5. BASE-STOCK SYSTEM FOR PATIENT CUSTOMER WITH DEMAND
DISTRIBUTION UNDERGOING A CHANGE
5.1 INTRODUCTION
In this chapter, the base-stock for patient customer is studied. The
base-stock system for patient customer is a different type of inventory policy
in which an ordering mechanism of a new type is introduced. Under the basestock systems, the total inventory on hand is to be taken as the sum of the
actual inventory on ground and inventory due to orders for replenishment. In
this model, the inventory process starts with initial inventory of size 𝐵.
Whenever a customer order is received, it is supplied immediately and
at the same time a replenishment order is placed immediately. The
replenishment takes place after a lead time 𝐿. If the demand exceeds this
stock level on hand then customer do not leave, but they wait till supply is
received. For this reason the customer are called patient customer. In this
case there is no shortage cost, but some concession is shown to the
customer and it is a denoted as a shortage cost. The total inventory is
denoted as 𝐵, which is the sum of the inventory on hand, and inventory on
order. This is called Base-Stock. Here the demand during the period [0, 𝑡] is
taken to be a random variable.
In this model it is assumed that, the distribution of the random variable
denoting the demand undergoes a change in the distribution after a change
or truncation point. The demand distributions in this model is distributed as
an exponential before the truncation point and distributed as Erlang2 after the
truncation point. Truncated exponential distribution discussed in chapter 4 is
used to obtain the optimal expected cost of base-stock system for patient
customer.
The objective is to derive the expression for optimal base-stock and
also numerical illustration is provided. If lead-time demand is denoted as 𝑄
then 𝑓(𝑄) is the probability density of this random variable. Under these
circumstances the equation for the expected cost can be written in the form
91
𝐵
∞
𝐸(𝐶) = ℎ ∫(𝐵 − 𝑄) 𝑓(𝑄)𝑑𝑄 + 𝑑 ∫ (𝑄 − 𝐵) 𝑓(𝑄)𝑑𝑄
0
𝐵
(5.1)
The optimal value of 𝐵 is to be determined by taking
𝑑𝐸(𝐶)
𝑑𝐵
= 0 and can
𝑑
be shown that the optimal value 𝐵 is one such that 𝐹(𝐵̂ ) = 𝑑+ℎ where 𝐵̂ is
optimal 𝐵 and 𝐹(𝑄) is the cumulative distribution of the random variable 𝑄.
This model has been discussed in Hanssman F [33]. The base stock system
for patient customer has been initially discussed by Gaver D.P [21] and a
modification of this model has been attempted by Ramanarayanan R [54]
In this chapter, a new model is developed by assuming that during the
lead time 𝐿 which is deterministic, there are 𝐾 different demand epochs and
the demand during these epochs are denoted as 𝑄1 , 𝑄2 , 𝑄3 , … , 𝑄𝑘 , which are
identically independent random variables. The inter-arrival times between the
demand epochs are also random variables which are identically independent
with the density function 𝑔(. ) and the distribution function 𝐺(. ). The
probability there will be exactly n demands denoting the lead-time 𝐿 is given
as 𝑃[𝐾 = 𝑘⁄𝐿] = 𝐺𝑘 (𝐿) − 𝐺𝑘−1 (𝐿) by the renewal theory which is discussed in
chapter 4 where 𝐺𝑘 (𝐿) is 𝑛-fold convolution of G with itself. Therefore the
probability of total demand is utmost 𝑞 during 𝐿 is
𝑘
𝑃(𝑄 ≤ 𝑞) = ∑∞
𝑘=0 𝑃(𝑄1 + 𝑄2 + ⋯ + 𝑄𝑘 ≤ 𝑞) . 𝑃 [𝐾 = ⁄𝐿 ]
(5.2)
where 𝑘 denotes the number of demand epochs during 𝐿
𝑃[𝑄 ≤ 𝑞] = ∑∞
𝑘=1[𝐺𝑘 (𝐿) − 𝐺𝑘+1 (𝐿)] . 𝐹𝑘 (𝑞)
(5.3)
𝐹𝑘 (𝑞) is the 𝑘 –fold convolution of 𝐹 with itself.
