7. THE MULTI-PERIOD MODEL WITH TWO VARYING DEMANDS
7.1 INTRODUCTION
So far in the above chapters, the single period stochastic models are
discussed. In this chapter, the multi-period stochastic model is discussed. The
key difference between single-period model and multi-period model is that, in
single period stock left over will not be carried over to the next period which
means profit is loss. In case of the multi-period, the model may involve stock
leftovers from previous periods, which makes the optimal choice of order
quantities more complicated. It may be observed that in many situations, the
demand for a product cannot be below a particular level.
Another aspect of consideration in the representation of the demand with
a suitable probability distribution is that, the demand size with past has impact or
influence over the demand at a future points of time. Hence, it should be
represented as not having the Lack of Memory Property (LMP). Hence, it is
proposed to use the random variable which follows Erlang 2 distribution that
does not satisfy the LMP property. The random variable ∑𝑛𝑖=1 𝑄𝑖 follows
exponential distribution with parameter 𝜃1 prior to the truncation point 𝑄0 and it
follows truncated exponential distribution with parameter 𝜃2 after the truncation
point.
Sakaguchi M et.al [59] studied the probabilistic inventory models of multiperiod in which some conditions are reviewed to help getting an optimal policy
provided that the total cost function of single period is known very well. A model
with exponential demand is studied in Sakaguchi M et.al [60], since it is easy
when demand subjects to an exponential distribution.
In this Chapter, two varying demand model is discussed. Under model
7.4, demand and lead time is a constant. In model 7.5, demand and lead time is
considered as random variable. In obtaining the expected total cost, the
probability of having exactly ‘Nth’ demand epochs in the interval (0, 𝑇) is taken
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under consideration. In model 7.6, the optimal one time supply during the
interval (0, 𝑇) using the generalized gamma distribution with bessel’s function is
discussed, where else model 7.7 deals with a multi-commodity inventory system
with periodic review operating under a stationary policy using the exponential
order statistics.
The expected optimal ordering shown in figure 7.1 indicates a
point at which there is a requirement of reorder. Hence this point is considered
to be the truncation point.
Figure 7.1: Optimal expected ordering when the truncation occurs
The optimal inventory level or the reorder point is determined form the
Figure 7.1 for the multi-period demands. Also adequate numerical analysis
shows its effectiveness.
7.2 BASIC MODEL
In this chapter, a modified version of the model discussed in Sehik
Uduman P.S et.al [64] is considered under the assumption that the random
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variable denoting demand ∑𝑛𝑖=1 𝑄𝑖 undergoes a change in the distribution after a
change point or truncation point denoted as 𝑄0 . Hence, the use of change of the
distribution after a change point is justified by the fact that demand for any
product over the time interval (0, 𝑇) is not fixed. If the demand for the product is
according to some probability distribution initially and it is very likely that after a
certain point the demand may undergo some changes and the increase in
demand or decrease in demand will undergo considerable change. Hence, to
depict the demand as a random variable undergoing a change of distribution
after the particular magnitude 𝑄0 is quite reasonable. The concept of change of
distribution at a change point was discussed by Suresh Kumar R [72]. In the
present model, the expected total cost is given as
𝑍
∞
𝜓(𝐶) = 𝜑1 ∫0 (𝑍 − ∑𝑛𝑖=1 𝑄𝑖 ) 𝑓(𝑄𝑖 )𝑑𝑄𝑖 + 𝜑2 ∫𝑍 (∑𝑛𝑖=1 𝑄𝑖 − 𝑍) 𝑓(𝑄𝑖 )𝑑𝑄𝑖
(7.1)
Since, equation 7.1 is in form of the differentiation of an integral with respect to
the variable 𝑍, i.e., 𝑍 is as the integrand, as well as in the limits of integration.
Hence, differential of integral formula is used to solve the result which is
discussed as equation 3.7 of chapter 3. This implies that, the optimal value of 𝑍
is one which satisfies the equation
𝑧
𝜑1
∫0 𝑓(𝑄𝑖 )𝑑𝑄𝑖 = 𝐹(𝑍̂) = 𝑃[∑𝑛𝑖=1 𝑄𝑖 < 𝑍] = 𝜑 +𝜑
1
2
(7.2)
Given the values of the inventory holding cost 𝜑1 , the shortage cost 𝜑2 and the
probability distribution 𝑓(𝑄𝑖 ) of the random variables ∑𝑛𝑖=1 𝑄𝑖 denoting Multiproduct demands, the optimal 𝑍̂ can be determined. This was a basic procedure
for solving the model in Hanssman F [33].
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7.3 NOTATIONS AND ASSUMPTIONS
𝑚
∑ 𝑄𝑖
𝑖=1
-
A continuous identically independent random variable denoting
the demand at the Nth epoch, N = 1,2,…, and 𝑄𝑖 has PDF 𝑓(. )
with CDF 𝐹(. )
𝜑1
- Inventory holding cost / unit
𝜑2
- Shortage cost / unit
𝜃1
- Time variable constant before the truncation point
𝜃2
- Time variable constant after the truncation point
𝑍
- The supply size or initial stock level
𝑄0
- The change point or truncation point
𝑇
- Total lead time
𝑍̂
- Optimal value of Z
𝑈𝑖
- The inter arrival times between successive demand epochs.
