INTRODUCTION
Competition features of 90’s have evolved into TimeBased Manufacturing
(TBM). The two time elements considered in TBM are the replenishment lead
time to supply a product to the customer for a specific order and the
time to develop a product from concept to final market delivery. Therefore,
reducing lead time on product supply is the strategic objective of the
TBM company (Bockerstette and Shell, 1993). Although the time compression
will inevitably raise the cost, a customer will pay a premium to the supplier
who can furnish its product faster and more reliably than the competition
and the premium may be respectable.
Silver et al. (1998) defined the replenishment lead time as the
time elapsed from the moment at which it is decided to place an order,
until it is physically on the shelf to satisfy customer demands. Although
lead time can be a constant or a random variable, it is often treated
as a prescribed parameter (Silver et al., 1998) thus not controllable.
However, lead time can be reduced at extra cost and shorter lead time
is the primary driver to achieving customer satisfaction in successful
TBM operations (Bockerstette and Shell, 1993). The benefits resulting
from reduced lead time include lower cost, less waste and less obsolescence,
greater flexibility to response to change, closely linked organization
priorities to customers’ needs, improved service, quality and reliability
and substantially accelerated supply system improvements (Blackburn, 1991).
Tersine (1994) and Vollmann et al. (1992) attributed the replenishment
lead time mostly to manufacturing considerations and addressed some guidelines
for its reduction. Liao and Shyu (1991) suggested that lead time could
be decomposed into n components each having a different crashing
cost for reduction. BenDaya and Raouf (1994) generalized the Liao and
Shyu model (1991) by considering both lead time and the order quantity
as decision variables.
Ouyang et al. (1996) extended the BenDaya and Raouf’s model (1994)
to include a mixture of backorder and lost sales in the model by assuming
a predetermined service level with both reorder point and backorder rate
being fixed. Moon and Choi (1998) suggested that it was not appropriate
to include the service constraint if the shortage cost was explicitly
specified and claimed that a better solution could be obtained by allowing
the reorder point to be variable. Hariga and BenDaya (1999) also revised
Ouyang et al. (1996) model by including the reorder point as a
decision variable. Pan and Hsiao (2001) presented two inventory models
where shortage was allowed with controllable back ordering.
This study considers a continuous review inventory system in which the
lead time is controllable and can be decomposed into several components
each having a crashing cost function. In addition, shortage is permitted
and the total amount of stockout is a combination of backorder and lost
sale. It is further assumed that the patient customers with outstanding
orders during the shortage period are offered a backorder price discount
and consequently the backorder ratio is proportional to the magnitude
of this discount (Pan and Hsiao, 2001). Since the shortage cost is explicitly
included, the reorder point is also treated as a decision variable in
this study (Moon and Choi, 1998).
There is form of lead time demand considered following a normal distribution
in the study. In this models, the objectives are to simultaneously optimize
the order quantities, back order discounts, reorder points and lead times
such that the total expected annual costs are minimized. Furthermore,
an iterative algorithm is applied to find the optimal solution for the
case where the lead time demand follows a normal distribution. Our model
serves as a pioneering work investigating the effects backorder discounts
and safety factor have on the integrated inventory system.
NOTATIONS AND ASSUMPTIONS
The following notations are used throughout the study.
L 
= 
The length of lead time (decision variable) 
Q 
= 
The order quantity (decision variable) 
π_{x} 
= 
The backorder price discount offered by the supplier per unit (decision
variable) 
k 
= 
The safety factor (decision variable) 
r 
= 
The reorder point 
π_{0} 
= 
The gross marginal profit per unit 
D 
= 
The average demand per year 
A 
= 
The fixed ordering cost per order 
h 
= 
The inventory holding cost per unit per year 
β 
= 
The backorder ratio 
β_{0} 
= 
The upper bound of the backorder ratio 
φ 
= 
The standard normal distribution; 
Φ 
= 
The standard normal cumulative distribution function 
SS 
= 
The safety stock 
B(r) 
= 
The expected shortage of a cycle 
R(L) 
= 
The total crashing cost of a cycle 
The assumptions made in the research are defined as follow:
• 
The reorder point r = expected demand during lead time
+ safety stock, that is, r = DL + kσ√L, where k is a safety
factor 
• 
The lead time L has n mutually independent components, where the
ith component has a normal duration T_{i} and a minimum duration
t_{i}, i = 1, 2, ..., n and a crashing cost per unit time
a_{i}. These a_{i}’s are arranged such that a_{1}
≤ a_{2} ≤ ... ≤ a_{n}. The lead times are
crashed that should be first on component 1 and then component 2 and
so on. 
• 
Let L_{i} denote the length of lead time with component
1, 2, ...n, i crashed to their minimum values, then L_{i}
can be expressed as: 

and L_{0} is not cashed the length of lead time. Thus, the
lead time crashing cost R(L) per replenishment cycle is given by: 