The expression for the expected cost is given as
92
𝐵
∞
𝐸(𝐶) = ℎ ∫(𝐵 − 𝑄) 𝑃(𝑄 ≤ 𝑞)𝑑𝑞 + 𝑑 ∫ (𝑄 − 𝐵)𝑃(𝑄 ≤ 𝑞) 𝑑𝑞
0
𝐵
(5.4)
where the assumption are followed below. To find the optimal level 𝐵
satisfies the following equation in accordance with the equation for the
optimal base-stock. Now
𝐵
∞
𝐸(𝐶) = ℎ ∑[𝐺𝑘 (𝐿) − 𝐺𝑘+1 (𝐿)] ∫(𝐵 − 𝑄) 𝑃(𝑄 ≤ 𝑞)𝑑𝑞
𝑘=1
0
∞
∞
+ 𝑑 ∑[𝐺𝑘 (𝐿) − 𝐺𝑘+1 (𝐿)] ∫ (𝑄 − 𝐵)𝑃(𝑄 ≤ 𝑞) 𝑑𝑞
𝑘=1
𝐵
(5.5)
Using the equation 3.7 of chapter 3, following equation is obtained
∞
𝐸(𝐶) = ℎ ∑[𝐺𝑘 (𝐿) − 𝐺𝑘+1 (𝐿)] 𝐹𝑘 (𝐵)
𝑘=1
∞
+ 𝑑 ∑[𝐺𝑘 (𝐿) − 𝐺𝑘+1 (𝐿)] 𝐹𝑘 (𝐵)[1 − 𝐹𝑘 (𝐵)] = 0
𝑘=1
(5.6)
𝑑
Hence ∑∞
𝑘=1[𝐺𝑘 (𝐿) − 𝐺𝑘+1 (𝐿)] 𝐹𝑘 (𝐵) = 𝑑+ℎ
(5.7)
on simplification since ∑∞
𝑘=1[𝐺𝑘 (𝐿) − 𝐺𝑘+1 (𝐿)] = 1, then equation 5.7
𝑑
becomes 𝐹𝑘 (𝐵̂ ) = 𝑑+ℎ. From the model discussed above, a new model is
developed by considering the fact that the demand distribution undergoes a
change after a change point. This assumption of the demand distribution
undergoing a change is valid, since the demand distribution has the very
basic nature that the probability that a random variable denoting the demand
taking a value beyond a certain level may undergo change in its structure.
93
5.2 ASSUMPTIONS
i) The total demand is a constant, which under goes a change of
distribution after a change point 𝑄0 .
ii) The distribution of the total demand follows exponential with
parameter 𝜃1 and becomes Erlang2 with parameter 𝜃2 after the change point.
iii) Also the demand is considered to be truncated exponential
distribution.
5.3 NOTATIONS
ℎ
= Inventory holding cost /unit
𝑑
= Shortage cost/unit
𝐹𝑘 (𝑄) = k fold convolution of 𝑓(𝑄)
𝑞
= The total demand
5.4 ERLANG2 DISTRIBUTION FOR OPTIMAL BASE STOCK
Assuming that the distribution of 𝑓(𝑄) of the random variable 𝑄
denoting the demand undergoes a change of distribution in the sense that
𝑓(𝑄) = 𝑓1 (𝑄) if 𝑄 ≤ 𝑄0
= 𝑓2 (𝑄) if 𝑄 > 𝑄0
Where 𝑄0 is called the change point. Let the change of distribution in the
expression for expected total cost is incorporated. In doing so, 𝑄0 becomes
less than base stock. Hence considering the model when 𝑄0 < 𝐵
94
If 𝑄0 < 𝐵
𝑄0
𝐵
𝐸(𝐶) = ℎ ∫ (𝐵 − 𝑄) [𝑓1
(𝑄)]𝑘
𝑑𝑞 + ℎ ∫(𝐵 − 𝑄) [𝑓2 (𝑄)]𝑘 𝑑𝑞
0
𝑄0
∞
+ 𝑑 ∫ (𝑄 − 𝐵) [𝑓2 (𝑄)]𝑘 𝑑𝑞
𝐵
(5.8)
by the formulation of rule discussed in chapter 3 as equation 3.7, the
following result is obtained
𝑄0
𝐵
𝑑𝐸(𝐶)
𝑑
=
{ℎ ∫ (𝐵 − 𝑄) [𝑓1 (𝑄)]𝑘 𝑑𝑞 + ℎ ∫(𝐵 − 𝑄) [𝑓2 (𝑄)]𝑘 𝑑𝑞
𝑑𝐵
𝑑𝐵
0
𝑄0
∞
+ 𝑑 ∫ (𝑄 − 𝐵) [𝑓2 (𝑄)]𝑘 𝑑𝑞}
𝐵
(5.9)
𝑄0
𝐵
𝑑𝐸(𝐶)
𝑑
𝑑
=
[ℎ ∫ (𝐵 − 𝑄) [𝑓1 (𝑄)]𝑘 ] 𝑑𝑞 +
[ℎ ∫(𝐵 − 𝑄) [𝑓2 (𝑄)]𝑘 ] 𝑑𝑞
𝑑𝐵
𝑑𝐵
𝑑𝐵
0
𝑄0
∞
𝑑
+
[𝑑 ∫ (𝑄 − 𝐵) [𝑓2 (𝑄)]𝑘 ] 𝑑𝑞
𝑑𝐵
𝐵
= [𝐼1 + 𝐼2 + 𝐼3 ]
(5.10)
𝑑𝐼1
To find
𝑑𝐵
𝑄0
𝑄0
0
0
𝑑𝐼1
= ℎ ∫ (𝐵 − 𝑄) [𝑓1 (𝑄)]𝑘 𝑑𝑞 = ℎ𝜃1 ∫ (𝐵 − 𝑄)𝑒 −𝜃1 𝑘𝑞 𝑑𝑞
𝑑𝐵
𝑘
= ℎ𝜃1 {− [(𝐵 − 𝑄).
