )𝑓(𝑅1 )𝑑𝑅1 + 𝐶1 ∫
0
𝑆
∞
+ 𝐶2 ∫
𝑆
𝑆 𝑡1
𝑓(𝑅1 )𝑑𝑅1
2𝑇
(𝑅2 − 𝑆) 𝑡2
𝑓(𝑅2 )𝑑𝑅2
𝑅2
𝑇
(3.6)
47
̂
𝑑𝐸(𝑆)
To find optimal 𝑆̂, 𝑑𝑠 = 0 is considered. Since the limit of the integral
involves 𝑆 which is also in the integrand, the differential of integral is applied
𝑑
𝑑𝑥
𝜓(𝑥)
𝜓(𝑥) 𝑑
= ∫𝜑(𝑥) 𝑓(𝑥, 𝑡)𝑑𝑡 = 𝜓′(𝑥) 𝑓(𝑥, 𝜓(𝑥)) − 𝜑 ′(𝑥) 𝑓(𝑥, 𝜑(𝑥)) + ∫𝜑(𝑥)
𝑑𝑥
𝑓(𝑥, 𝑡)𝑑𝑡
(3.7)
which can be proved to result in the following equation
𝐶2
𝐹(𝑆̂) = 𝑃(𝑅 ≤ 𝑆̂) = 𝐶 +𝐶
1
2
(3.8)
Given the probability distribution of the demand 𝑅1 , 𝑅2 and using the
expression for 𝐹(𝑆̂), the optimal 𝑆̂ is determined. This basic Newsboy
problem is quite similar to the one discussed in Hanssman F [33].
Figure 3.1: Shortage curve under negative inventory level
The probability distribution function defined above satisfies the SCBZ
property and the optimal 𝑆̂ is to be derived. Now using the equation 3.4 and
3.6, the total expected cost is given as
𝑥0
𝑆
𝐸(𝐶) = 𝐶1 ∫ (𝑆 − 𝑅1 )𝑓(𝑅1 , 𝜃1 )𝑑𝑅1 + 𝐶1 ∫(𝑆 − 𝑅1 )𝑓(𝑅2 , 𝜃2 )𝑑𝑅1
0
𝑥0
𝑥0
∞
𝑆 𝑡1
𝑆 𝑡1
+ 𝐶1 ∫
𝑓(𝑅1 , 𝜃1 )𝑑𝑅1 + 𝐶1 ∫
𝑓(𝑅2 , 𝜃2 )𝑑𝑅1
2𝑇
2𝑇
𝑆
𝑥0
(3.9)
48
Case i) when the holding cost before and after the truncation point 𝑥0 is
considered and using equation 3.3, the result obtained is as follows
𝑥0
𝑆
𝐸(𝐶1 ) = 𝐶1 ∫ (𝑆 − 𝑅1 )𝜃1 𝑒
−𝜃1 𝑅1
𝑑𝑅1 + 𝐶1 ∫(𝑆 − 𝑅1 )𝑒 𝑥0 (𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑅2 𝑑𝑅1
0
𝑥0
𝑥0
∞
𝑆 𝑡1
𝑆 𝑡1 𝑥 (𝜃 −𝜃 )
+ 𝐶1 ∫
𝜃1 𝑒 −𝜃1 𝑅1 𝑑𝑅1 + 𝐶1 ∫
𝑒 0 2 1 𝜃2 𝑒 −𝜃2 𝑅2 𝑑𝑅1
2𝑇
2𝑇
𝑆
𝑥0
(3.10)
𝑥0
𝑆
𝐸(𝐶1 ) = 𝐶1 𝜃1 ∫ (𝑆 − 𝑅1 )𝑒 −𝜃1 𝑅1 𝑑𝑅1 + 𝐶1 𝑒 𝑥0 (𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑅2 ∫(𝑆 − 𝑅1 )𝑒 −𝜃2 𝑅2 𝑑𝑅1
0
𝑥0
𝑥0
∞
𝑆 𝑡1 −𝜃 𝑅
𝑆 𝑡1
+ 𝐶1 𝜃1 ∫
𝑒 1 1 𝑑𝑅1 + 𝐶1 𝑒 𝑥0 (𝜃2 −𝜃1) 𝜃2 ∫
𝑑𝑅
2𝑇
2𝑇 1
𝑆
𝑥0
(3.11)
𝑑𝐸(𝐶1 )
Now to find
𝑑𝑆
= 0 the following substitution is carried
𝑥
Let 𝐾1 = 𝐶1 𝜃1 ∫0 0(𝑆 − 𝑅1 )𝑒 −𝜃1 𝑅1 𝑑𝑅1
(3.12)
𝑆
𝐾2 = 𝐶1 𝑒 𝑥0 (𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑅2 ∫𝑥 (𝑆 − 𝑅1 )𝑑𝑅1
(3.13)
0
𝑥 𝑆𝑡
𝐾3 = 𝐶1 𝜃1 ∫𝑆 0 2 𝑇1 𝑒 −𝜃1 𝑅1 𝑑𝑅1
(3.14)
∞ 𝑆 𝑡1
𝐾4 = 𝐶1 𝑒 𝑥0 (𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑅2 ∫𝑥
0
2𝑇
𝑑𝑅1
(3.15)
Now solving the equation the above equation for
geometry
𝑡1
𝑍
= 𝑄 and
𝑇
𝑡2
𝑇
=
𝑄−𝑍
𝑄
𝑑𝐾1 𝑑𝐾2 𝑑𝐾3 𝑑𝐾4
𝑑𝑆
, 𝑑𝑆 ,
𝑑𝑆
,
𝑑𝑆
using the
and by the rule given by equation 3.7. Hence
the following result is obtained
𝑥0
𝑥0
𝑥0
𝑑𝐾1
−(𝑆 − 𝑅1 )𝑒 −𝜃1 𝑅1
𝑒 −𝜃1 𝑅1
= 𝐶1 𝜃1 ∫ (𝑆 − 𝑅1 )𝑒 −𝜃1 𝑅1 𝑑𝑅1 = 𝐶1 𝜃1 {[
] +[
2 ] }
𝑑𝑆
𝜃1
𝜃
1
0
0
0
49
= 𝐶1 𝜃1 {[
−𝑒 −𝜃1 𝑅1
𝜃1
1
] + [𝜃 ]} = 𝐶1 [1 − 𝑒 −𝜃1 𝑅1 ]
(3.16)
1
𝑆
𝑑𝐾2
= 𝐶1 𝑒 𝑥0 (𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑅2 ∫(𝑆 − 𝑅1 )𝑑𝑅1
𝑑𝑆
𝑥0
= 𝐶1 𝑒 𝑥0 (𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑅2 [𝑆𝑅1 −
𝑅1 2
2
𝑆
]
𝑆2
𝑥0
= 𝐶1 𝑒 𝑥0 (𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑅2 [2 ]
(3.17)
𝑥0
𝑥0
𝑆
𝑆
𝑥0
𝑑𝐾3
𝑆 𝑡1 −𝜃 𝑅
𝑆 𝑡1
𝑆 𝑡1 𝑒 −𝜃1 𝑅1
−𝜃1 𝑅1
1
1
= 𝐶1 𝜃1 ∫
𝑒
𝑑𝑅1 = 𝐶1 𝜃1
∫𝑒
𝑑𝑅1 = 𝐶1 𝜃1
[
]
𝑑𝑆
2𝑇
2𝑇
2 𝑇 −𝜃1 𝑆
𝑆 𝑡1 −𝑒 −𝜃1 𝑆 + 𝑒 −𝜃1 𝑥0
𝑆2
[
] = 𝐶1
[−𝑒 −𝜃1𝑆 + 𝑒 −𝜃1 𝑥0 ]
2𝑇
𝜃1
2𝑅
= 𝐶1 𝜃1
(3.18)
𝑥0
𝑥0
𝑆
𝑆
𝑑𝐾4
𝑆 𝑡1
𝑆 𝑡1
= 𝐶1 𝑒 𝑥0 (𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑅2 ∫
𝑑𝑅1 = 𝐶1 𝑒 𝑥0 (𝜃2 −𝜃1) 𝜃2 𝑒 −𝜃2 𝑅2
∫ 𝑑𝑅1
𝑑𝑆
2𝑇
2𝑇
𝑆2
= 𝐶1 𝑒 𝑥0 (𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑅2 2𝑅 [𝑆 − 𝑥0 ]
(3.19)
Hence the total expected cost is given by
𝐸(𝐶1 ) =
𝑑𝐾1
𝑑𝑆
+
𝑑𝐾2
𝑑𝑆
+
𝑑𝐾3
𝑑𝑆
