Lead time plays an important role and has been a topic of
interest for many authors in inventory management. The length of
lead time directly affects the size of order quantity, the
service level for customer, and the competitive ability in
business. Almost all inventory models assume that lead time is a
prescribed constant or a stochastic variable. But in many real
situations, lead time can be reduced at an added crashing cost;
in other words, it is controllable. Therefore, many firms felt
doubtful about the uncontrollability of lead time. In addition,
for the traditional inventory models, many researchers studied
the (Q,r) inventory model which considers the case that all
demand during the stockout period is either backordered or lost
sales. In practical market situations, some customers maintain
the loyalty and trust for specific supplier though it has many
competitors, and the customers are also willing to wait whenever
a shortage occurs. Hence, the total amount of stockout during
the stockout period is worth to be considered a mixture of
backorders and lost sales. This paper focuses on the
controllable lead time and presents six mathematical inventory
models, in which both lead time and order quantity are the
decision variables. In addition, during the stockout period, the
total amount of stockout considers a mixture of backorders and
lost sales to obtain the optimal solution. In this thesis, we
first assume that the demand of lead time follows normal
distribution and the lead time consists of n components which
have different normal durations, minimum durations and crashing
costs. The objective is to determine the optimal lead time and
the order quantity pair so as to minimize the expected total
cost which is composed of the ordering cost, the expected
carrying cost, the expected shortage cost and the crashing cost.
Second, we assume that the distribution of lead time demand is
free. This model is an extended model of the previous inventory
model. We consider the cumulative distribution function of the
lead time demand which has only known mean and variance, but
make no assumption on the distributional form, and try to apply
the minimax distribution free approach criterion to find out the
optimal solution. Next, we discuss the mixture inventory model
with a service level constraint. In many practical problems, the
stockout cost includes: the penalty of contract, the damage of
goodwill, loss of goodwill for customer, the reduced potential
of the future demand, and so on. Since it is difficult to
estimate an exact value of the stockout cost, we try to replace
the stockout cost in the objective function by the service level
constraint. The service level constraint here indicates that the
stockout level per cycle is bounded. Finally, we discuss an
arrival order which contains some defective units in mixture
inventory model. Traditionally, inventory models almost assume
that an arrival order has no defective units. In fact, this is
impossible. As a result of imperfect production of the supplier,
and/or damage in transit, an arrival order often contains some
defective units. If there are defective units in orders, there
will be impact on the on-hand inventory level, the number of
shortage and the frequency of orders in inventory system.
Therefore, ordering policies determined by conventional
inventory models may be inappropriate for the defective
inventory situation. Hence, in this situation, we change the
previous inventory models and assume that an arrival order may
contain some defective units. And the number of defective units
in an arrival order is a random variable with binomial
probability distribution. We also assume that the purchaser
inspects the entire items which are assumed to be non-
destructive, and the defective units in each lot which can not
be repaired will be returned to the vendor at the time of
delivery of the next lot. In all mathematical inventory models,
we discuss the effects of parameters and give economic
interpretation of the circumstances.