𝑒 −𝜃1 𝑘𝑞
𝜃1 𝑘
𝑄0
]
𝑒 −𝜃1 𝑘𝑞
+[
0
(𝜃1 𝑘 )
2
95
𝑄0
] }
0
𝑑𝐼1
𝑑𝐵
= ℎ𝜃1 𝑘 [[
To find
1
𝜃1
𝑘
]−
𝑒 −𝜃1 𝑘𝑄0
𝜃1 𝑘
]
(5.11)
𝑑𝐼2
𝑑𝐵
𝐵
𝐵
𝑄0
𝑄0
𝑑𝐼2
= ℎ ∫(𝐵 − 𝑄) [𝑓2 (𝑄)]𝑘 𝑑𝑞 = ℎ ∫ 𝜃2 2𝑘 (𝑞 − 𝑄0 )𝑘 . 𝑒 −𝜃2 𝑘(𝑞−𝑄0) . 𝑒 −𝜃1 𝑘𝑄0 𝑑𝑞
𝑑𝐵
𝐵
2𝑘
= ℎ𝜃2 . 𝑒
−𝜃1 𝑘𝑄0
𝐵
𝑒 −𝜃2 𝑘(𝑞−𝑄0 )
𝑒 −𝜃2 𝑘(𝑞−𝑄0)
[(𝑞 − 𝑄0 ) .
] − [𝑘(𝑞 − 𝑄0 )𝑘−1
]
2 2
−𝜃2 𝑘
𝜃
𝑘
2
𝑄
𝑄
𝑘
0
+ [𝑘(𝑘 − 1)(𝑞 − 𝑄0 )
𝑘−2
0
𝑒 −𝜃2 𝑘(𝑞−𝑄0)
−𝜃2 3 𝑘 3
𝐵
]
−⋯
𝑄0
𝑑𝐼2
𝑒 −𝜃2 𝑘(𝐵−𝑄0)
= ℎ𝜃2 2𝑘 . 𝑒 −𝜃1 𝑘𝑄0 {[−(𝐵 − 𝑄0 )𝑘 .
]
𝑑𝐵
𝜃2 𝑘
− [𝑘(𝐵 − 𝑄0 )𝑘−1
𝑒 −𝜃2 𝑘(𝐵−𝑄0)
𝜃2 2 𝑘 2
+ [𝑘(𝑘 − 1)(𝐵 − 𝑄0 )𝑘−2
]
𝑒 −𝜃2 𝑘(𝐵−𝑄0 )
−𝜃2 3 𝑘 3
]−⋯}
(5.12)
∞
𝑑𝐼3
= 𝑑 ∫ (𝑄 − 𝐵) [𝑓2 (𝑄)]𝑘 𝑑𝑞
𝑑𝐵
𝐵
∞
= −𝑑 ∫ 𝜃2 2𝑘 (𝑞 − 𝑄0 )𝑘 . 𝑒 −𝜃2 𝑘(𝑞−𝑄0 ) . 𝑒 −𝜃1 𝑘𝑄0 𝑑𝑞
𝐵
∞
2𝑘 −𝜃1 𝑘𝑄0
= −𝑑𝜃2 𝑒
∞
𝑒 −𝜃2 𝑘(𝑞−𝑄0 )
𝑒 −𝜃2 𝑘(𝑞−𝑄0)
𝑘−1
{[(𝑞 − 𝑄0 ) .