• 
The backorder ratio β is variable and is in proportion to the
backorder price discount offered by the supplier per unit π_{x};
thus, β = β_{0}π_{x}/π_{0},
for 0 <β_{0}≤1, 0≤π_{x} ≤ π_{0}
(Pan and Hsiao, 2001) 
NORMAL DEMAND MODEL
The problem under study is to minimize the following total expected annual
cost:
EAC(Q, π_{x}, k, L) = OC + HC +
SC + CC 
Where:
OC 
= 
Stands for ordering cost 
HC 
= 
Holding cost 
SC 
= 
Shortage cost 
CC 
= 
Crashing cost 
The annual expected ordering cost can be expressed as:
The lead time demand X is assumed to follow the normal distribution with
mean μL and standard deviation σ√L. Since shortage is
allowed, the expected inventory shortage at the end of a cycle is given
by:
E(Xr)^{+} = B(r) = σ√LΨ(k),

Where:
Ψ(k) 
= 
φ(k)k[1  Φ(k)] 
φ, Φ 
= 
Standard normal distribution and cumulative distribution function,
respectively (Ravindran et al., 1987) 
For backorder ratio β, the expected inventory at the end of a cycle
is βB(r) and the expected lost sale is (1β)B(r). Therefore,
the average inventory of a cycle is:
Hence, the expected annual holding cost is:
The backorder price discount of a cycle is βπ_{x}B(r)
and the expected lost sales is (1β)B(r), so the profit lost due
to shortage is π_{0}(1β)B(r) and the expected annual
shortage cost can be expressed as:
Following the definition of L_{i}, the lead time crashing cost
R(L) per replenishment cycle is given by:
Hence, the expected annual crashing cost is:
Consequently, the total expected annual cost EAC(Q, π_{x},
k, L) is:
Taking the first partial derivatives of EAC(Q, π_{x}, k,
L) with respect to Q, π_{x}, k and L, respectively, it follows
that:
and
Setting Eq. 3 to zero and solving for π_{x},
it follows that:
Setting Eq. 2 to zero and substituting Eq.
6 into 2 to solve for Q, thus:
Setting Eq. 4 to zero and substituting Eq.
6 into 4 to solve for k, then:
It can be shown from Eq. 8 that hQ/D≤ π_{0},
so Φ(k)≥0.5 holds for nonnegative safety factor k. Consequently,
the value of π_{x} derived in Eq. 6 will
automatically lie between 0 and π_{0} as specified in assumption
4th.
For fixed values of Q, π_{x} and k, EAC(Q, π_{x},
k, L) is concave in the range L∈(L_{i}, L_{i1}],
since:
For fixed L∈(L_{i}, L_{i1}], denote the values
of Q, π_{x} and k found from Eq. 68
by Q*, πx* and k*, respectively. In addition, for fixed L∈(L_{i},
L_{i1}], the determinant of Hessian matrix of EAC(Q, π_{x},
k, L) is positive definite at (Q*, π_{x}*, k*) as shown in
Appendix.
The following algorithm can be used to find the optimal values of the
order quantity, backorder discount, reorder point and lead time.
Step 1: For i = 0, 1, 2, ..., n
(i) 
Set k_{io} = 0 (implies Ψ(k_{io})
= 0.39894) 
(ii) 
Substitute Ψ(k_{io}) into Eq. 8 to
evaluate Q_{io} 
(iii) 
Use Q_{io} to determine Φ(k_{in}) from Eq.
8, then find kin from Φ(k_{in}) by checking the normal
table. Let k_{io} = k_{in} 
(iv) 
Repeat (ii) and (iii) until no change occurs in the values of Q_{i}
and k_{i}. Denote these resulting solutions by Q_{i }and
k_{i}. 
(v) 
Use Q_{i }and Eq. 7 to compute the backorder
price discount π_{xi}. 
(vi) 
Use Eq. 1 to compute the expected total annual
cost EAC(Q_{i}, π_{xi}, k_{i}, L_{i}). 
Step 2: Set EAC(Q*, π_{x}*, k*, L^{*}) =
Min{EAC(Q_{i}, π_{xi}, k_{i}, L_{i}),
i = 0, 1, 2, ..., n }.
Step 3: (Q*, π_{x}*, k*, L*) is a set of optimal
solutions.
NUMERICAL EXAMPLE AND ITS SENSITIVITY ANALYSIS
Consider an inventory system with the following data: D = 600 units/year,
A = $200 per order, h = $20 per unit per year, π0 = $150 per unit,
σ = 7 unit/week and the lead time has three components with data
shown in Table 1 (Pan and Hsiao, 2001). Apply the proposed
algorithm to solve the problem for the upper bound of the backorder ratio
β_{0} = 0.95, 0.80, 0.65, 0.50, 0.35 and 0.20, respectively.
The resulting optimal solutions are summarized in Table
2. Also included in Table 2 are the results obtained
from the associated Pan and Hsiao model (2001) by setting k fixed at 0.85,
along with the corresponding saving on the total expected annual cost
of the proposed model over that of Pan and Hsiao (2001). It is interesting
to observe that the saving increases as β_{0} decreases.
Next, we study the effect of change in the model parameters such as D,
A, h, σ and π_{0} under β_{0} = 0.5, the
optimal order quantity (Q* = 121), the optimal backorder price discount
(π_{x}* = 77.0157), the optimal safety factor (k* = 1.88),
the optimal lead time (L* = 4) and the corresponding total cost (EAC*
= EAC(Q*, π_{x}*, k*, L^{*}) = 2947.72) keeping the
same parameter values in Example. The sensitivity analysis is performed
by changing each of the parameter by 50, 25, +25 and +50%, taking one
at a time while keeping remaining unchanged. The results are shown in
Table 3. Let EAC(δ) denote the percentage difference
between the new total expected annual cost obtained from changing the
values of these factors and the original total expected annual cost, that
is, EAC(δ) = (the new total expected annual costthe original j total
expected annual cost)/(the original total expected annual cost).
Makes the sensitivity analysis in view of various parameters, the obtained
data result has the following discovery.
• 
k*, Q* and EAC* all increases while π_{x}*
decreases with an increase in the value of the model parameter demand
rate D. The obtained results show that π_{x}* and k*
are lowly sensitive and Q* and EAC* are middling sensitive to changes
in the value of D. Moreover, L_{i}* is unchanged in D. 
• 
π_{x}*, Q* and EAC* increases while k* decreases with
an increase in the value of the model parameter A. The obtained results
show that π_{x}* and k* are lowly sensitive and Q* and
EAC* are middling sensitive to changes in the value of A. Moreover,
L_{i}* is unchanged in A. 
• 
π_{x}* and EAC* increases while Q* and k* decreases
with an increase in the value of the model parameter h. The obtained
results show that π_{x}* and k* are lowly sensitive and
Q* and EAC* are middling sensitive to changes in the value of h. Moreover,
L_{i}* is unchanged in h. 
Table 1: 
Lead time data of the examples 