+
𝑑𝐾4
𝑑𝑆
1
𝑆
= 𝜃 − 𝑒 𝜃1 𝑥
1
0
(3.20)
Case ii) When the shortage cost before and after the truncation point 𝑥0 is
considered and using equation 3.3 the following result is obtained
𝑥0
𝐸(𝐶2 ) = 𝐶2 ∫
𝑆
(𝑅2 − 𝑆) 𝑡2
𝜃 𝑒 −𝜃1 𝑅1 𝑑𝑅2
𝑅2
𝑇 1
∞
+ 𝐶2 ∫
𝑥0
(𝑅2 − 𝑆) 𝑡2 𝑥 (𝜃 −𝜃 )
𝑒 0 2 1 𝜃2 𝑒 −𝜃2 𝑅2 𝑑𝑅2
𝑅2
𝑇
(3.21)
50
𝑥0
𝐸(𝐶2 ) = 𝐶2 𝜃1 𝑒 −𝜃1 𝑅1 ∫
𝑆
(𝑅2 − 𝑆) 𝑡2
𝑑𝑅2
𝑅2
𝑇
∞
+ 𝐶2 𝑒
𝑥0 (𝜃2 −𝜃1 )
𝜃2 ∫
𝑥0
(𝑅2 − 𝑆) 𝑡2 −𝜃 𝑅
𝑒 2 2 𝑑𝑅2
𝑅2
𝑇
(3.22)
Similarly to find
𝑑𝐸(𝐶2 )
𝑑𝑠
= 0 the above procedure is followed and hence taking
𝑥 (𝑅2 −𝑆) 𝑡2
𝐼1 = 𝐶2 𝜃1 𝑒 −𝜃1 𝑅1 ∫𝑆 0
𝑅2
𝑇
∞ (𝑅2 −𝑆) 𝑡2
𝐼2 = 𝐶2 𝑒 𝑥0 (𝜃2 −𝜃1 ) 𝜃2 ∫𝑥
𝑑𝑅2 and
the result is as follows 𝐸(𝐶2 ) = 𝜃
1
1
𝑥
𝑒 𝜃1 𝑥 0
+ 𝑒 𝜃10𝑥0
0
𝑅2
𝑇
𝑒 −𝜃2 𝑅2 𝑑𝑅2
(3.23)
Considering the case when the supply and the level of inventory are
same and all the other cases are considered negligible. On substitution the
equation 3.20 and 3.23 becomes
1
𝑆
𝐸(𝐶) = 𝜃 − 𝑒 𝜃1 𝑥 + 𝜃
1
0
1
𝜃 𝑥
1𝑒 1 0
𝑥
+ 𝑒 𝜃10𝑥0
(3.24)
Therefore after substituting the value for 𝜃1 , 𝑥0 , 𝜃2 in 𝐸(𝐶), the optimal
solution of the expected cost is obtained as 𝐸(𝐶) =19. In contrary to the
above model 3.4 another model 3.5 is developed in order to the test the
hypothesis in case of single individual demand which will be a base for the
following model 3.5.
3.5 OPTIMALITY OF TOTAL EXPECTED COST USING SCBZ PROPERTY
The case discussed in the model 3.4 is modified and a study over a
single demand is carried out. Here the assumptions are as given in the model
3.4. Now a change to the model in form of cost function is that here in this
model the cost function is denoted as 𝜓(𝑍).Total cost incurred during the
interval T given is given as
𝜑1 (𝑍 − 𝑄)𝑇
, 𝑄≤𝑍
𝜑2
𝐶(𝑇) = {𝜑1
𝑍𝑡1 + 2 (𝑄 − 𝑍)𝑡2 , 𝑄 > 𝑍
2
Where 𝑡1 and 𝑡2 are defined as in Figure 3.2
51
(3.25)
Figure 3.2: Supply size against the time t
Therefore the expected total cost function per unit time is given in the form
𝑍
∞
∞
(𝑄 − 𝑍) 𝑡2
𝑍 𝑡1
𝜓(𝑍) = 𝜑1 ∫(𝑍 − 𝑄)𝑓(𝑄)𝑑𝑄 + 𝜑1 ∫
𝑓(𝑄)𝑑𝑄 + 𝜑2 ∫
𝑓(𝑄)𝑑𝑄
2𝑇
2
𝑇
0
𝑍
𝑍
(3.26)
From the geometry it follows that
𝑡1
𝑇
𝑍
= 𝑄 and
𝑡2
𝑇
=
𝑄−𝑍
𝑄
(3.27)
The uncertainty here is related to a well known property called as SCBZ
Property. SCBZ property which is defined in model 3.4 is applied in this
model 3.5. Accordingly SCBZ property is defined by the PDF as
𝑓(𝑄) = {
𝜃1 𝑒 −𝜃1 𝑄
; 𝑄 ≤ 𝑄0
𝑄0 (𝜃2 −𝜃1 )
−𝜃2 𝑄
𝑒
𝜃2 𝑒
; 𝑄 > 𝑄0
(3.28)
where 𝑄0 is constant denoting truncation point. The probability distribution
function is denoted as 𝑓(𝑄, 𝜃1 ) if 𝑄 ≤ 𝑄0 and 𝑓(𝑄, 𝜃2 ) if 𝑄 > 𝑄0
(3.29)
Sathiyamoorthy R et.al [63] introduced the concept of SCBZ property in
inventory. The probability distribution function defined above satisfies the
SCBZ property under the above assumptions and the optimal 𝑍̂ is derived.
Now, the total expected cost given in equation 3.26 is written as
𝜓(𝑍) = 𝜑1 𝑘1 + 𝜑1 𝑘2 + 𝜑2 𝑘3
52
(3.30)
𝑍
∞ 𝑍 𝑡1
where 𝑘1 = ∫0 (𝑍 − 𝑄)𝑓(𝑄)𝑑𝑄 , 𝑘2 = ∫𝑍
2 𝑇
∞ (𝑄−𝑍) 𝑡2
𝑓(𝑄)𝑑𝑄 , 𝑘3 = ∫𝑍
2
𝑇
𝑓(𝑄)𝑑𝑄
(3.31)
The solution of equation 3.31 is dealt in form two model i.e., model 3.6
and model 3.7 due to the complexity involved while using the limit. In solving
model 3.5 following is study carried out. Using equation 3.7
𝑄0
𝑍
𝑑𝜓(𝑍)
= 𝜑1 ∫ (𝑍 − 𝑄)𝑓(𝑄, 𝜃1 )𝑑𝑄 + 𝜑1 ∫(𝑍 − 𝑄)𝑓(𝑄, 𝜃2 )𝑑𝑄
𝑑𝑍
0
𝑄0
𝑄0
∞
𝑍 𝑡1
𝑍 𝑡1
+ 𝜑1 ∫