] − [𝑘(𝑞 − 𝑄0 )
]
−𝜃2 𝑘
𝜃2 2 𝑘 2
𝐵
𝐵
𝑘
+ [𝑘(𝑘 − 1)(𝑞 − 𝑄0 )
𝑘−2
𝑒 −𝜃2 𝑘(𝑞−𝑄0 )
96
−𝜃2 3 𝑘 3
∞
] −⋯}
𝐵
𝑑𝐼3
𝑒 −𝜃2 𝑘(𝐵−𝑄0 )
2𝑘 −𝜃1 𝑘𝑄0
𝑘
= −𝑑𝜃2 𝑒
{[(𝐵 − 𝑄0 ) .
]
𝑑𝐵
𝜃2 𝑘
+ [𝑘(𝐵 − 𝑄0 )
𝑘−1
𝑒 −𝜃2 𝑘(𝐵−𝑄0)
𝜃2 2 𝑘 2
+ [𝑘(𝑘 − 1)(𝐵 − 𝑄0 )
𝑘−2
]
𝑒 −𝜃2 𝑘(𝐵−𝑄0 )
𝜃2 3 𝑘 3
]−⋯}
(5.13)
Substituting equation 5.11, 5.12 and 5.13 in equation 5.10, the following
result is obtained
𝑑𝐸(𝐶)
1
𝑒 −𝜃1 𝑘𝑄0
= ℎ𝜃1 𝑘 [[ 𝑘 ] −
]
𝑑𝐵
𝜃1
𝜃1 𝑘
2𝑘
−ℎ𝜃2 . 𝑒
−𝜃1 𝑘𝑄0
{[(𝐵 − 𝑄0 )𝑘 .
𝑒 −𝜃2 𝑘(𝐵−𝑄0 )
𝑒 −𝜃2 𝑘(𝐵−𝑄0)
𝑘−1
)
] + [𝑘(𝐵 − 𝑄0
]
𝜃2 𝑘
𝜃2 2 𝑘 2
− [𝑘(𝑘 − 1)(𝐵 − 𝑄0
−𝑑𝜃2 2𝑘 𝑒 −𝜃1 𝑘𝑄0 {[(𝐵 − 𝑄0 )𝑘 .
)𝑘−2
𝑒 −𝜃2 𝑘(𝐵−𝑄0 )
−𝜃2 3 𝑘 3
]+⋯}
𝑒 −𝜃2 𝑘(𝐵−𝑄0)
𝑒 −𝜃2 𝑘(𝐵−𝑄0 )
] + [𝑛(𝐵 − 𝑥0 )𝑛−1
]
𝜃2 𝑘
𝜃2 2 𝑘 2
+ [𝑘(𝑘 − 1)(𝐵 − 𝑄0 )𝑘−2
𝑒 −𝜃2 𝑘(𝐵−𝑄0 )
𝜃2 3 𝑘 3
]−⋯} = 0
(5.14)
Any value of 𝐵 which satisfies equation 5.14 is the optimal base stock
namely 𝐵̂ .
5.4.1 Numerical illustration
Considering the value 𝑄0 = 10, 𝜃1 = 1.7 𝜃2 = 2, ℎ = 10, 𝑘 = 5
97
Table 5.1: Shortage variability for base stock
𝑑
10
20
30
40
50
𝐵
7.7
7.9
8.0
8.1
8.2
Figure 5.1: Base-stock with the shortage cost
5.4.2 Inference
In fig.5.1, as the value of the shortage cost ‘𝑑’ increases,
a larger
inventory size is suggested as in the case of all other models discussed
earlier by many authors the above curve obtained in the figure is valid and is
similar to the one obtained earlier by the other authors.
5.4.3 Numerical illustration
Considering the value 𝑄0 = 10, 𝜃1 = 1.7 𝜃2 = 2, 𝑑 = 10, 𝑘 = 5
98
Table 5.2: Holding variability for base stock
ℎ
5
10
15
20
𝐵
7.7
7.4
7.0
6.8
Figure 5.2: Base-stock with holding cost
5.4.4 Inference
In figure 5.2, as the inventory holding cost ‘ℎ’ increases then this
model suggests a smaller inventory size to be stocked, which is common to
all inventory models.