Table 2: 
Summary of the results for example (L_{i} in
weeks) 

Table 3: 
Effect of changes in the parameters of the continuous
review inventory models 

• 
π_{x}*, Q* and EAC* increases while k*and
L_{i}* decreases with an increase in the value of the model
parameter σ. The obtained results show that π_{x}*
and k* are lowly sensitive and Q* and EAC* are middling sensitive
to changes in the value of σ. Moreover, L_{i}* is highly
sensitive to changed in σ. 
• 
π_{x}*, k* and EAC* all increase with an increase in
the value of the model parameter π_{0}. The obtained
results show that EAC* are lowly sensitive, k* are middling sensitive
and π_{x}* highly sensitive to changes in the value of
π_{0}. Moreover, L_{i}*and Q* is unchanged in
π_{0}. 
CONCLUSIONS
This study extends the Pan and Hsiao model (2001) to study the impact of safety
factor on the continuous review inventory model involving controllable lead
time and backorder price discount with mixture of backorder and partial lost
sales. The objective is to minimize the expected total annual cost by simultaneously
optimizing order quantity, backorder price discount, safety factor and lead
time. It is found that the expected total inventory cost tends to decrease as
the upper bound of the backorder ratio increases while all the other parameters
stay fixed. The results of the illustrative example also show that the expected
total inventory cost and the safety factor increase for a given the upper bound
of the backorder ratio decreases. The savings of the total cost on considering
safety factor that is decision variable are demonstrated in the examples as
well.
APPENDIX
The Hessian matrix H of EAC(Q, π_{x}, k, L) for a given
value of L is:
Where:
and
Next, evaluate the principal minor of H at point (Q*, πx*, k*).
The first principal minor of H is
Since from Eq. 7 that π_{x}* = hQ*/2D +
π_{0}/2, the second principal minor of H is:
Therefore, after substituting π_{x}* from Eq.
6 and Φ(k*) from Eq. 8, we obtain
The third principal minor of H is:
Where, F(k*) = β_{0}(4β_{0})φ(k*)Ψ(k*)
 2β_{0}(1Φ(k*))^{2}
For ∀ k* ∈ [0, ∞) and 0 < β_{0} ≤1,
F(k*) is positive. Hence, we have  H_{33}  >0.
Consequently, the results from 1113 show that the Hessian matrix H is
positive define at (Q*, π_{x}*, k*).