𝑓(𝑄, 𝜃1 )𝑑𝑄 + 𝜑1 ∫
𝑓(𝑄, 𝜃2 )𝑑𝑄
2𝑇
2𝑇
𝑍
𝑄0
𝑄0
∞
(𝑄 − 𝑍) 𝑡2
(𝑄 − 𝑍) 𝑡2
+ 𝜑2 ∫
𝑓(𝑄, 𝜃1 )𝑑𝑄 + 𝜑2 ∫
𝑓(𝑄, 𝜃2 )𝑑𝑄
2
𝑇
2
𝑇
𝑍
𝑄0
(3.32)
𝑄0
𝑍
𝑑𝜓(𝑍)
= 𝜑1 ∫ (𝑍 − 𝑄)𝜃1 𝑒 −𝜃1 𝑄 𝑑𝑄 + 𝜑1 ∫(𝑍 − 𝑄)𝑒 𝑄0 (𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑄 𝑑𝑄
𝑑𝑍
0
𝑄0
𝑄0
∞
𝑍 𝑡1
𝑍 𝑡1 𝑄 (𝜃 −𝜃 )
+ 𝜑1 ∫
𝜃1 𝑒 −𝜃1 𝑄 𝑑𝑄 + 𝜑1 ∫
𝑒 0 2 1 𝜃2 𝑒 −𝜃2 𝑄 𝑑𝑄
2𝑇
2𝑇
𝑍
𝑄0
𝑄0
+ 𝜑2 ∫
𝑍
∞
+ 𝜑2 ∫
𝑄0
(𝑄 − 𝑍) 𝑡2
𝜃 𝑒 −𝜃1 𝑄 𝑑𝑄
2
𝑇 1
(𝑄 − 𝑍) 𝑡2 𝑄 (𝜃 −𝜃 )
𝑒 0 2 1 𝜃2 𝑒 −𝜃2 𝑄 𝑑𝑄
2
𝑇
(3.33)
53
𝑄0
𝑄0
0
0
𝑍
𝑑𝜓(𝑍)
= 𝜑1 [𝜃1 ∫ 𝑍 𝑒 −𝜃1 𝑄 𝑑𝑄 − 𝜃1 ∫ 𝑄 𝑒 −𝜃1 𝑄 𝑑𝑄 + 𝜃2 ∫ 𝑍𝑒 𝑄0(𝜃2 −𝜃1 ) 𝑒 −𝜃2 𝑄 𝑑𝑄
𝑑𝑍
𝑄0
𝑍
− 𝜃2 ∫ 𝑄𝑒 𝑄0 (𝜃2 −𝜃1 ) 𝑒 −𝜃2 𝑄 𝑑𝑄 ]
𝑄0
𝑄0
∞
𝑍 𝑡1
𝑍 𝑡1 𝑄 (𝜃 −𝜃 )
+ 𝜑1 [∫
𝜃1 𝑒 −𝜃1 𝑄 𝑑𝑄 + ∫
𝑒 0 2 1 𝜃2 𝑒 −𝜃2 𝑄 𝑑𝑄 ]
2𝑇
2𝑇
𝑍
𝑄0
𝑄0
+ 𝜑2 [∫
𝑍
∞
+ ∫
𝑄0
(𝑄 − 𝑍) 𝑡2
𝜃 𝑒 −𝜃1 𝑄 𝑑𝑄
2
𝑇 1
(𝑄 − 𝑍) 𝑡2 𝑄 (𝜃 −𝜃 )
𝑒 0 2 1 𝜃2 𝑒 −𝜃2 𝑄 𝑑𝑄]
2
𝑇
(3.34)
using equation 3.27, the following equation are obtained
𝑄0
𝑄0
0
0
𝑍
𝑑𝜓(𝑍)
= 𝜑1 [𝜃1 𝑍 ∫ 𝑒 −𝜃1 𝑄 𝑑𝑄 − 𝜃1 ∫ 𝑄 𝑒 −𝜃1 𝑄 𝑑𝑄 + 𝜃2 𝑍𝑒 𝑄0 (𝜃2 −𝜃1 ) ∫ 𝑒 −𝜃2 𝑄 𝑑𝑄
𝑑𝑍
𝑄0
𝑍
− 𝜃2 𝑒 𝑄0(𝜃2 −𝜃1 ) ∫ 𝑄𝑒 −𝜃2 𝑄 𝑑𝑄 ]
𝑄0
𝑄0
∞
𝑍2
+ 𝜑1
[∫ 𝜃1 𝑒 −𝜃1 𝑄 𝑑𝑄 + ∫ 𝑒 𝑄0(𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑄 𝑑𝑄]
2𝑄
𝑍
𝑄0
𝑄0
∞
(𝑄 − 𝑍)2
+ 𝜑2
[𝜃1 ∫ 𝑒 −𝜃1 𝑄 𝑑𝑄 + 𝑒 𝑄0 (𝜃2 −𝜃1 ) 𝜃2 ∫ 𝑒 −𝜃2 𝑄 𝑑𝑄 ]
2𝑄
𝑍
𝑄0
(3.35)
54
𝑄0
𝑄0
𝑑𝜓(𝑍)
𝑒 −𝜃1 𝑄
𝑒 −𝜃1 𝑄
= 𝜑1 [𝜃1 𝑍 [
] −𝜃1 (𝑄 [
] − 𝑒 −𝜃1 𝑄 (1))
𝑑𝑍
−𝜃1 0
−𝜃1 0
𝑍
+ 𝜃2 𝑍𝑒
𝑄0 (𝜃2 −𝜃1 )
𝑒 −𝜃2 𝑄
[
]
−𝜃2 𝑄
0
𝑍
− 𝜃2 𝑒
𝑄0 (𝜃2 −𝜃1 )
𝑒 −𝜃2 𝑄
(𝑄 [
] − 𝑒 −𝜃2 𝑄 (1))]
−𝜃2 𝑄
0
+ 𝜑1
𝑍2
𝑄0
∞
[[𝑒 −𝜃1 𝑄 ]𝑍 + 𝑒 𝑄0 (𝜃2 −𝜃1 ) [−𝑒 −𝜃2 𝑄 ]𝑄 ]
0
2𝑄
(𝑄 − 𝑍)2
𝑄0
∞
+ 𝜑2
[[−𝑒 −𝜃1 𝑄 ]𝑍 + 𝑒 𝑄0(𝜃2 −𝜃1 ) [−𝑒 −𝜃2 𝑄 ]𝑄 ]
0
2𝑄
(3.36)
𝑑𝜓(𝑍)
𝑄0
𝑄0
= 𝜑1 [𝑍[−𝑒 −𝜃1 𝑄 ]0 − (𝑄[−𝑒 −𝜃1 𝑄 ]0 − 𝑒 −𝜃1 𝑄 (1))
𝑑𝑍
𝑍
𝑍
+ 𝑍𝑒 𝑄0(𝜃2 −𝜃1 ) [−𝑒 −𝜃2 𝑄 ]𝑄 − 𝑒 𝑄0(𝜃2 −𝜃1 ) (𝑄[ 𝑒 −𝜃2𝑄 ]𝑄 − 𝑒 −𝜃2 𝑄 (1))]
0
0
2
+ 𝜑1
𝑍
𝑄0
∞
[[𝑒 −𝜃1 𝑄 ]𝑍 + 𝑒 𝑄0 (𝜃2 −𝜃1 ) [−𝑒 −𝜃2 𝑄 ]𝑄 ]
0
2𝑄
+ 𝜑2
(𝑄 − 𝑍)2
𝑄0
∞
[[−𝑒 −𝜃1 𝑄 ]𝑍 + 𝑒 𝑄0(𝜃2 −𝜃1 ) [−𝑒 −𝜃2 𝑄 ]𝑄 ]
0
2𝑄
(3.37)
𝑑𝜓(𝑍)
= 𝜑1 [𝑍(1−𝑒 −𝜃1 𝑄0 ) − (𝑄(1 − 𝑒 −𝜃1 𝑄0 ) − 𝑒 −𝜃1 𝑄 )
𝑑𝑍
+ 𝑍𝑒 𝑄0(𝜃2 −𝜃1 ) (𝑒 −𝜃2 𝑄0 − 𝑒 −𝜃2 𝑍 )
− 𝑒 𝑄0 (𝜃2 −𝜃1 ) (𝑄( 𝑒 −𝜃2 𝑄0 − 𝑒 −𝜃2 𝑍 ) − 𝑒 −𝜃1 𝑄 )]
𝑍2
+ 𝜑1
[(𝑒 −𝜃1 𝑍 −𝑒 −𝜃1 𝑄0 ) + 𝑒 𝑄0 (𝜃2 −𝜃1 ) 𝑒 −𝜃1 𝑄0 ]
2𝑄
+ 𝜑2
(𝑄 − 𝑍)2
[(𝑒 −𝜃1𝑍 − 𝑒 −𝜃1 𝑄0 ) + 𝑒 𝑄0 (𝜃2 −𝜃1 ) 𝑒 −𝜃2 𝑄0 ]
2𝑄
(3.38)
55
𝑑𝜓(𝑍)
= 𝜑1 [(1−𝑒 −𝜃1 𝑄0 )(𝑍 − 𝑄) − 𝑄𝑒 −𝜃1 𝑄
𝑑𝑍
+ (𝑒 −𝜃2 𝑄0 − 𝑒 −𝜃2 𝑍 )(𝑍𝑒 𝑄0(𝜃2 −𝜃1 ) − 𝑒 𝑄0(𝜃2 −𝜃1 ) ) − 𝑒 −𝜃2 𝑄 ]
+ 𝜑1
(𝑄 − 𝑍)2 −𝜃 𝑍
𝑍2
[(𝑒 −𝜃1 𝑍 )] + 𝜑2
[𝑒 1 ]
2𝑄
2𝑄
(3.39)
The following assumption is considered while solving equation 3.39
that the lead time is zero and single period inventory model will be used with
the time horizon considered to as finite. When the supply and the level of
inventory are same and all the other cases are considered zero. Since
analytical solutions to the problem are difficult to obtain. Equation3.39 is
solved using Maple 13 and using equation 3.7. Hence,
𝑑𝜓(𝑍)
1
𝑍
1
𝑄0
= 𝜓(𝑍̂) = 𝑍 − − 𝜃 𝑄 +
+ 𝜃𝑄
𝜃
𝑄
𝑑𝑍
𝜃1 𝑒 1 0 𝜃1 𝑒 1 0 𝑒 1 0
𝑄𝑒 𝜃2 𝑄0
𝑒 𝜃1 𝑄0
𝑒 𝜃1 𝑄0 𝑒 𝜃2 𝑧
1
𝑍
1
𝑄0
=𝑍− − 𝜃 𝑄 +
+
𝜃1 𝑒 1 0 𝜃1 𝑒 𝜃1 𝑄0 𝑒 𝜃1 𝑄0
−𝑍𝑄0 −
𝑄
− 𝑍2 +
(3.40)
The optimal solution is obtained using the numerical illustration by
substituting the value for 𝜃1 , 𝑄0 , 𝑍. 𝜓(𝑍̂) is obtained in form of numerical
illustration 3.5.1 shown below. The Table 3.1 and Figure 3.3, shows the
numerical illustration for model 3.5 when the shortage cost is permitted in the
interval(0, 𝑇).