𝑛1 , 𝑛2 and 𝑛 - All are nonnegative, and their inequality is
𝑛1 + 𝑛2
𝑎1 , … , 𝑎𝑛
- the stock level
=𝑎
𝜇
- Location parameter
121
max(𝑛1 , 𝑛2 ) ≤ 𝑛 ≤
7.4 THE MULTI-DEMAND TRUNCATED EXPONENTIAL DISTRIBUTION
In this model an extension of the work done by Deemer W.L et.al [18]
form of the truncated exponential distribution considered under the two
parameters 𝜃1 , 𝜃2
𝜃1 𝑒 −𝑄0 𝜃1 , 0 < 𝜃1 < 𝑄0
𝑓(𝑄𝑖 , 𝜃), = {𝜃2 𝑒 −𝑄0𝜃2 (1 − 𝑒 −𝑄0 𝜃2 ), 𝑄0 < 𝜃2 < 𝑍
𝑂 , 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
(7.3)
When 𝑄0 is a constant, then following cases arises. (i) 𝑄0 ≤ 𝑍 and (ii) 𝑄0 > 𝑍
Case i): 𝑄0 ≤ 𝑍
𝑍
𝑛
∞
𝑛
𝜓(𝐶) = 𝜑1 ∫ (𝑍 − ∑ 𝑄𝑖 ) 𝑓(𝑄𝑖 , 𝜃)𝑑𝑄𝑖 + 𝜑2 ∫ (∑ 𝑄𝑖 − 𝑍) 𝑓(𝑄𝑖 , 𝜃)𝑑𝑄𝑖
0
𝑍
𝑖=1
𝑖=1
(7.4)
𝑄0
𝜓(𝐶) = 𝜑1 ∫
0
𝑛
(𝑍 − ∑ 𝑄𝑖 ) 𝑓(𝑄𝑖 , 𝜃)𝑑𝑄𝑖
𝑖=1
𝑛
𝑍
∞
𝑛
+ 𝜑1 ∫ (𝑍 − ∑ 𝑄𝑖 ) 𝑓(𝑄𝑖 , 𝜃)𝑑𝑄𝑖 + 𝜑2 ∫ (∑ 𝑄𝑖 − 𝑍) 𝑓(𝑄𝑖 , 𝜃)𝑑𝑄𝑖
𝑄0
𝑍
𝑖=1
𝑖=1
(7.5)
Applying differential of integral equation 3.7 of chapter 3, the equation 7.5 is given
as
𝑛
𝑛
𝑖=1
𝑖=1
𝑄0
𝑍
𝑑𝜓(𝐶)
= 𝜑1 (∫ (𝑍 − ∑ 𝑄𝑖 ) 𝑓(𝑄𝑖 , 𝜃)) 𝑑𝑄𝑖 + 𝜑1 (∫ (𝑍 − ∑ 𝑄𝑖 ) 𝑓(𝑄𝑖 , 𝜃)) 𝑑𝑄𝑖
𝑑𝑍
0
𝑄0
∞
𝑛
+ 𝜑2 (∫ (∑ 𝑄𝑖 − 𝑍) 𝑓(𝑄𝑖 , 𝜃)) 𝑑𝑄𝑖
𝑍
𝑖=1
(7.6)
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Hence
𝑑𝜓(𝐶)
𝑑𝑍
𝑑
= 𝑑𝑍 (𝐼1 + 𝐼2 + 𝐼3 )
(7.7)
Using the equation 7.3, the expression for expected cost is written as
𝑑𝐼1
𝑑𝑍
𝑄
𝑄
= 𝜑1 ∫0 0(𝑍 − ∑𝑛𝑖=1 𝑄𝑖 ) 𝑓(𝑄𝑖 , 𝜃1 )𝑑𝑄𝑖 = 𝜑1 ∫0 0(𝑍 − ∑𝑛𝑖=1 𝑄𝑖 ) 𝜃1 𝑒 −𝑄0𝜃1 𝑑𝑄𝑖
𝑄
𝑑𝐼1
= 𝜑1 𝜃1 𝑒 −𝑄0 𝜃1 ∫0 0(𝑍 − ∑𝑛𝑖=1 𝑄𝑖 )𝑑𝑄𝑖 = 𝑄0 𝜑1 𝜃1 𝑒 −𝑄0 𝜃1 (𝑄0 −
𝑑𝑍
2
(∑𝑛
𝑖=1 𝑄0 )
2
)
(7.8)
𝑍
𝑑𝐼2
= 𝜑1 ∫𝑄 (𝑍 − ∑𝑛𝑖=1 𝑄𝑖 ) 𝑓(𝑄𝑖 , 𝜃2 )𝑑𝑄𝑖
𝑑𝑍
0
𝑍
= 𝜑1 ∫𝑄 (𝑍 − ∑𝑛𝑖=1 𝑄𝑖 ) 𝜃2 𝑒 −𝑄0 𝜃2 (1 − 𝑒 −𝑄0 𝜃2 )𝑑𝑄𝑖
0
𝑑𝐼2
𝑑𝑍
𝑍
= 𝜑1 (1 − 𝑒 −𝑄0𝜃2 )𝜃2 𝑒 −𝑄0𝜃2 ∫𝑄 (𝑍 − ∑𝑛𝑖=1 𝑄𝑖 ) 𝑑𝑄𝑖
0
2
(∑𝑛
𝑖=1 𝑍)
= 𝜑1 𝜃2 (1 − 𝑒 −𝑄0 𝜃2 )𝑒 −𝑄0 𝜃2 {(𝑍 − 𝑄0 ) − (
𝑑𝐼3
𝑑𝑍
2
∞
2
−
(∑𝑛
𝑖=1 𝑄0 )
2
)}
(7.9)
∞
= 𝜑2 ∫𝑍 (∑𝑛𝑖=1 𝑄𝑖 − 𝑍) 𝑓(𝑄𝑖 , 𝜃)𝑑𝑄𝑖 = 𝜑2 ∫𝑍 (∑𝑛𝑖=1 𝑄𝑖 − 𝑍) 0 = 0
(7.10)
Hence using equation 7.8, 7.9 and 7.10, the following result is obtained
𝑛
2
(∑
𝑄 )
𝜓(𝐶̂ ) = 𝑄0 𝜑1 𝜃1 𝑒 −𝑄0 𝜃1 (𝑄0 − 𝑖=12 0 )
+𝜑1 𝜃2 (1 − 𝑒
−𝑄0 𝜃2
)𝑒
−𝑄0 𝜃2
2
(∑𝑛
𝑖=1 𝑍)
{(𝑍 − 𝑄0 ) − (
2
2
−
(∑𝑛
𝑖=1 𝑄0 )
2
)} = 0
(7.11)
̂
Any value of Z, which satisfies equation 7.11 is the optimal 𝑍.
7.4.1 Numerical illustration
Considering the numerical example when the value of 𝜑1 , 𝜃1 , 𝜃2 are fixed
and the values of 𝑄0 , ̂𝑍 are varied accordingly. Let 𝜑1 = 5, 𝜃1 = 10, and 𝜃2 = 20
then the following Table7.1 is obtained.
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Table 7.1: Numerical value for 𝑄0 for obtaining 𝑍̂
𝑄0
1.5
2.0
2.5
3.0
3.5
𝑍̂
0.7348
0.5511
0.4359
0.3649
0.3109
Figure 7.2: Optimal supply z against the truncation point 𝑄0 when 𝑄0 ≤ 𝑍
7.4.2 Inference
For the case, when 𝑄0 is a constant, the condition 𝑄0 ≤ 𝑍 is independent
of the functions of the parameter 𝑄0 , 𝜑1 , 𝜃1 , 𝜃2 . It is observed that as 𝑄0
increases, the value of 𝑍̂ decreases. This is due to the fact that ∑𝑛𝑖=1 𝑄𝑖 is the
parameter of the exponential distribution that denotes the demand. Also various
points at which there is a fluctuation in demand is shown in the figure 7.2.
