Inventory Models
Inventory Models
Inventory Models
Inventory Models
The purpose of inventory theory is to determine rules that management can use to
minimize the costs associated with maintaining inventory and meeting customer
demand. Inventory models answer the following questions
Inventory models involve some or all of the following variables: We list the factors that
are important in making decisions related to inventories.
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Operations
Research
Ordering cost (c(z))This is the cost of placing an order to an outside supplier or re-
leasing a production order to a manufacturing shop. The amount ordered is z and
the function c(z) is often nonlinear.
Setup cost (K)A common assumption is that the ordering cost consists of a fixed cost,
that is independent of the amount ordered, and a variable cost that depends on
the amount ordered. The fixed cost is called the setup cost.
Product cost (c)This is the unit cost of purchasing the product as part of an order. If
the cost is independent of the amount ordered, the total cost is cz, where c is the
unit cost and z is the amount ordered. Alternatively, the product cost may be a
decreasing function of the amount ordered.
Holding cost (h)This is the cost of holding an item in inventory for some given unit of
time. It usually includes the lost investment income caused by having the asset
tied up in inventory. This is not a real cash flow, but it is an important
component of the cost of inventory. If c is the unit cost of the product, this
component of the cost is cα , where α is the discount or interest rate. The holding
cost may also include the cost of storage, insurance, and other factors that are
proportional to the amount stored in inventory.
Shortage cost (p)When a customer seeks the product and finds the inventory empty,
the demand can either go unfulfilled or be satisfied later when the product
becomes available. The former case is called a lost sale, and the latter is called a
backorder. Although lost sales are often important in inventory analysis, they are
not considered in this section, so no notation is assigned to it. The total
backorder cost is assumed to be proportional to the number of units backordered
and the time the customer must wait. The constant of proportionality is p, the
per unit backorder cost per unit of time.
Demand rate (d)This is the constant rate at which the product is withdrawn from
inventory. (units / time)
Lot/Order Size (Q)This is the fixed quantity received at each inventory replenishment.
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Order level (S)The maximum level reached by the inventory is the order level. When
backorders are not allowed, this quantity is the same as Q. When backorders are
allowed, it is less than Q.
Cycle time (τ )The time between consecutive inventory replenishments is the cycle
time. For the models of this section τ = Q/d
Cost per time (T )This is the total of all costs related to the inventory system that
are affected by the decision under consideration.
Optimal Quantities (Q∗ , S ∗ , τ ∗ , T ∗ )The quantities defined above that maximize profit
or minimize cost for a given model are the optimal solution.
Constant Lead Time The lead time for each order is a known constant, L. By the
lead time we mean the length of time between the instant when an order is placed
and the instant at which the order arrives.
Continuous Ordering An order may be placed at any time. Inventory models that
allow this are called continuous review models. If the amount of on-hand
inventory is reviewed periodically and orders may be placed only periodically, we
are dealing with a periodic review model.
2.In an order of any size (say q units is placed, an ordering and setup cost K is
incurred.
3.The lead time for each order is zero. This means orders arrivve on
5.The cost per unit-year of holding inventory is h. This implies that if I units are
held for T years, a holding cost of ITh is incurred.
Given these assumptions, the EOQ model determines an ordering policy that minimizes
the yearly sum of ordering cost, purchasing cost, and holding cost.
• We should never place an order when I, the inventory level, is greater than zero; if
we place an order then the we are incurring an unnecessary holding cost.
• Each time an order is placed (when I = 0) we should order the same quantity, Q.
• To determine the annual holding cost, we need to examine the behavior of I over
time.
• Definition: Any interval of time that begins with the arrival of an order and ends
the instant before the next order is received is called a cycle.
• The average inventory during any cycle is simply half of the maximum inventory
level attained during the cycle.
• This result will hold in any model for which demand occurs at a constant rate.
(4.1)
total annual cost. Substituting the EOQ into the total cost expression gives
. .
hK d hK d √
T∗ = + pd + = pd + 2hKd (4.3)
2 2
Example 4.1. Braneast Airlines uses 500 taillights per year. Each time an order for
taillights is placed, an ordering cost of $5 is incurred. Each light cost 40cents, and the
holding cost is 8cents/light/year. Assume that demand occurs at a constant rate and
shortages are not allowed. What is the EOQ? How many orders will be placed each
year? How much time will elapse between the placement of orders?
solution
We are given that K = $5, h = 0.08/light/year, and d = 500 lights/year. The EOQ is
. .
2K d 2 × 5 × 500
Q∗ = = = 250
h 0.08
Hence, the airline should place an order for 250 taillights each time that inventory
reaches zero.
The time between placement (or arrival) of orders is simply the length of a cycle. Since
the length of each cycle is , τ = Q∗/d, the time between orders will be
τ = Q = 250 = 1
∗
d per year
500 2
Example 4.2. Paul Peterson is the inventory manager for Office Supplies, Inc., a large
office supply warehouse. The annual demand for paper punches is 20,000 units. The
ordering cost is $100 per order, and the carrying (holding) cost is $5 per unit per year.
Determine the EOQ
solution . .
2K d 2 × 100 × 20, 000
Q∗ =
h = 5 = 894