3.5.1 Numerical illustration
In this Numerical illustration the value for 𝑍 = 1, 2, 3, … ,6. 𝜃1 =
0.5 𝑡𝑜 3, and 𝑄0 = 5,10,15. . ,30 is evaluated and the graph representing these
values are given below which is obtained to get the optimal expected cost
𝜓(𝑍̂). This numerical illustration provides a clear idea of the increased profit
form curve.
56
Table 3.1 Numerical tabulation for obtaining optimal supply
𝟏
𝜽𝟏
𝒆𝜽𝟏
𝒁
𝒆𝜽𝟏 𝑸𝟎
𝟏
𝜽𝟏 𝒆𝜽𝟏 𝑸𝟎
𝑸𝟎
𝜽
𝒆 𝟏 𝑸𝟎
̂)
𝝍( 𝒁
𝒁
𝜽𝟏
𝑸𝟎
1
0.5
5
2
1.648
12.18249
0.1213
0.16417
0.410425 -0.54
2
1
10
1
2.718
22026.47
0.0735
4.54E-05
0.000454 0.926
3
1.5
15
0.6
4.481
5.91E+09 0.0446
1.13E-10
2.54E-09
2.288
4
2
20
0.5
7.389
2.35E+17 0.0270
2.12E-18
8.5E-17
3.472
5
2.5
25
0.4
12.182
1.39E+27 0.0164
2.88E-28
1.8E-26
4.583
6
3
30
0.3
20.085
1.14E+26 0.0149
2.92E-27
1.75E-25
5.651
𝒆𝜽𝟏 𝑸𝟎
Figure 3.3 Expected profit curve
57
Figure 3.4 Optimal supply vs truncation point
3.5.2 Inference:
Supply against 𝜓(𝑍̂) is shown in the Figure 3.3. When the supply size is
increased according to the demand then there a profit or otherwise
instantaneous increase in 𝜓(𝑍̂) is noted. This model shows a sharp increase
in the cost curve is obtained. Figure 3.4 shows a instantaneous decrease in
optimal supply.
3.5.3 Numerical illustration
A comparative study was carried out with the data value available from
Sehik Uduman P.S et.al [65] to check the optimality if the cost curve unique.
Table 3.2 and Figure 3.5 shows the comparative result for supply size and
the shortage graph. When there is shortage in the supply size then there is a
decrease in the expected cost which leads to the profit loss for the company.
58
Table 3.2 Comparative result for supply size
𝒁
𝜽𝟏
𝑸𝟎
0.5 0.5 5
𝒆𝜽𝟏
𝟏⁄
𝜽𝟏 𝒆𝜽𝟏 𝑸𝟎
𝟏⁄
𝒁
𝜽𝟏 ⁄𝒆𝜽𝟏 𝑸𝟎
𝒆𝜽𝟏 𝑸𝟎
𝑸𝑶 /𝒆𝜽𝟏 𝑸𝟎
̂)
𝝍(𝒁
1.648 12.18249
2
0.0606
0.164169997 0.410425 -0.986
10 2.718 22026.47
1
0.0257
4.53999E-05
0.000454 -0.325
0.9 1.5 15 4.481 5.91E+09 0.66
0.0133
1.12793E-10
2.54E-09
0.2199
1.1 2
20 7.389 2.35E+17 0.5
0.0074
2.12418E-18
8.5E-17
0.5925
1.3 2.5 25 12.18 1.39E+27 0.4
0.0042
2.87511E-28
1.8E-26
0.8957
1.5 3
0.0037
2.91884E-27
1.75E-25
1.1629
0.7 1
20 20.08 1.14E+26 0.33
[1]
[2]
CTED
1.0
0.5
0.0
-0.5
-1.0
EXPE
2.0
1.5
COST
3.0
2.5
1.6
1.4
C
1.2
1.0
SU
0.8
PP
B
0.6
LY
0.4
SI
ZE
Figure 3.5 Comparative graph
3.5.4 Inference
A comparative study on the optimal expected profit curve with that of a
Sehik Uduman P.S et.al [65] is shown in form of the Figure 3.5 where curve
[1] is a curve as in the existing model Sehik Uduman P.S et.al [65] and curve
[2] is a new curve for model 3.5. It observed that there is an increase in the
59
profit from negative to positive value leading to an instantaneous increase in
supply size and increased profit. The curve shows the optimality and validity
of this model.
3.6 OPTIMALITY FOR HOLDING COST USING SCBZ PROPERTY
A model is developed to study the optimality for holding cost using
SCBZ property. In such case the units of items unsold at the end of the
season if any are removed from the retail shop to the outlet discount store
and are sold at a lowest price than the cost price of the item which is known
as the salvage loss. A situation is discussed when there is a holding cost
occurred and there is no shortage allowed. In this case the attention can be
restricted to the consideration of the part when the holding cost 1
is
involved, in which case there is an immense loss to the organisation leading
to the setup cost and the cost of holding the item. From Hanssman F [33], the
expected holding cost is given as follows and this cost is truncated before
and after the particular event in the interval (0, 𝑇)
𝑄0
𝑄
𝜓1 (𝑍) = 𝜑1 ∫ (𝑍 − 𝑄)𝑓(𝑄, 𝜃)𝑑𝑄 + 𝜑1 ∫(𝑍 − 𝑄)𝑓(𝑄, 𝜃)𝑑𝑄
0
𝑄0
𝑄0
𝑄
𝑍 𝑡1
𝑍 𝑡1
+ 𝜑1 ∫
𝑓(𝑄, 𝜃)𝑑𝑄 + 𝜑1 ∫
𝑓(𝑄, 𝜃)𝑑𝑄
2𝑇
2𝑇
0
𝑄0
(3.41)
Using the rule given in equation 3.7 and substituting equation 3.28, the
equation 3.41 changes as follows
𝑄0
𝑄
𝑑𝜓1 (𝑍)
= 𝜑1 ∫ (𝑍 − 𝑄)𝜃1 𝑒 −𝜃1 𝑄 𝑑𝑄 + 𝜑1 ∫(𝑍 − 𝑄)𝑒 𝑄0 (𝜃2 −𝜃1) 𝜃2 𝑒 −𝜃2 𝑄 𝑑𝑄
𝑑𝑧
0
𝑄0
𝑄0
𝑄
𝑍 𝑡1
𝑍 𝑡1 𝑄 (𝜃 −𝜃 )
+ 𝜑1 ∫