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Case ii): When 𝑄0 > 𝑍, then from equation 7.4
𝑛
𝑍
∞
𝑛
𝜓(𝐶) = 𝜑1 ∫ (𝑍 − ∑ 𝑄𝑖 ) 𝑓(𝑄𝑖 , 𝜃)𝑑𝑄𝑖 + 𝜑2 ∫ (∑ 𝑄𝑖 − 𝑍) 𝑓(𝑄𝑖 , 𝜃)𝑑𝑄𝑖
0
𝑍
𝑖=1
𝑖=1
𝑛
𝑍
𝜓(𝐶) = 𝜑1 ∫ (𝑍 − ∑ 𝑄𝑖 ) 𝑓(𝑄𝑖 , 𝜃1 )𝑑𝑄𝑖
0
𝑖=1
𝑛
𝑄0
+ 𝜑2 ∫
∞
𝑛
(∑ 𝑄𝑖 − 𝑍) 𝑓(𝑄𝑖 , 𝜃1 )𝑑𝑄𝑖 + 𝜑2 ∫ (∑ 𝑄𝑖
𝑍
𝑄0
𝑖=1
𝑖=1
− 𝑍) 𝑓(𝑄𝑖 , 𝜃2 )𝑑𝑄𝑖
(7.12)
Applying differential of integral equation 3.7 of chapter 3, the equation 7.12 is
given as,
𝑑𝜓(𝐶)
𝑑
(𝑀 + 𝑀2 + 𝑀3 )
=
𝑑𝑍
𝑑𝑍 1
𝑛
𝑛
𝑖=1
𝑖=1
𝑍
𝑍
𝑍
𝑍
𝑑𝑀1
= 𝜑1 ∫ (𝑍 − ∑ 𝑄𝑖 ) 𝑓(𝑄𝑖 , 𝜃1 )𝑑𝑄𝑖 = 𝜑1 ∫ (𝑍 − ∑ 𝑄𝑖 ) 𝜃1 𝑒 −𝑄0 𝜃1 𝑑 𝑄𝑖
𝑑𝑍
0
0
𝑍
= 𝜑1 𝜃1 𝑒 −𝑄0 𝜃1 ∫0 (𝑍 − ∑𝑛𝑖=1 𝑄𝑖 )𝑑 𝑄𝑖 = 𝜑1 𝜃1 𝑒 −𝑄0 𝜃1 [∫0 𝑍 𝑑 𝑄𝑖 − ∫0 ∑𝑛𝑖=1 𝑄𝑖 𝑑 𝑄𝑖 ]
𝑑𝑀1
𝑑𝑍
1
= 𝜑1 𝜃1 𝑒 −𝑄0𝜃1 [𝑍𝑍𝑖 − ∑𝑛𝑖=1 (2 𝑍 2 ) ]
(7.13)
𝑖
𝑛
𝑛
𝑖=1
𝑖=1
𝑄0
𝑄0
𝑑𝑀2
= 𝜑2 ∫ (∑ 𝑄𝑖 − 𝑍) 𝑓(𝑄𝑖 , 𝜃1 )𝑑𝑄𝑖 = 𝜑2 ∫ (∑ 𝑄𝑖 − 𝑍) 𝜃1 𝑒 −𝑄0𝜃1 𝑑 𝑄𝑖
𝑑𝑍
𝑍
𝑍
𝑄
= 𝜑2 𝜃1 𝑒 −𝑄0𝜃1 ∫𝑍 0(∑𝑛𝑖=1 𝑄𝑖 − 𝑍)𝑑 𝑄𝑖
𝑄
𝑄
= 𝜑2 𝜃1 𝑒 −𝑄0 𝜃1 [∫𝑍 0 ∑𝑛𝑖=1 𝑄𝑖 𝑑 𝑄𝑖 − ∫𝑍 0 𝑍 𝑑 𝑄𝑖 ]
125
𝑑𝑀2
𝑑𝑍
1
= 𝜑2 𝜃1 𝑒 −𝑄0 𝜃1 ∑𝑛𝑖=1 [(2 (𝑄0 2 − 𝑍 2 )) − 𝑍(𝑍 − 𝑄0 )𝑖 ]
𝑖
(7.14)
𝑛
𝑛
𝑖=1
𝑖=1
∞
∞
𝑑𝑀3
= 𝜑2 ∫ (∑ 𝑄𝑖 − 𝑍) 𝑓(𝑄𝑖 , 𝜃2 )𝑑𝑄𝑖 = 𝜑2 ∫ (∑ 𝑄𝑖 − 𝑍) 𝜃2 𝑒 −𝑄0𝜃2 (1
𝑑𝑍
𝑄0
𝑄0
− 𝑒 −𝑄0 𝜃2 )𝑑𝑄𝑖
𝑛
∞
𝑑𝑀3
= 𝜑2 𝜃2 𝑒 −𝑄0 𝜃2 (1 − 𝑒 −𝑄0 𝜃2 ) ∫ (∑ 𝑄𝑖 − 𝑍) 𝑑𝑄𝑖
𝑑𝑍
𝑄0
𝑖=1
∞ 𝑛
= 𝜑2 𝜃2 𝑒
−𝑄0 𝜃2
(1 − 𝑒
−𝑄0 𝜃2
∞
) [∫ ∑ 𝑄𝑖 𝑑 𝑄𝑖 − ∫ 𝑍 𝑑 𝑄𝑖 ]
𝑄0 𝑖=1
𝑄0
𝑛
∞
∞
𝑑𝑀3
= 𝜑2 𝜃2 𝑒 −𝑄0 𝜃2 (1 − 𝑒 −𝑄0 𝜃2 ) [∫ ∑ 𝑄𝑖 𝑑 𝑄𝑖 − ∫ 𝑍 𝑑 𝑄𝑖 ]
𝑑𝑍
𝑄0
𝑄0
𝑖=1
𝑑𝑀3
𝑑𝑍
= 𝜑2 𝜃2 𝑒 −𝑄0𝜃2 (1 − 𝑒 −𝑄0 𝜃2 ) [(−
𝑄0 2
2
) + 𝑍(𝑄0 )𝑖 ]
𝑖
(7.15)
𝑑
Now, 𝑑𝑍 (𝑀1 + 𝑀2 + 𝑀3 ) = 0 gives,
𝜓(𝐶) = 𝜑1 𝜃1 𝑒 −𝑄0𝜃1 [𝑍𝑍𝑖
𝑛
𝑛
𝑖=1
𝑖=1
1
1
− ∑ ( 𝑍 2 ) ] +𝜑2 𝜃1 𝑒 −𝑄0 𝜃1 ∑ [( (𝑄0 2 − 𝑍 2 )) − 𝑍(𝑍 − 𝑄0 )𝑖 ]
2
2
𝑖
𝑖
+ 𝜑2 𝜃2 𝑒 −𝑄0 𝜃2 (1 − 𝑒 −𝑄0 𝜃2 ) [(−
𝑄0 2
2
) + 𝑍(𝑄0 )𝑖 ]
𝑖
(7.16)
Any value of 𝑍 which satisfies equation 7.16 is the optimal value ‘ Ẑ ’.