99
5.5 TRUNCATED EXPONENTIAL DISTRIBUTION FOR PATIENT
CUSTOMER
Maintaining inventories is necessary in order to meet the demand of
stocks for a given period of time which may be either finite or infinite. An
optimal base-stock inventory policies using finite horizon is examined. In
Hanssman F [33] the basic model for the base-stock systems is discussed.
Sachithanantham S et.al [58] had discussed the model of base stock system
for patient customers with lead time distribution undergoing a parametric
change. Suresh Kumar R [72] showed how by applying the threshold, a
Shock model had a change of distribution after a change point. A modified
model had been attempted by Sachithanantham S et.al [58].
The base-stock for Patient customer model discussed in model 5.2 is
evaluated using the Truncated Exponential Distribution. Since among the
parametric models, the exponential distribution is perhaps the most widely
applied statistical approach in several fields. Hence, it is justified to apply the
truncated exponential distribution approach. In this model demand during the
period [0, t] is taken to be a random variable and truncated exponential
distribution satisfies the base-stock policy for the patient customer as a
continuous model. Whenever a customer orders for 𝑄 units is received it is
supplied immediately and at the same time a replenishment order for Q units
is placed immediately. The replenishment takes place after a lead-time L.
From Hanssman F [33]
∞
𝐵
𝐸(𝐶) = ℎ ∫(𝐵 − Q)𝑓(𝑄)𝑑𝑄 + 𝑑 ∫(Q − 𝐵) 𝑓(𝑄)𝑑𝑄
0
𝐵
(5.15)
Assuming the PDF 𝑓(𝑄) of the random variable 𝑄 denoting the demand
undergoes a change of distribution in the sense that
𝑓(𝑄) = 𝑓1 (𝑄) 𝑖𝑓𝑄 ≤ 𝑄0
= 𝑓2 (𝑄) 𝑖𝑓𝑄 > 𝑄0
(5.16)
100
Where 𝑄0 is called the change point. The following equation is obtained while
incorporating the change of distribution in equation 5.15,
𝑄0
∞
𝐵
𝐸(𝐶) = ℎ ∫ (𝐵 − 𝑄)𝑓1 (𝑄)𝑑𝑄 + ℎ ∫(𝐵 − 𝑄)𝑓1 (𝑄)𝑑𝑄 + 𝑑 ∫(𝑄 − 𝐵)𝑓2 (𝑄)𝑑𝑄
0
𝑄0
𝐵
(5.17)
Equation 5.17 is differentiated by using the differential of integral method as
discussed in equation 3.7 of chapter 3 and is solved as follows
𝑄0
𝐵
𝑑𝐸(𝐶)
𝑑
=
{ℎ ∫ (𝐵 − 𝑄)𝑓1 (𝑄)𝑑𝑄 + ℎ ∫(𝐵 − 𝑄)𝑓1 (𝑄)𝑑𝑄
𝑑𝐵
𝑑𝐵
0
𝑄0
∞
+ 𝑑 ∫(𝑄 − 𝐵)𝑓2 (𝑄)𝑑𝑄}
𝐵
(5.18)
Now
𝑄0
𝐵
𝑑𝐸(𝐶)
𝑑
𝑑
=ℎ
∫ (𝐵 − 𝑄)𝑓1 (𝑄)𝑑𝑄 + ℎ
∫(𝐵 − 𝑄)𝑓1 (𝑄)𝑑𝑄
𝑑𝐵
𝑑𝐵
𝑑𝐵
0
𝑄0
∞
𝑑
+𝑑
∫(𝑄 − 𝐵)𝑓2 (𝑄)𝑑𝑄
𝑑𝐵
𝐵
(5.19)
Hence
𝑑𝐸(𝐶)
𝑑𝐵
= [𝐾1 + 𝐾2 + 𝐾3 ] = 0
(5.20)
𝑄0
𝑑
𝐾1 = ℎ
∫ (𝐵 − 𝑄)𝑓1 (𝑄)𝑑𝑄
𝑑𝐵
0
𝐵
𝑑