𝜃1 𝑒 −𝜃1 𝑄 𝑑𝑄 + 𝜑1 ∫
𝑒 0 2 1 𝜃2 𝑒 −𝜃2 𝑄 𝑑𝑄
2𝑇
2𝑇
0
𝑄0
60
(3.42)
𝑄0
𝑄
𝑑𝜓1 (𝑍)
= 𝜑1 [∫ (𝑍 − 𝑄)𝜃1 𝑒 −𝜃1 𝑄 𝑑𝑄 + ∫(𝑍 − 𝑄)𝑒 𝑄0 (𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑄 𝑑𝑄 ]
𝑑𝑧
0
𝑄0
𝑄0
𝑄
𝑍 𝑡1
𝑍 𝑡1 𝑄 (𝜃 −𝜃 )
+ 𝜑1 [∫
𝜃1 𝑒 −𝜃1 𝑄 𝑑𝑄 + ∫
𝑒 0 2 1 𝜃2 𝑒 −𝜃2 𝑄 𝑑𝑄 ]
2𝑇
2𝑇
0
𝑄0
(3.43)
𝑄0
𝑄
𝑑𝜓1 (𝑍)
= 𝜑1 [𝜃1 ∫ (𝑍 − 𝑄)𝑒 −𝜃1 𝑄 𝑑𝑄 + 𝜃2 𝑒 𝑄0(𝜃2 −𝜃1 ) ∫(𝑍 − 𝑄)𝑒 −𝜃2 𝑄 𝑑𝑄 ]
𝑑𝑧
0
𝑄0
𝑄0
𝑄
𝑍 𝑡1
+ 𝜑1
[𝜃 ∫ 𝑒 −𝜃1 𝑄 𝑑𝑄 + 𝜃2 ∫ 𝑒 𝑄0 (𝜃2 −𝜃1 ) 𝑒 −𝜃2 𝑄 𝑑𝑄]
2𝑇 1
0
𝑄0
(3.44)
𝑄0
𝑄0
0
0
𝑑𝜓1 (𝑍)
= 𝜑1 {𝜃1 [∫ 𝑍𝑒 −𝜃1 𝑄 𝑑𝑄 − ∫ 𝑄𝑒 −𝜃1 𝑄 𝑑𝑄 ]
𝑑𝑧
𝑄
𝑄
+ 𝜃2 𝑒 𝑄0(𝜃2 −𝜃1 ) [ ∫ 𝑍𝑒 −𝜃2 𝑄 𝑑𝑄 − ∫ 𝑄𝑒 −𝜃2 𝑄 𝑑𝑄]}
𝑄0
𝑄0
𝑄0
𝑄
𝑍 𝑡1
+ 𝜑1 {
[𝜃 ∫ 𝑒 −𝜃1 𝑄 𝑑𝑄 + 𝜃2 𝑒 𝑄0 (𝜃2 −𝜃1 ) ∫ 𝑒 −𝜃2 𝑄 𝑑𝑄 ]}
2𝑇 1
0
𝑄0
(3.45)
61
𝑄0
𝑄0
0
0
𝑑𝜓1 (𝑍)
= 𝜑1 {𝜃1 [𝑍 ∫ 𝑒 −𝜃1 𝑄 𝑑𝑄 − ∫ 𝑄𝑒 −𝜃1 𝑄 𝑑𝑄 ]
𝑑𝑧
𝑄
𝑄
+ 𝜃2 𝑒 𝑄0(𝜃2 −𝜃1 ) [𝑍 ∫ 𝑒 −𝜃2 𝑄 𝑑𝑄 − ∫ 𝑄𝑒 −𝜃2𝑄 𝑑𝑄]}
𝑄0
𝑄0
𝑄0
𝑄
𝑍 𝑡1
+ 𝜑1 {
[𝜃1 ∫ 𝑒 −𝜃1 𝑄 𝑑𝑄 + 𝜃2 𝑒 𝑄0 (𝜃2 −𝜃1 ) ∫ 𝑒 −𝜃2 𝑄 𝑑𝑄 ]}
2𝑇
0
𝑄0
(3.46)
𝑄0
𝑄0
𝑑𝜓1 (𝑍)
𝑒 −𝜃1 𝑄
𝑒 −𝜃1 𝑄
= 𝜑1 {𝜃1 [𝑍 [
] − (𝑄 [
] − 𝑒 −𝜃1 𝑄 )]
𝑑𝑧
−𝜃1 0
−𝜃1 0
𝑄
+ 𝜃2 𝑒
𝑄0 (𝜃2 −𝜃1 )
𝑄
𝑒 −𝜃2 𝑄
𝑒 −𝜃2 𝑄
[𝑍 [
] − (𝑄 [
] − 𝑒 −𝜃1 𝑄 )]}
−𝜃2 𝑄
−𝜃2 𝑄
0
0
𝑄0
𝑄
𝑍 𝑡1
𝑒 −𝜃1 𝑄
𝑒 −𝜃2 𝑄
+ 𝜑1 {
[𝜃1 [
] + 𝜃2 𝑒 𝑄0 (𝜃2 −𝜃1 ) [
] ]}
2𝑇
−𝜃1 0
−𝜃2 𝑄
0
(3.47)
𝑑𝜓1 (𝑍)
𝑑𝑧
𝑍
1
1
1
= 𝜑1 {[𝜃1 {𝜃 (𝑒 −𝜃1𝑄0 − 1)} − 𝑄𝑒 −𝜃1 𝑄0 − 𝜃 (𝑒 −𝜃2 𝑄0 − 1) + 𝑄𝑒 −𝜃1 𝑄0 ]}
(3.48)
𝑑𝜓1 (𝑍)
𝑑𝑧
1
= 𝜓1 (𝑍̂) = 𝜑1 [𝑍(𝑒 −𝜃1 𝑄0 − 1) − 𝜃 (𝑒 −𝜃2 𝑄0 − 1)]
1
(3.49)
Hence the expected cost obtained is given as equation 3.49.
3.6.1 Numerical illustration
A load of items from 1 tonnes to 6 tonnes is varied accordingly and the
result shows an increase of the expected profit curve which is given by Table
3.3 and Figure 3.6.
62
̂)
Table 3.3 Load of data for 1 to 6 tonnes for finding 𝝍(𝒁
𝒁
𝜽𝟏
𝟏⁄
𝜽𝟏
𝑸𝟎
𝒆𝜽𝟏 𝑸𝟎
1
0.05
20
5
1.28
0.2840
2
0.03 33.33
10
1.34
3
0.07 14.28
15
4
0.09 11.11
5
0.08
6
𝑸𝟎 (𝒆𝜽𝟏 𝑸𝟎 − 𝟏)
̂)
𝝍(𝒁
0.2840
5.6808
-5.3968
0.3498
0.6997
11.661
-11.312
2.85
1.8576
5.5729
26.537
-24.680
20
6.04
5.0496
20.198
56.107
-51.057
12.5
25
7.38
6.3890
31.945
79.863
-73.474
0.09 11.11
20
6.04
5.0496
30.297
56.107
-51.057
(𝒆𝜽𝟏 𝑸𝟎 − 𝟏) 𝒁(𝒆𝜽𝟏 𝑸𝟎 − 𝟏)
7
S
U 6
P
5
P
L 4
Y
3
S 2
I
Z 1
E
0
1
2
3
4
5
EXPECTED HOLDING COST
Figure 3.6 Supply against holding cost
63
6
A state space representation when the supply size is excess then the
total expected cost rules out and this state of condition is shown in the figure
3.7
S
U
P
P
L
Y
S
I
Z
E
3
2.5
2
1.5
1
s
0.5
E(C)
0
-0.5 1
2
3
4
5
6
-1
-1.5
EXPECTED COST
Figure 3.7 State space for the expected total profit
3.7 OPTIMALITY IN CASE OF PLANNED SHORTAGE USING PARTIAL
BACKLOGGING
Planned Shortages or backordering model is illustrated in very few text
books (Anderson et.al [5] and Vora N.D [75]). In literature, few authors use
term "back ordering" while many authors prefer "planned shortages" to
describe this model. Notable work is observed in partial backordering. The
backlogging phenomenon is modelled without using the backorder cost and
the lost sale cost as these costs are not easy to estimate in practice. Abad. P
[1] had studied a continuous review inventory control system over an infinitehorizon with deterministic demand where shortage is partially backlogged.
Khouja M [37] had discussed the state of condition in which a single
period imperfect inventory model with price dependent stochastic demand
and partial backlogging was considered. Mainly there are two types of
shortages, inventory followed by shortages and shortages followed by
inventory. Occurrence of shortage may be either due to the presence of the
64
defective items in the ordered lot or due to the uncertainty of demand. The
shortage cost is assumed proportional to the area under the negative part of
the inventory curve.
The following are the assumptions which are relative to Vora N.D [75]
used in this model.
a) The demand for the item is taken to be constant and continuous.
b) The replenishment for order quantity is done when shortage level
reaches planned shortage level.
c) Stock outs are permitted and shortage or backordering cost per unit is
known and is constant.