7.4.3 Numerical Illustration
Let us take the numerical example when the value of 𝜑1 , 𝜃1 , 𝜃2 and 𝜑2
are fixed and the values of 𝑄0 , ̂𝑍 are varied accordingly. Let us assume that
126
𝜃1 = 1.0, 𝜃2 = 2.0, 𝜑1 = 5, 𝜑2 = 10. Hence, Table 7.2 is obtained as follows,
Table 7.2: Tabulation for obtaining 𝑍̂
𝑄0
𝑍̂
25
30
35
40
45
4.4804 7.3137 10.0035 12.6383 15.2433
Figure 7.3: Optimal supply z against the truncation point 𝑄0 when 𝑄0 > 𝑍
7.4.4 Inference
It is observed that as the value of the truncation point 𝑄0 increases, the
size of the optimal inventory 𝑍̂ also increases. This is due to fact that prior to the
truncation point the demand is distributed as exponential with parameter 𝜃1 .
After the truncation point the demand is distributed as truncated exponential
distribution with parameter 𝜃2 . Also the average of demand before 𝑄0 is
exponential and after 𝑄0 varies according to the situation. Hence in figure 7.3,
127
the variations of the demand at three points are depicted with different colour
arrows. Therefore the optimal inventory is also increasing.
7.5 Nth EPOCH TWO COMMODITY MODEL
In real life situation demand is always assumed to be random. In this
model, demand and the lead time is considered to be random N th epoch. Let
there be Nth demand epochs in (0, 𝑇), i.e., 𝑄1 , 𝑄2 , 𝑄3 … , 𝑄𝑁 be the random
demands and 𝑄𝑖 , i = 1, 2…N are identically independent random variables.
It may be noted that, if ∑𝑘𝑖=1 𝑄𝑖 ≤ 𝑍, then inventory holding occurs and if
∑𝑘𝑖=1 𝑄𝑖 > 𝑍 then shortage occurs.
Since ∑𝑘𝑖=1 𝑄𝑖 is sum of the identically independent random variable and
its PDF is given by 𝑓𝑁 (𝑄), which is the Nth convolution of 𝑓(𝑄). Hence
∞
𝑍
𝜓(𝐶) = 𝜑1 ∫0 (𝑍 − ∑𝑘𝑖=1 𝑄𝑖 )𝑓𝑁 (𝑄)𝑑𝑄 + 𝜑2 ∫𝑍 (∑𝑘𝑖=1 𝑄𝑖 − 𝑍)𝑓𝑁 (𝑄)𝑑𝑄
(7.17)
The probability of having exactly ‘N’ demand epochs in (0,T) is given
by renewal theory as 𝑃[𝑁(𝑇) = 𝑘] = 𝐺𝑁 (𝑇) − 𝐺𝑁+1 (𝑇) = 1 and this statement is
proved in chapter 5 and 𝑈𝑖 is the inter arrival times between successive demand
epochs. The determination of 𝑍̂ , which is the optimal value of 𝑍 is possible
using equation 7.17, provided the number of demand epochs in (0, 𝑇) which is
namely ‘N’ is known. But, in practice the value of N is not a predetermined
constant. It is also of random character. But from Nabil S Faour [47], it is
possible to have an approximate value for N. The author discussed that, If N is
taken to be a particular value then using incomplete gamma function values the
optimal 𝑍̂ can be obtained. If 𝑍 > 0 units are ordered, the fixed cost will be a
function of 𝑍, say 𝑁(𝑍). In general, 𝑁(𝑍) will take as many different values as
the number of alternative different ordering decisions.
128
𝑛1
𝑛
For the two commodity problem 𝑁(𝑍) = { 2
𝑛
0
𝑖𝑓 𝜃1 > 0 𝑎𝑛𝑑 𝜃2 = 0
𝑖𝑓 𝜃1 = 0 𝑎𝑛𝑑𝜃2 > 0
𝑖𝑓 𝜃1 > 0 𝑎𝑛𝑑 𝜃2 > 0
𝑖𝑓𝜃1 = 0 𝑎𝑛𝑑 𝜃2 = 0
(7.18)
Where 𝑛1 , 𝑛2 𝑎𝑛𝑑 𝑛 are all nonnegative, and the inequality max(𝑛1 , 𝑛2 ) ≤ 𝑛 ≤
𝑛1 + 𝑛2 is satisfied. It may be observed 𝑛1 and 𝑛2 are obtained on the basis of
an average which is taken by variable of occurrences of demand value before
𝑄0 and after 𝑄0 respectively. It is assumed that there are 𝑛1 random epochs at
𝑄 ≤ 𝑍 and at 𝑛2 epochs 𝑄 > 𝑍 and 𝑛 = 𝑛1 + 𝑛2 . Using the property of 7.18 in
7.17, the expected optimal profit is given by
𝑄0
𝑍
𝜓(𝐶) = 𝜑1 ∫ (𝑍 − 𝑛1 𝑄)𝑓𝑁 (𝑄, 𝜃1 )𝑑𝑄 + 𝜑1 ∫ (𝑍 − 𝑛2 𝑄)𝑓𝑁 (𝑄, 𝜃2 )𝑑𝑄
0
𝑄0
∞
+𝜑2 ∫𝑍 (𝑛 𝑄 − 𝑍)𝑓𝑁 (𝑄, 𝜃)𝑑𝑄
(7.19)
Let 𝜓(𝐶) be continuous and twice differentiable. The function 𝜓(𝐶̂ ) is the cost
charged over a given period of time excluding the ordering cost and in general it
is the holding and shortage costs for each item 𝑖 = 1,2,3 … , 𝑚 in a linear form.