𝐾2 = ℎ
∫(𝐵 − 𝑄)𝑓1 (𝑄)𝑑𝑄
𝑑𝐵
𝑄0
101
∞
𝑑
𝐾3 = 𝑑
∫ (𝑄 − 𝐵)𝑓2 (𝑄)𝑑𝑄
𝑑𝐵
𝐵
Deemer W.L et.al [18] derived the maximum likelihood estimator of
the parameter 𝜃 in the truncated exponential distribution as
𝑓(𝑄, 𝜃) = {
𝜃 𝑒𝑥𝑝(−𝜃𝑄) (1 − 𝑒 −𝜃𝑄0 ), 0 < 𝑄 < 𝑄0
0
, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(5.21)
Applying the equation 5.21, the equation 5.20 becomes as follows,
𝑄0
𝑑
𝐾1 = ℎ
∫ (𝐵 − 𝑄)𝜃 exp(−𝜃𝑄) (1 − 𝑒 −𝜃𝑄0 )𝑑𝑄
𝑑𝐵
0
𝑄0
𝑑
𝐾1 =
(ℎ(1 − 𝑒 −𝜃𝑄0 ) ∫ (𝐵 − 𝑄)𝜃 exp(−𝜃𝑄) 𝑑𝑄 )
𝑑𝐵
0
−𝐵(𝑒 −𝜃𝑄0 − 1) 𝜃𝑄0 𝑒 −𝜃𝑄0 + 𝑒 −𝜃𝑄0 − 1
𝐾1 = ℎ𝜃(1 − 𝑒 −𝜃𝑄0 ) (
+
)
𝜃
𝜃2
(5.22)
𝐵
𝑑
𝐾2 = ℎ
∫(𝐵 − 𝑄)𝜃 exp(−𝜃𝑄) (1 − 𝑒 −𝜃𝑄0 )𝑑𝑄
𝑑𝐵
𝑄0
𝐵
𝑑
𝐾2 =
(ℎ(1 − 𝑒 −𝜃𝑄0 ) ∫(𝐵 − 𝑄)𝜃 exp(−𝜃𝑄) 𝑑𝑄 )
𝑑𝐵
𝑄0
𝐵(−𝑒 −𝜃𝐵 +𝑒 −𝜃𝑄0 )
𝜃
𝐾2 = ℎ 𝜃(1 − 𝑒 −𝜃𝑄0 )
+
𝜃𝑄0 𝑒 −𝜃𝑄0 + 𝑒 −𝜃𝑄0 − 𝜃𝐵𝑒 −𝜃𝐵 − 𝑒 −𝜃𝐵
(
)
𝜃2
(5.23)
102
∞
𝑑
𝐾3 = 𝑑
∫ (𝑄 − 𝐵)0𝑑𝑄 = 0
𝑑𝐵
𝐵
(5.24)
Hence substituting the equation 5.22, 5.23 and 5.24 in equation 5.20, the
result obtained is
𝑑𝐸(𝐶)
𝑑𝐵
−𝑒
= 𝐵̂ = ℎ(1 − 𝑒 −𝜃𝑄0 ) [
−𝜃𝑄0 −1−𝑒 −𝜃𝐵 +𝑒 −𝜃𝑄0
𝜃
]
(5.25)
The model discussed above is formulated numerically. Therefore by
substituting the value for the truncation point, holding cost, base stock and
time variant which is denoted as follows 𝑄0 , ℎ, 𝐵, 𝜃, the optimal base stock in
case of the patient customer is obtained. Also, this result is compared with
that of earlier models.
5.5.1 Numerical illustration
The following table 5.3 and figure 5.3 shows the numerical existence
of the model developed.
Table 5.3: Optimal base stock case with varying 𝑄0 , ℎ, 𝐵, 𝜃
𝑸𝟎
𝒉
𝑩
𝜽
−𝟏 − 𝒆−𝜽𝑩
𝟏−𝒆−𝜽𝑸𝟎
̂
𝑩
1
10
6
0.5
-1.049
0.393
-4.130
1.5
20
6.2
0.7
-1.013
0.650
-13.17
2
30
6.4
0.9
-1.003
0.834
-25.11
103
Figure 5.3: Base-stock curve for truncation point.
5.5.2 Inference
In figure 5.3, as the inventory holding cost 𝒉 is monotonically
increasing, then this model demands for stocking of a limited or very fewer
inventory.
5.6 CONCLUSION
In this chapter, the demand during the period [0, t] is taken to be a
constant. In theoretical and applied work, the truncated exponential
distribution plays a crucial role due to its application to real life. The idea of
inventory decisions could be applied to production systems with several
machines and impatient customers. The model presented in this chapter can
be extended to system with both customer impatience and allocation of
hospital bed. This direction of research is taken into study in the next chapter
6.
104