From (a) and (b), the limit of integral is considered to be (𝑍, ∞). Now
from equation 3.32, the expected holding cost is given as follows and this
cost is truncated before and after the particular event in the interval (𝑍, ∞)
𝑄0
∞
(𝑄 − 𝑍) 𝑡2
(𝑄 − 𝑍) 𝑡2
𝜓2 (𝑍) = 𝜑2 ∫
𝑓(𝑄, 𝜃1 )𝑑𝑄 + 𝜑2 ∫
𝑓(𝑄, 𝜃2 )𝑑𝑄
2
𝑇
2
𝑇
𝑍
𝑄0
(3.50)
Using equation 3.28 in equation 3.50, the following equation is obtained
𝑄0
∞
(𝑄 − 𝑍) 𝑡2
(𝑄 − 𝑍) 𝑡2 𝑄 (𝜃 −𝜃 )
𝜓2 (𝑍) = 𝜑2 ∫
𝜃1 𝑒 −𝜃1 𝑄 𝑑𝑄 + 𝜑2 ∫
𝑒 0 2 1 𝜃2 𝑒 −𝜃2 𝑄 𝑑𝑄
2
𝑇
2
𝑇
𝑍
𝑄0
(3.51)
Using equation 3.27 in equation 3.51, the following observation is carried out
𝑄0
∞
(𝑄 − 𝑍)2
(𝑄 − 𝑍)2 𝑄 (𝜃 −𝜃 )
−𝜃
𝑄
1
𝜓2 (𝑍) = 𝜑2 ∫
𝜃1 𝑒
𝑑𝑄 + 𝜑2 ∫
𝑒 0 2 1 𝜃2 𝑒 −𝜃2 𝑄 𝑑𝑄
2𝑄
2𝑄
𝑍
𝑄0
(3.52)
65
𝑄0
∞
(𝑄 − 𝑍)2 −𝜃 𝑄
(𝑄 − 𝑍)2 −𝜃 𝑄
𝑄0 (𝜃2 −𝜃1 )
1
(𝑍)
𝜓2
= 𝜑2 {𝜃1 ∫
𝑒
𝑑𝑄 + 𝜃2 𝑒
∫
𝑒 2 𝑑𝑄}
2𝑄
2𝑄
𝑍
𝑄0
(3.53)
𝑄0
𝑄
𝑍2
𝜓2 (𝑍) = 𝜑2 {𝜃1 ∫ ( − 𝑍 + ) 𝑒 −𝜃1 𝑄 𝑑𝑄
2
2𝑄
𝑍
∞
+ 𝜃2 𝑒
𝑄0 (𝜃2 −𝜃1 )
𝑄
𝑍2
∫ ( − 𝑍 + ) 𝑒 −𝜃2 𝑄 𝑑𝑄}
2
2𝑄
𝑄0
Using the equation 3.7, the equation 3.53 changes as follows
𝑄
𝜓2 (𝑍) = 𝜑2 (∑𝑍 0 {−2
𝑒 −𝜃1 𝑄
𝑍2
1
(𝑄 + 𝜃 ) + 2
𝑒 −𝜃1 𝑄
1
𝑍
1
(−𝑄 − 𝜃 )} − 𝑒 −𝜃1 𝑄 )
1
(3.54)
From equation 3.49 and equation 3.54 the following result is obtained
𝜓(𝑍̂) = 𝜓1 (𝑍) + 𝜓2 (𝑍)
(3.55)
The solution of equation 3.55 requires the basic property 𝑍 = 𝑍̂ and
from equation 3.32 it suggests a general principle of balancing the shortage
and overage which shall have an occasion to be applied repeatedly. By
recalling the standard notations generally a control variable 𝑍 and a random
variable 𝑄 with known density which was earlier introduced and two functions
𝜓1 (𝑍, 𝑄) ≥ 0 and 𝜓2 (𝑍, 𝑄) ≤ 0 which may be interpreted as overage and
shortage levels respectively. Assuming the fundamental property of linear
control:
𝑑
𝑑𝑄
[𝜓1 (𝑍)+𝜓2 (𝑍)] = α
(3.56)
where 𝜓 denotes the expected value and α is a constant.
For minimizing a cost of the form 𝜓(𝑍̂) = 𝜑1 𝜓1 (𝑍) + 𝜑2 𝜓2 (𝑍) using equation
3.56 by differentiated with respect to 𝑄, thus following equation is obtained
𝑑𝜓(𝑍̂)
𝑑𝑄
= 𝜑1
𝑑𝜓1 (𝑍)
𝑑𝑄
− 𝜑2 [−
𝑑𝜓1 (𝑍)
𝑑𝑄
+ 𝛼]
(3.57)
The following condition for the optimal valve 𝑍̂ is obtained
𝑑𝜓1 (𝑍)
𝑑𝑄
=
α𝜑2
(3.58)
𝜑1 +𝜑2
66
In other words the derivative of the expected overage must be equal to the
characteristics cost ratio in the equation 3.58. Accordingly, the SCBZ
Property satisfies the existence of the solution hence
𝜓1 (𝑍) = (𝑍 − 𝑄)+
(3.59)
when α = 1 and by computing the general principle of balancing shortage
and overage the following is computed
𝑑𝜓(𝑍)
= 𝜓(𝑍̂)
𝑑𝑄
= 𝜑1 [𝑍(𝑒 −𝜃1 𝑄0 − 1) −
𝑄0
+ 𝜑2 (∑ {−2
𝑍
1 −𝜃 𝑄
(𝑒 2 0 − 1)]
𝜃1
𝑒 −𝜃1 𝑄
1
𝑒 −𝜃1 𝑄
1
(𝑄
+
)
+
2
(−𝑄
−
)} − 𝑒 −𝜃1 𝑄 )
2
𝑍
𝜃1
𝑍
𝜃1
(3.60)
Thus the optimal solution is given by the following
α𝜑2
𝜓(𝑍̂) = 1 − 𝑒 −θ1 Q (𝜑 +𝜑
)
1
(3.61)
2
by considering a linear control with α=1 equation 3.61 to the following result
𝜑2
𝜓(𝑍̂) = 1 − 𝑒 −θ1 Q (𝜑 +𝜑
)
1
(3.62)
2
3.7.1 Inference
The necessity of storage of items cannot be ignored and emphasis
should be given whether the storage is needed or not in the context of
deteriorating items and allowing shortages.
So far in the previous models of this thesis the newsboy problem for
single and double demand was considered, now the generalisation of
newsboy problem is discussed in model 3.8 using SCBZ property.
67
3.8 GENERALIZATION OF NEWSBOY PROBLEM WITH DEMEND
DISTRIBUTION SATISFYING THE SCBZ PROPERTY
The basic Newsboy model has been discussed in Hanssman F [33].
According to the review, the researchers have followed two approaches to
solve the newsboy problems. In the first approach, the expected cost is
overestimated and demand was underestimated. In the second approach,
the expected profit is maximized. But, both the approach yields the same
result. In this chapter, the first approach is used to solve the newsboy
problem. By an appropriate demand decision, the expected cost due to lost
sale could be minimised.
Amy Lau et.al [4] studied the price dependent demand in the Newsboy
problem, Chin Tsai et.al [14] studied the generalisation of Chang and Lin’s
model in a multi location Newsboy problem in which the actual model of
Chang and Lin’s model was extended by adding the delay supply product
cost. In Chin Tsai et.al [14], the Newsboy problem was solved in the
centralised and decentralised system. Nicholas A. Nechval et.al [49] showed
how the statistical inference equivalence principle could be employed in a
particular case of finding the effective statistical solution for the multiproduct
Newsboy problem with constraints.
In this chapter, a generalisation of the actual problem as discussed in
Sehik Uduman P.S [65] is derived using the demand distribution which
satisfies the SCBZ property. This chapter aims to show how SCBZ property
is applied in case of single period, single product inventory model with
several individual source of demand. The objective is to derive the optimal
stock level or the optimal reorder level. Hence the optimal order quantity is
derived. Numerical illustrations are also provided as an example for the
validation of this model.
3.8.1 Basic model
The concept of decentralized inventory system is introduced. The
decentralized inventory system is a system in which a separate inventory is
68
kept to satisfy the several individual source of demand and there is no
reinforcement between locations of demands. Its aim is to minimize the
expected total cost 𝐸(𝐶) and hence the expected total cost function is given
in the form
𝑆𝑖
∞
𝐸(𝐶) = ∫ 𝐶1 (𝑄𝑖 )(𝑆𝑖 − 𝑄𝑖 )𝑓𝑖 (𝑄𝑖 )𝑑𝑄𝑖 + ∫ 𝐶2 (𝑄𝑖 )(𝑄𝑖 − 𝑆𝑖 )𝑓𝑖 (𝑄𝑖 )𝑑𝑄𝑖
0
𝑆𝑖
(3.63)
where 𝑖 = 1,2, … 𝑛
The basic newsboy problem is derived in Hanssman F [33] and hence
adopting the expected total cost given in this model, which is as follows
𝑆
∞
𝐸(𝐶) = 𝐶1 ∫(𝑆 − 𝑋) 𝑓(𝑋)𝑑𝑋 + 𝐶2 ∫ (𝑋 − 𝑆)𝑓(𝑋)𝑑𝑋
0
𝑆
(3.64)
To find optimal 𝑆̂,
𝑑𝐸(𝐶)
𝑑𝑠
= 0 is considered. Since the limit of the integral
involves 𝑆 which is also in the integrand, the differential of integral is applied
as given in equation 3.7. Hence it is proved to result in the equation 3.8.
Given the probability distribution of the demand 𝑋 using the expression
for 𝐹(𝑆̂), the optimal 𝑆̂ was determined. This was the basic Newsboy problem
discussed by Hanssman F [33]. In this model, the SCBZ property is
reformulated as
𝑓𝑖 (𝑄𝑖 ) = {
𝜃1 𝑒 −𝜃1 𝑄𝑖
; 𝑄𝑖 ≤ 𝑄0
𝑄0 (𝜃2 −𝜃1 )
−𝜃2 𝑄𝑖
𝑒
𝜃2 𝑒
; 𝑄𝑖 > 𝑄0
(3.65)
where 𝑄0 is constant denoting truncation point. The probability distribution
function is denoted as
𝑓1 (𝑄𝑖 , 𝜃1 ) if 𝑄𝑖 ≤ 𝑄0
𝑓2 (𝑄𝑖 , 𝜃2 ) if 𝑄𝑖 > 𝑄0
(3.66)
69
The probability distribution function defined above satisfies the SCBZ
property under the above assumptions and the optimal 𝑆𝑖 is to be derived.