From Nabil S Faour [47], considering the case of the two commodity, the optimal
expected cost is obtained for the 𝑁 th demand epoch,
𝜃1 𝑒 −𝑄0 𝜃1 , 0 < 𝜃1 < 𝑄0
𝑓𝑁 (𝑄, 𝜃), = { 𝜃2 𝑒 −𝑄0𝜃2 (1 − 𝑒 −𝑄0 𝜃2 )2 , 𝑄0 < 𝜃2 < 𝑍
𝑂,
𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
(7.20)
Considering 𝑄 ≤ 𝑍 and by solving the equation 7.19 with respect to equation
7.20
129
𝑄0
𝑑𝜓(𝐶)
= 𝜑1 𝜃1 ∫ (𝑍 − 𝑛1 𝑄)𝑒 −𝑄0 𝜃1 𝑑𝑄
𝑑𝑍
0
𝑍
∞
2
+𝜑1 𝜃2 ∫ (𝑍 − 𝑛2 𝑄)𝑒 −𝑄0 𝜃2 (1 − 𝑒 −𝑄0 𝜃2 ) 𝑑𝑄 + 𝜑2 ∫ (𝑛 𝑄 − 𝑍)0𝑑𝑄
𝑄0
𝑍
(7.21)
𝑄0
𝑄0
𝑑𝜓(𝐶)
= 𝜑1 𝜃1 [∫ (𝑍)𝑒 −𝑄0𝜃1 𝑑𝑄 − ∫ (𝑛1 𝑄)𝑒 −𝑄0 𝜃1 𝑑𝑄]
𝑑𝑍
0
0
𝑍
2
+ 𝜑1 𝜃2 [∫ (𝑍)𝑒 −𝑄0 𝜃2 (1 − 𝑒 −𝑄0 𝜃2 ) 𝑑𝑄
𝑄0
𝑍
2
− ∫ (𝑛2 𝑄)𝑒 −𝑄0 𝜃2 (1 − 𝑒 −𝑄0 𝜃2 ) 𝑑𝑄 ]
𝑄0
(7.22)
𝑄0
𝑄0
𝑑𝜓(𝐶)
−𝑄0 𝜃1
= 𝜑1 𝜃1 𝑒
[𝑍 ∫ 𝑑𝑄 − 𝑛1 ∫ 𝑄𝑑𝑄 ]
𝑑𝑍
0
0
+ 𝜑1 𝜃2 𝑒
−𝑄0 𝜃2
(1 − 𝑒
−𝑄0 𝜃2 2
𝑍
𝑍
) [𝑍 ∫ 𝑑𝑄 − 𝑛2 ∫ 𝑄𝑑𝑄 ]
𝑄0
𝑄0
𝑑𝜓(𝐶)
𝑄0 2
= 𝜑1 𝜃1 𝑒 −𝑄0𝜃1 [𝑍𝑄0 − 𝑛1 (
)]
𝑑𝑍
2
+ 𝜑1 𝜃2 𝑒
−𝑄0 𝜃2
(1 − 𝑒
𝑄 2 𝑄0 2
) [𝑍(𝑄 − 𝑄0 ) − 𝑛2 ( −
)]
2
2
−𝑄0 𝜃2 2
(7.23)
𝑑𝜓(𝐶)
𝑑𝑍
𝑄0 2
= 𝜑1 𝑒 −𝑄0𝜃1 {𝜃1 [𝑍𝑄0 − 𝑛1 (
𝑄2
𝑛2 ( 2 −
𝑄0
2
2
2
2
)] + 𝜃2 (1 − 𝑒 −𝑄0𝜃2 ) [𝑍(𝑄 − 𝑄0 ) −
)]}
(7.24)
130
Any value of 𝑍 which satisfies the equation 7.24 for the given values of
𝜃1 , 𝑄0 , 𝜃2 , 𝑛1 , 𝑛2 , 𝜑1 gives the optimal 𝑍 namely 𝑍̂.
7.5.1 Conclusion
Thus this model provides an insight on the optimal supply using the
truncated exponential distribution when the demand over 𝑄 is a constant and
the demand over 𝑄0 is a Nth demand random epoch. Future work may involve
the use of truncation demand distribution when demand over 𝑄 > 𝑍 is a Nth
demand random epoch.
7.6 GENERALIZED GAMMA BESSEL MODEL
In Nicy Sebastian [50], a new probability density function associated with
a Bessel function was introduced, which is the generalization of a gamma-type
distribution. Some of its special cases were mentioned. Multivariate analogue,
conditional density, best predictor function, Bayesian analysis, etc., connected
with this new density are also introduced and suitability of this density as a good
model in Bayesian inference and regression theory was also discussed in their
work.
Hence in this model, generalized gamma distribution with Bessel function
is used and the optimal supply at (0, 𝑇) is determined using this function. Under
these assumptions two different approaches are used in analyzing this model. In
model 7.6 using generalized gamma distribution with Bessel function, the
probability density function is derived and hence optimal supply size is obtained.
In model 7.7, a multi commodity inventory system with periodic review operating
under a stationary policy is considered using the exponential order statistics and
this methodology is applied in the well known Hanssman F [33] model. Hence
131
the optimal 𝑍̂ is obtained for both the cases and adequate numerical example is
provided.
7.6.1 Basic model
The basic model is adopted from the Hanssman F [33] and the notation of
the model is as follows, 𝑄 is considered a continuous random variable
representing demand and 𝑄 ~ generalized gamma distribution with Bessel
function and truncated at ‘𝛼’ in the left and at ‘𝛽’ in the right, with parameter ‘𝑡’.
The general form for the total expected cost given in Hanssman F [33] is
𝑍
𝛽
𝜓(𝐶) = 𝜑1 ∫𝛼 (𝑍 − 𝑄)𝑓(𝑄)𝑑𝑄 + 𝜑2 ∫𝑍 (𝑄 − 𝑍)𝑓(𝑄)𝑑𝑄
(7.25)
To find the optimal inventory, equation 7.25 is evaluated for
𝑑𝜓(𝐶)
𝑑𝑍
=0
The solution of Involves the concept differentiation of an integral given by
equation 3.7 of chapter 3, because variable ‘𝑍’ is in limit as well as in the
𝜑2
integrand. Hence 𝐹(𝑍̂) = 𝜑 +𝜑
. This implies that quantity
1
2
Ẑ
is determined using
the demand distribution function 𝑓(𝑄) such that 𝑃[𝑄 < 𝑍̂] = 𝜑
𝜑2
1 +𝜑2
A different variation of this basic model is attempted in above chapters. But in
this chapter, an attempt to solve the above model using generalized gamma
distribution with bessel function is made.
In model 7.6, the ordering decision in each period is affected by a single
setup cost ‘k’ and expected holding and shortage cost function 𝜓(𝐶). Condition
for being in stock level is given as 𝑎1 , … , 𝑎𝑛 = 𝑎, at the beginning of a period,
𝜓(𝐶) is assumed to be twice differentiable. Demand 𝑄𝑖 for the item over a
sequence of period 𝑖 = 1,2, …, is assumed to be independently and identically
distributed, continuous non-negative random variable with continuous joint
density function 𝑓(𝑄). Let some constraints be placed on the limits of the
132
demand distribution. Hence the PDF of the generalized gamma distribution
with bessel function is given in the form
(𝛿𝑄)𝐾
𝑓(𝑄) = 𝐾𝑒 −𝛼𝑄 𝑄𝛽−1 ∑∞
𝑘=0 (𝛽)
(7.26)
𝐾 𝐾!
Now to prove the validity for the equation 7.26 to be the probability density
function and hence the aim is to find 𝑘
∞
(𝛿𝑄)𝑘
∞
∫0 𝑓(𝑄) = 1 = 𝑘 ∫0 𝑒 −𝛼𝑄 𝑄𝛽−1 ∑∞
𝑘=0 (𝛽)
Consider (𝛽)𝑘 =
(7.27)
𝑘 𝑘!
𝛤(𝛽+𝑘)
(7.28)
𝛤(𝛽)
Using equation 7.28 in equation 7.27
∞
𝑘 ∫0 𝑒 −𝛼𝑄 𝑄𝛽−1 ∑∞
𝑘=0
(𝛿𝑄)𝑘 𝛤(𝛽)
𝛤(𝛽+𝑘)𝑘!
𝑑𝑄 = 𝑘 ∑∞
𝑘=0
(𝛿𝑄)𝑘 𝛤(𝛽)
𝛤(𝛽+𝑘)𝑘!
𝛿
𝛿
∞
∞
𝑘=0
𝑘=0
∞
∫0 𝑒 −𝛼𝑄 𝑄𝛽−1 𝑑𝑄 = 1
(𝛼)𝑘 𝛤(𝛽) 𝑒 𝛼 𝛤(𝛽)
(𝛿)𝑘 𝛤(𝛽)𝛤(𝛽 + 𝑘)
𝑘
𝑘∑
=
∑
⇒
𝑘=1
𝑘!
(𝛼)𝛽+𝑘 𝛤(𝛽 + 𝑘)𝑘! (𝛼)𝛽+𝑘
𝛼𝛽
(7.29)
𝛼𝛽
Hence the value of 𝑘 is obtained as 𝑘 =
𝛿
The PDF for the above model is
𝑒 𝛼 Γ(β)
obtained as
𝛼𝛽
𝛿
𝑒 𝛼 Γ(β)
𝑓(𝑄) = {
(𝛿𝑄)𝑘
𝑒 −𝛼𝑄 𝑄𝛽−1 ∑∞
𝑘=0 (𝛽)
𝑘 𝑘!
; 𝛼 > 0, 𝛽, 𝛿 > 0, 𝑄 ≥ 0
0 ; 𝑒𝑙𝑠𝑒 𝑤ℎ𝑒𝑟𝑒
Hence the expected total cost in this case can be written as
133
(7.30)
𝛼𝛽
𝑍
𝜓(𝐶) = 𝜑1 ∫𝛼 (𝑍 − 𝑄)
(𝛿𝑄)𝑘
𝛿
𝑒 𝛼 𝛤(𝛽)
𝑒 −𝛼𝑄 𝑄𝛽−1 ∑∞
𝑘=0 (𝛽)
𝛼𝛽
𝛽
+𝜑2 ∫𝑍 (𝑄 − 𝑍)
𝑘 𝑘!
𝑑𝑄
(𝛿𝑄)𝑘
𝛿
𝑒 𝛼 𝛤(𝛽)
𝑒 −𝛼𝑄 𝑄𝛽−1 ∑∞
𝑘=0 (𝛽)
𝑘 𝑘!
𝑑𝑄
(7.31)
𝑑𝜓(𝐶)
To obtain optimal 𝑍̂, 𝑑𝑍 = 0
𝑑𝜓(𝐶)
𝑑𝑍
𝛼𝛽
𝑑
= 𝑑𝑍 {𝜑1
𝛿
𝑒 𝛼 Γ(β)
𝑘 𝑘!
𝛼𝛽
𝑑
+ 𝑑𝑍 {𝜑2
(𝛿𝑄)𝑘
𝑍
[∫𝛼 (𝑍 − 𝑄)𝑒 −(𝛼+𝜇)𝑄 𝑄𝛽−1 ∑∞
𝑘=0 (𝛽)
(𝛿𝑄)𝑘
𝛽
𝛿
𝑒 𝛼 Γ(β)
𝑑𝑄 ]}
[∫𝑍 (𝑄 − 𝑍)𝑒 −(𝛼+𝜇)𝑄 𝑄𝛽−1 ∑∞
𝑘=0 (𝛽)
𝑘 𝑘!
𝑑𝑄 ]}
(7.32)
Hence
𝑑𝜓(𝐶)
𝑑𝑍
= 𝑇1 + 𝑇2 . So, the Laplace transform for the above model is given as
𝐿𝑓 (𝜇) = 𝐸(𝑒 −𝜇𝛼 ) =
=
𝛼𝛽
𝛿
𝑒 𝛼 𝛤(𝛽)
𝑒 −𝜇𝑄 𝛼𝛽 𝑒 −𝛼𝑄
𝛿
𝑒 𝛼 𝛤(𝛽)
𝑄𝛽−1 ∑∞
𝑘=0
(𝛿)𝑘 𝛤(𝛽) ∞ −(𝛼+𝜇)𝑄
∑∞
𝑘=0 𝛤(𝛽+𝐾)𝐾! ∫0 𝑒
𝑄
𝛽+𝑘+1
(𝛿𝑄)𝑘 𝛤(𝛽)
𝛤(𝛽+𝐾)𝐾!