Now, the total expected cost from equation 3.63 is given by
𝑆𝑖
𝑄0
𝐸(𝐶) = 𝐶1 (𝑄𝑖 ) ∫ (𝑆𝑖 − 𝑄𝑖 )𝑓1 (𝑄𝑖 , 𝜃1 )𝑑𝑄𝑖 + 𝐶1 (𝑄𝑖 ) ∫ (𝑆𝑖 − 𝑄𝑖 )𝑓2 (𝑄𝑖 , 𝜃2 )𝑑𝑄𝑖
0
𝑄0
𝑄0
+ 𝐶2 (𝑄𝑖 ) ∫ (𝑄𝑖 − 𝑆𝑖 )𝑓1 (𝑄𝑖 , 𝜃1 )𝑑𝑄𝑖
𝑆𝑖
∞
+ 𝐶2 (𝑄𝑖 ) ∫ (𝑄𝑖 − 𝑆𝑖 )𝑓2 (𝑄𝑖 , 𝜃2 )𝑑𝑄𝑖
𝑄0
(3.67)
Using the PDF given of equation 3.65 for 𝑓1 (𝑄𝑖 , 𝜃1 ) and 𝑓2 (𝑄𝑖 , 𝜃2 ) in equation
3.67, the following is obtained
𝑄0
𝐸(𝐶) = 𝐶1 (𝑄𝑖 ) ∫ (𝑆𝑖 − 𝑄𝑖 )𝜃1 𝑒 −𝜃1 𝑄𝑖 𝑑𝑄𝑖
0
𝑆𝑖
+ 𝐶1 (𝑄𝑖 ) ∫(𝑆𝑖 − 𝑄𝑖 )𝑒 𝑄0 (𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑄𝑖 𝑑𝑄𝑖
𝑄0
𝑄0
+ 𝐶2 (𝑄𝑖 ) ∫ (𝑄𝑖 − 𝑆𝑖 )𝜃1 𝑒 −𝜃1 𝑄𝑖 𝑑𝑄𝑖
𝑆𝑖
∞
+ 𝐶2 (𝑄𝑖 ) ∫ (𝑄𝑖 − 𝑆𝑖 )𝑒 𝑄0(𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑄𝑖 𝑑𝑄𝑖
𝑄0
(3.68)
Using the equation 3.7, the equation 3.68 is solved as follows
70
𝑄0
𝑑𝐸(𝐶)
𝑑
= 𝐶1 (𝑄𝑖 ) [
(∫ (𝑆𝑖 − 𝑄𝑖 )𝜃1 𝑒 −𝜃1 𝑄𝑖 𝑑𝑄𝑖
𝑑𝑆𝑖
𝑑𝑠𝑖
0
𝑆𝑖
+ ∫ (𝑆𝑖 − 𝑄𝑖 )𝑒 𝑄0 (𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑄𝑖 𝑑𝑄𝑖 )]
𝑄0
𝑄0
𝑑
+ 𝐶2 (𝑄𝑖 ) [
(∫ (𝑄𝑖 − 𝑆𝑖 )𝜃1 𝑒 −𝜃1 𝑄𝑖 𝑑𝑄𝑖
𝑑𝑠𝑖
𝑆𝑖
∞
+ ∫ (𝑄𝑖 − 𝑆𝑖 )𝑒 𝑄0 (𝜃2 −𝜃1 ) 𝜃2 𝑒 −𝜃2 𝑄𝑖 𝑑𝑄𝑖 )]
𝑄0
(3.69)
To find
𝑑𝐸(𝐶)
𝑑𝑆𝑖
= 0. Assuming
𝑑𝐸(𝐶)
𝑑𝑆𝑖
= 𝐼1 + 𝐼2 = 0
𝑆𝑖
𝑄0
𝐼1 = 𝐶1 (𝑄𝑖 ) [𝜃1 ∫ (𝑆𝑖 − 𝑄𝑖 )𝑒 −𝜃1 𝑄𝑖 𝑑𝑄𝑖 + 𝜃2 ∫ (𝑆𝑖 − 𝑄𝑖 )𝑒 𝑄0(𝜃2 −𝜃1 ) 𝑒 −𝜃2 𝑄𝑖 𝑑𝑄𝑖 ]
0
𝑄0
(3.70)
𝑄0
∞
𝐼2 = 𝐶2 (𝑄𝑖 ) (𝜃1 ∫ (𝑄𝑖 − 𝑆𝑖
)𝑒 −𝜃1 𝑄𝑖
𝑑𝑄𝑖 + 𝜃2 ∫ (𝑄𝑖 − 𝑆𝑖 )𝑒 𝑄0 (𝜃2 −𝜃1 ) 𝑒 −𝜃2 𝑄𝑖 𝑑𝑄𝑖 )
𝑆𝑖
𝑄0
(3.71)
𝑑𝐼
To find 𝑑𝑆1
𝑖
𝑄0
𝐶1 (𝑄𝑖 ) {𝜃1 ∫ (𝑆𝑖 − 𝑄𝑖 )𝑒 −𝜃1 𝑄𝑖 𝑑𝑄𝑖 }
0
𝑄0
𝑄0
−(𝑆𝑖 − 𝑄𝑖 )𝑒 −𝜃1 𝑄𝑖
𝑒 −𝜃1 𝑄𝑖
= 𝐶1 (𝑄𝑖 )𝜃1 {[
] +[
2 ] }
𝜃1
𝜃
1
0
0
= 𝐶1 (𝑄𝑖 )(1 − 𝑒 −𝜃1 𝑄0 )
(3.72)
71
𝑆𝑖
𝐶1 (𝑄𝑖 ) {𝜃2 ∫ (𝑆𝑖 − 𝑄𝑖 )𝑒 𝑄0 (𝜃2 −𝜃1 ) 𝑒 −𝜃2 𝑄𝑖 𝑑𝑄𝑖 }
𝑄0
𝑆𝑖
= 𝐶1 (𝑄𝑖 ) {𝜃2 ∫ 𝑒 𝑄0 (𝜃2 −𝜃1 ) 𝑒 −𝜃2 𝑄𝑖 𝑑𝑄𝑖 }
𝑄0
= 𝐶1 (𝑄𝑖 ) {𝜃2 𝑒 𝑄0 (𝜃2 −𝜃1 ) [
= 𝐶1 (𝑄𝑖 ) {𝜃2 𝑒
𝑄0 (𝜃2 −𝜃1 )
𝑒 −𝜃2 𝑄𝑖
𝜃2
𝑆𝑖
] }
𝑄0
𝑒 −𝜃2 𝑄𝑖 − 𝑒 −𝜃2 𝑄0
[
]} = 𝐶1 (𝑄𝑖 ){𝑒 𝑄0(𝜃2 −𝜃1 ) [𝑒 −𝜃2 𝑆𝑖 − 𝑒 −𝜃2 𝑄0 ]}
−𝜃2
(3.73)
Adding equation 3.72 and equation 3.73
𝑑𝐼1
𝑑𝑆𝑖
= 𝐶1 (𝑄𝑖 )[1 − 𝑒 −𝜃1 𝑄0 . 𝑒 𝜃2 (𝑄0−𝑆𝑖 ) ]
Similarly to find
(3.74)
𝑑𝐼2
𝑑𝑆𝑖
𝑄0
∞
𝑑𝐼2
= 𝐶2 (𝑄𝑖 ) (𝜃1 ∫ (𝑄𝑖 − 𝑆𝑖 )𝑒 −𝜃1 𝑄𝑖 𝑑𝑄𝑖 + 𝜃2 ∫ (𝑄𝑖 − 𝑆𝑖 )𝑒 𝑄0 (𝜃2 −𝜃1 ) 𝑒 −𝜃2 𝑄𝑖 𝑑𝑄𝑖 )
𝑑𝑆𝑖
𝑆𝑖
𝑄0
(3.75)
𝑄0
𝑄0
𝐶2 (𝑄𝑖 ) {𝜃1 ∫ (𝑄𝑖 − 𝑆𝑖 )𝑒 −𝜃1 𝑄𝑖 𝑑𝑄𝑖 } = 𝐶2 (𝑄𝑖 )𝜃1 { ∫ 𝑒 −𝜃1 𝑄𝑖 𝑑𝑄𝑖 }
𝑆𝑖
𝑆𝑖
𝑄0
𝑒 −𝜃1 𝑄𝑖
(𝑄
)𝜃
= −𝐶2 𝑖 1 [
] = 𝐶2 (𝑄𝑖 )[𝑒 −𝜃1 𝑄0 − 𝑒 −𝜃1 𝑆𝑖 ]
−𝜃1 𝑆
𝑖
(3.76)
72
∞
𝐶2 (𝑄𝑖 ) {𝜃2 ∫ (𝑄𝑖 − 𝑆𝑖 )𝑒 𝑄0(𝜃2 −𝜃1 ) 𝑒 −𝜃2 𝑄𝑖 𝑑𝑄𝑖 }
𝑄0
∞
= 𝐶2 (𝑄𝑖 ) {𝜃2 𝑒
𝑄0 (𝜃2 −𝜃1 )
∫ (𝑄𝑖 − 𝑆𝑖 )𝑒 −𝜃2 𝑄𝑖 𝑑𝑄𝑖 }
𝑄0
∞
= 𝐶2 (𝑄𝑖 ) {𝜃2 𝑒
𝑄0 (𝜃2 −𝜃1 )
(𝑄0 − 𝑆𝑖 )𝑒 −𝜃2 𝑄0
𝑒 −𝜃2 𝑄𝑖
([
]−[
] )}
𝜃2
𝜃2 2 𝑥
0
(3.77)
𝑑𝐼
Hence by the equation 3.76 and equation 3.77, 𝑑𝑆2 is obtained as follows,
𝑖
𝑄0
∞
𝑑𝐼2
= 𝐶2 (𝑄𝑖 ) (𝜃1 ∫ (𝑄𝑖 − 𝑆𝑖 )𝑒 −𝜃1 𝑄𝑖 𝑑𝑄𝑖 + 𝜃2 ∫ (𝑄𝑖 − 𝑆𝑖 )𝑒 𝑄0 (𝜃2 −𝜃1 ) 𝑒 −𝜃2 𝑄𝑖 𝑑𝑄𝑖 )
𝑑𝑆𝑖
𝑆𝑖
𝑄0
= −𝐶2 (𝑄𝑖 )𝑒 −𝜃1 𝑆𝑖
(3.78)
Hence
𝑑𝐸(𝐶)
𝑑𝑆𝑖
Therefore
𝑑𝐼
𝑑𝐼
= 0 ⇒ 𝑑𝑆2 + 𝑑𝑆2 = 0
𝑖
𝑖
𝐶1 (𝑄𝑖 )[1 − 𝑒 −𝜃1 𝑄0 . 𝑒 𝜃2 (𝑄0 −𝑆𝑖 ) ] − 𝐶2 (𝑄𝑖 )𝑒 −𝜃1 𝑆𝑖 = 0
𝐶1 (𝑄𝑖 )[1 − 𝑒 −𝜃2 𝑆𝑖 +𝑄0 (𝜃2 −𝜃1 ) ] − 𝐶2 (𝑄𝑖 )(𝑒 −𝜃1 𝑆𝑖 ) = 0
𝐶1 (𝑄𝑖 ) − 𝐶1 (𝑄𝑖 )𝑒 −𝜃2 𝑆𝑖 +𝑄0 (𝜃2 −𝜃1 ) − 𝐶2 (𝑄𝑖 )(𝑒 −𝜃1 𝑆𝑖 ) = 0
𝐶1 (𝑄𝑖 ) = 𝐶1 (𝑄𝑖 )𝑒 −𝜃2 𝑆𝑖 +𝑄0(𝜃2 −𝜃1 ) + 𝐶2 (𝑄𝑖 )(𝑒 −𝜃1 𝑆𝑖 )
(3.79)
Taking log on both sides
𝑙𝑜𝑔𝐶1 (𝑄𝑖 ) = 𝑙𝑜𝑔𝐶1 (𝑄𝑖 ) − 𝜃2 𝑆𝑖 + 𝑄0 (𝜃2 − 𝜃1 ) + 𝑙𝑜𝑔𝐶2 (𝑄𝑖 ) − 𝜃1 𝑆𝑖
(3.80)
𝑙𝑜𝑔𝐶1 (𝑄𝑖 ) − 𝑙𝑜𝑔𝐶1 (𝑄𝑖 ) + 𝑄0 (𝜃2 − 𝜃1 ) + 𝑙𝑜𝑔𝐶2 (𝑄𝑖 ) = 𝜃2 𝑆𝑖 + 𝜃1 𝑆𝑖