𝑑𝑄 =
𝛼𝛽
(𝛿)𝑘
∞
𝛿 ∑𝑘=0 𝛤(𝛽+𝐾)𝐾!(𝛼+𝜇)𝛽+𝑘
𝑒𝛼
𝛿
𝛼𝛽 𝑒 𝛼+𝜇
=
𝛿
𝑒 𝛼 (𝛼+𝜇)𝛽
(7.33)
𝛿
Now 𝑇1 + 𝑇2 =
𝛼𝛽 𝑒 𝛼+𝜇
𝑑
𝛿
𝑒 𝛼 (𝛼+𝜇)𝛽
𝑑𝑍
𝑍
𝛽
(ℎ ∫𝛼 (𝑍 − 𝑄)𝑑𝑄 + 𝑑 ∫𝑆 (𝑄 − 𝑍) 𝑑𝑄)
𝛿
𝑑𝜓(𝐶)
=
𝑑𝑍
𝛼 𝛽 𝑒 𝛼+𝜇
𝛿
𝛼
𝑒 (𝛼 + 𝜇)
𝑑
ℎ 𝑑
𝛼
𝛽
[𝑍 2 ( + ) − 𝛼ℎ (𝑍 − ) + 𝛽𝑑 ( − 𝑍)]
2 2
2
2
𝛽 𝑑𝑍
(7.34)
134
𝛿
On simplification
𝑑𝜓(𝐶)
𝑑𝑍
𝛼𝛽 𝑒 𝛼+𝜇
=
𝛿
[𝑍(ℎ + 𝑑) − 𝛼ℎ − 𝛽𝑑]
(7.35)
𝑒 𝛼 (𝛼+𝜇)𝛽
To demonstrate that the objective function is convex, the second order for
equation 7.35 is carried out. Hence
𝑑2 𝜓(𝐶)
𝑑𝑍 2
𝛿
= 𝛼 𝛽 𝑒 𝛼+𝜇 (ℎ + 𝑑) = 0
(7.36)
Any value of 𝑍, which satisfies equation 7.35 for the given 𝜇, 𝛼, 𝛿, 𝛽, ℎ, 𝑑 is
the optimal supply size.
7.6.2 Numerical Illustration
When the demand is increased accordingly to the fixed values 𝜇 = 2,
ℎ = 2, 𝛼 = 1, 2, … 𝛽 = 2, 𝛿 = 1, then from equation 7.35 the following Figure 7.4
is obtained.
Figure 7.4: Optimal profit curve with respect to arrival of demand
135
7.6.3 Inference
When the demand is increased, the optimal profit decreases. Hence
increases in demand, corresponds to increase in supply size.
7.6.4 Numerical Illustration
When the lead time is increased accordingly for the fixed values 𝜇 = 2, ℎ = 2,
𝛼 = 1, 𝛽 = 1, 2,… 𝛿 = 1
then from equation 7.36 the following Figure7.5 is
obtained.
Figure7.5: Lead time with optimal supply size
7.6.5 Inference
When the lead time is increased, the optimal profit increases. Hence
increases in lead time, corresponds to increase in supply size.
136
7.7 A MULTI-COMMODITY EXPONENTIAL ORDER STATISTICS
A multi commodity inventory system with periodic review operating under
a stationary policy is considered. The ordering decision in each period is
affected by a single setup cost k and a linear variable ordering cost
𝑑1 , 𝑑2 , … , 𝑑𝑛 = 𝑑. At the beginning of a period, the stock level is 𝑎1 , … , 𝑎𝑛 = 𝑎. An
inventory system with time to shortage and holding of the items is our prime
interest. A single new component at time zero be started and replace it upon
loss by a new component and so on. This time to loss which is represented by
exponential order statistics ∑𝑛𝑖=1 𝑄𝑖 is independent and the key to model when
there is joint PDF is
𝑓(𝑄) =
𝛼𝛽
𝛿
𝑒
−𝛼𝑄
𝑘 −𝛼𝛽𝑄
(𝛿𝑄)
𝑄𝛽−1 ∑∞
𝑘=0 (𝛽) 𝑘
,0 < 𝑄 < ∞
(7.37)
𝑘
𝑒 𝛼 Γ(β)
Suppose 𝑄1:𝑛 , 𝑄2:𝑛 , 𝑄3:𝑛 , … 𝑄𝑛:𝑛 are the order statistics of a random variable of
size n arising from 𝑓(𝑄) interested with the distribution of
𝑄1:𝑛 , 𝑄2:𝑛 , 𝑄3:𝑛 , … 𝑄𝑛:𝑛 = 𝑑𝑟; 𝑟 = 1,2, … , 𝑛; 𝑄0:𝑛 = 0
(7.38)
Then 𝑑𝑟 will constitute the renewal process. Let us consider the joint PDF of all
order to be given by
𝑛
(𝑄 ,… ,𝑄 )
1
𝑛
𝑓1:2,…,𝑛:𝑛
= 𝑛! 𝑍
𝛼𝛽
𝛿
𝑒 𝛼 Γ(β)
𝑒 −𝛼𝑄 𝑄𝛽−1 ∑∞
𝑘=0
(𝛿𝑄)𝑘
−𝛼𝛽 ∑𝑖=1 𝑄𝑖
(7.39)
(𝛽)𝑘 𝑘
Let us define 𝑔(. ) as the length of time measured backwards from 1 to the last
renewal at or before n
𝑔(𝑑1 , 𝑑2 , … , 𝑑𝑛 ) = 𝑛! 𝑍
𝛼𝛽
𝛿
𝛼
𝑒 Γ(β)
∞
𝑒 −𝛼𝑄 𝑄𝛽−1 ∑
𝑘=0
137
(𝛿𝑄)𝑘
(𝛽)𝑘 𝑘
−𝛼𝛽[𝑛𝑑1 +⋯+(𝑛−𝑟+1)𝑑𝑟+⋯+2𝑑𝑛−1 +𝑑𝑛 ]
= 𝑛!
𝛼𝛽
𝛿
𝑒 𝛼 Γ(β)
𝑒 −𝛼𝑄 𝑄𝛽−1 ∑∞
𝑘=0
(𝛿𝑄)𝑘
(𝛽)𝑘 𝑘
∗ 𝑛𝑒 −𝑛𝑑1 … (𝑛 − 𝑟 + 1)𝑒 −(𝑛−𝑟+1)𝑑𝑟 ∗ … ∗
2𝑒 −2𝑑𝑛−1 ∗ 𝑒 −𝑑𝑛
(7.40)
where 𝑑𝑟 = 𝑄𝑟 − 𝑄𝑟−1 , 𝑟 = 1, … , 𝑛,
𝑑1 = 𝑄1 , 𝑑2 = 𝑄2 − 𝑄1 , … , 𝑑𝑛 = 𝑄𝑛 − 𝑄𝑛−1
hence 𝑄1 = 𝑑1 , 𝑄2 = 𝑑2 + 𝑑1 , …,
𝑄𝑟 = 𝛼𝛽 ∑𝑟𝑖=1 𝑑𝑖 , … , 𝑄𝑛 = 𝛼𝛽 ∑𝑛𝑖=1 𝑑𝑖
(7.41)
This proves that 𝑑1 , 𝑑2 , … , 𝑑𝑛 are all independently and exponentially distributed
and 𝑑𝑟 is distributed with an exponential distributed with scale parameter (𝑛 −
𝑟 + 1). From the earlier literature of exponential order statistics, following
equation is obtained
−𝜂
𝑓 (𝑄1 , … , 𝑄𝑛 ) = 𝐶𝑄1 𝛾1 𝑄2 𝛾2 … 𝑄𝑛 𝛾𝑛 [1 + (𝛼 − 1)(𝑎1 𝑄1 𝛿1 + ⋯ + 𝑎𝑛 𝑄𝑛 𝛿𝑛 ]𝛼−1
(7.42)
where 𝛼 > 0, 𝑄𝑖 > 0, 𝑎𝑖 > 0, 𝛿𝑖 > 0
lim 𝑓( 𝑄1 , … , 𝑄𝑛 ) = 𝑍𝐶𝑄1 𝛾1 𝑄2 𝛾2 … 𝑄𝑛 𝛾𝑛 ∗ 𝑒 −𝑎1 𝑄1
𝛿1
−⋯−𝑎𝑛 𝑄𝑛 𝛿𝑛
𝛼→1
= 𝑍𝐶𝑄1 𝛾1 𝑄2 𝛾2 … 𝑄𝑛 𝛾𝑛 ∗ 𝑒 −𝑎𝑛𝑄𝑛
𝛿𝑛
(7.43)
∞
−𝜂
𝑓(𝑄1 , … , 𝑄𝑛−1 ) = 𝐶 ∫ 𝑍𝑄1 𝛾1 𝑄2 𝛾2 … 𝑄𝑛 𝛾𝑛 [1 + (𝛼 − 1)(𝑎1 𝑄1 𝛿1 + ⋯ + 𝑎𝑛 𝑄𝑛 𝛿𝑛 ]𝛼−1 𝑑𝑄𝑛
0
138
∞
𝛾1
𝛾2
= 𝑍𝐶𝑄1 𝑄2 … 𝑄𝑛−1
𝛾𝑛−1
−𝜂
∫ 𝑄𝑛 𝛾𝑛 [ (𝛼 − 1)𝑎𝑛 𝑄𝑛 𝛿𝑛 ]𝛼−1
0
−𝜂
∗ [1 +
(𝛼 − 1)(𝑎1 𝑄1
𝛿1
+ ⋯ + 𝑎𝑛−1 𝑄𝑛−1
(𝛼 − 1)𝑎𝑛
−𝜂
(𝛼−1)
= 𝑍𝐶𝑄1 𝛾1 𝑄2 𝛾2 … 𝑄𝑛−1 𝛾𝑛−1 [(𝛼 − 1)𝑎𝑛 ]
𝛿𝑛−1
𝑄𝑛
−𝛿𝑛 𝛼−1
]
𝑑𝑄𝑛
∞
−𝜂
𝜂𝛿𝑛
∫ 𝑄𝑛 𝛾𝑛−𝛼−1 [1 + 𝑐𝑄𝑛 −𝛿𝑛 ]𝛼−1 𝑑𝑄𝑛
0
𝛾1
𝛾2
= 𝑍𝐶𝑄1 𝑄2 … 𝑄𝑛−1
𝛾𝑛−1
−𝛾
(𝛼−1)
[(𝛼 − 1)𝑎𝑛 ]
𝛾𝑛
𝜂
∞ 𝑍 𝛿 −(𝛼−1)
[1
∫0 [𝐶] 𝑛
𝜂
1
− 𝑍]−𝛼−1 𝑍 𝛿𝑛
−1
𝑑𝑍
(7.44)
Using equation 7.30 in equation 7.44, the following equation 7.45 is obtained
∞
1
−𝜂
𝐶𝑄1 𝛾1 𝑄2 𝛾2 … 𝑄𝑛−1 𝛾𝑛−1 [(𝛼 − 1)𝑎𝑛 ](𝛼−1)
= 𝑍 ∫ 𝑄𝑛
𝛾𝑛
𝜂
−
𝛿𝑛 (𝛼−1)
𝜂
[1 + 𝑐𝑄𝑛
−𝛿𝑛 −𝛼−1
]
𝑑𝑄𝑛
0
𝜂
=
1
−𝜂
𝐶𝑄1 𝛾1 𝑄2 𝛾2 …𝑄𝑛−1 𝛾𝑛−1 [(𝛼−1)𝑎𝑛 ](𝛼−1)
=𝑍
−
[1+𝑐𝑄𝑛 −𝛿𝑛 ] 𝛼−1
𝛾𝑛
𝜂
−
𝑄𝑛 𝛿𝑛 (𝛼−1)
(7.45)
Any value of 𝑍, which satisfies equation 7.45 is the optimal 𝑍̂. It may be noted
that the value of 𝑍̂ depends upon a number of parameter such as 𝐶,
𝑄1 𝛾1 , … 𝑄𝑛−1 𝛾𝑛−1 , 𝜂, 𝛿𝑛 . But for the use of this model in practical situations it
becomes necessary to estimate the value of 𝜂 on the basis of sample data.
𝐶, 𝑄1 𝛾1 , … 𝑄𝑛−1 𝛾𝑛−1 and 𝛿𝑛 are of deterministic character and hence they are
fixed. However in the determination of 𝑍̂, most vital values are obtained.
139
7.8 CONCLUSION
Under a stationary policy as discussed before either all items will be
ordered to bring to the inventory level to 𝑍, if 𝛼 is the inventory level at the
beginning of a period prior to making a decision, then after 𝛽 nothing is ordered.
The primary concern of this study is to find the optimality condition for
stationary policy. This is done by minimizing the expression for the stationary
total expected cost per period with respect to the decision variable that
characterizes the policy being used.
140