𝑙𝑜𝑔𝐶2 (𝑄𝑖 ) + 𝑄0 (𝜃2 − 𝜃1 ) = 𝑆𝑖 (𝜃2 + 𝜃1 )
𝑆𝑖 =
𝑙𝑜𝑔𝐶2 (𝑄𝑖 ) + 𝑄0 (𝜃2 − 𝜃1 )
(𝜃2 + 𝜃1 )
(3.81)
73
By substituting the values of 𝐶2 (𝑄𝑖 ), 𝑄0 , 𝜃1 and 𝜃2 , the value of 𝑆𝑖 is obtained
which satisfies equation 3.81. The value of 𝑆𝑖
is also evaluated using a
suitable computer program.
3.8.2 Numerical illustration
In the numerical example 𝑪𝟐 (𝑸𝒊 ) = 𝟓 is fixed and 𝜽𝟏 , 𝜽𝟐 , 𝑸𝟎 is varied
accordingly and these value are substituted in equation 3.81 to obtain the
value of 𝑺𝒊
𝑪𝟐 (𝑸𝒊 ) = 𝟓, 𝜽𝟐 = 𝟐 , 𝑸𝟎 = 𝟐𝟎
Case i)
Table 3.4: 𝜽𝟏 variation for obtaining 𝑺𝒊
𝜃1
1.0
1.5
2.0
𝑆𝑖
5.7
2.6
0.17
Figure 3.8: Supply against 𝛉𝟏 curve
74
𝑪𝟐 (𝑸𝒊 ) = 𝟓, 𝜽𝟏 = 𝟏. 𝟓, 𝑸𝟎 = 𝟐𝟎
Case ii)
Table 3.5: 𝜃2 variation for obtaining 𝑆𝑖
𝜃2
1.5
2.0
2.5
𝑆𝑖
0.23
3.56
6.89
Figure 3.9: Curve for supply 𝐒𝐢 against 𝛉𝟐
𝑪𝟐 (𝑸𝒊 ) = 𝟓, 𝜽𝟏 = 𝟏. 𝟓 , 𝜽𝟐 = 𝟐
Case iii)
Table 3.6: 𝑸𝟎 variation for obtaining 𝑺𝒊
𝑄0
10
15
20
𝑆𝑖
1.4
2.04
2.67
75
Figure 3.10: Curve for supply and truncation point Q0
3.8.3 Inference
From the numerical illustrations and corresponding figures the
following conclusions may be drawn.
Case i) If the parameter 𝜃1 of the demand distribution prior to the truncation
point 𝑄0 is varied exponentially then the expected demand will decrease
1
because 𝐸(𝑄𝑖 ) = 𝜃 this implies that whenever 𝜃1 increases the demand will
1
decrease and hence a smaller supply size 𝑆𝑖 for several individual demand is
suggested.
Case ii): When the parameter 𝜃1 is fixed and 𝜃2 which denotes the parameter
of the demand distribution posterior to the truncation point 𝑄0 is increased,
then a corresponding increase in supply size 𝑆𝑖 for several individual demand
is suggested.
Case iii): If both 𝜃1 and 𝜃2 are fixed and if the truncation point 𝑄0 increases
then there will be an increase in the supply size. The demand after 𝑄0 is
smaller. As the truncation point 𝑄0 increases then 𝑆𝑖 for several individual
demand increases and the demand is dominated by 𝜃1 .Therefore an
increased inventory is suggested.
76
3.9 CONCLUSION
In the most realistic setting, the variability of benefit in stochastic
inventory models cannot be ignored. This model is examined in form of time
point occurring before the truncation point, and the time point occurring after
the truncation point. In which case, the SCBZ property seems to be a useful
concept and needs further attention. Thus the demand may be stock
dependent up to certain time after that it is constant due to some good will of
the retailer. This model can be considered in future with deteriorating items.
Hence the optimal order level or Supply 𝑆𝑖 is less than the risk neutral
counterpart is applied as base stock policy which is discussed in chapter 5.
This model framework can be extended in several ways. An obvious
extension would be to consider this as newsvendor model for two products or
multiproduct, in which case the expression would be more complex and it will
have complex probability functions and integrations. So an attempt is made
to solve these models in form of chapter 7.
77