Inventory Models

3 Models with Deterministic Demand Model 1: Regular EOQ Model Model 2: EOQ with Backorder Model 3: EPQ with Production

2 Models with Stochastic Demand Model 5: Models with Discrete Probability Model 6: Model with Continuous

Probability

Deterministic Models:Model 1: The EOQ Model Prototype In the deterministic models under study,

the firm faces a known linear demand. As an example of deterministic models,

we will demonstrate in detail Model 1, the basic EOQ model, In this model the firm orders the product. The goal of the firm is to determine Q, the Economic Order Quantity that minimizes the firm’s total cost.

Determining the economic order quantity uniquely determines the cycle’s length T.

The determination depends on the relative cost of making an order relative to the cost of holding item in inventory

Types of costs

The firm has three types of costs Procurement Cost Ordering Cost Holding cost

Inputs of the EOQ Model

A Annual number of items demanded (Annual Demand)

k Fixed cost per order c Unit cost of procuring an item h Annual cost per dollar value of .

holding items in inventory.

Planning Period and holding item’s dollars in Inventory

Costs are computed for a planning period.

In the textbook, this planning period is a year.

Length of use Cycle ,T, (Time between orders) is then specified in yearly terms. Conversion to days would require multiplying the result by the number of days in a year.

Holding Cost

It is important to notice, the primary component in the holding cost is the cost of holding the monetary value of the item in inventory. Typically this cost accounts for about 80% of the holding cost.

The holding cost of the monetary value held in inventory is evaluated based on the opportunity cost for investment of the value held in inventory or based on the cost of borrowed funds needed to hold the item.

For example, a dollar invested may yield 10% return on investment or may be required borrowed fund at 10% interest cost. In these examples, the cost is .10 per dollar.

Other costs associated with holding are also therefore prorated per dollar value.

Therefore, in the model inputs we report, h, the annual cost per dollar value of holding the item in inventory. Hence, the corresponding holding cost for the item is hc.

Cost Computation is easy.

For every Cost Type the cost is,

Cost per unit of measurement times the Number of units

Units of Measurement and Number of Units for Type of Cost

Type of cost UOM Number of Units

Procurement One procured item Annual demand

Holding One item held in inventory for a year Average Inventory

Ordering One made order Number of orders in a year

Costs Per Unit

Type of Cost Cost Per Unit

Procurment c

Holing hc

Ordering k

Annual Number of Units per Type of Cost

Cost Type Number of Units Formula

Procurement Annual Demand A

Holding Average Number of Q/2 units in Inventory

Ordering Annual number of A/Q Orders

Some Outputs

The ordered (and use) Quantity per cycle, Q

Number of Orders, A/Q Length of order and use Cycle, Q/A Maximum Inventory level, Q Average Inventory level, Q/2 Annual Holding cost per item, hc

Total Cost

Total annual Cost is the sum of procurement cost ordering costs and holding cost.

= cA +k(A/Q) + hc(Q/2).

Since, the procurement cost cA is fixed, and does not affect optimization. The Relevant Cost for decision is then,

TC(Q) = k(A/Q) + hc(Q/2).

Model 2: Inventory with Backorder.

In Model 2. the firm designs an optimal order per cycle as well as optimal waiting list.

As soon as the order of size Q arrives, the firm supplies the waiting list’s demand. The stock, S, of the remaining units, is left as inventory to serve next customers not on the waiting list.

The stock S is depleted till it disappears. The firm is then beginning to collect orders on a waiting list, till the optimal level of the waiting list, Q-S, is reached. At which point a new order of size Q arrives.

Model 2, Inventory with Backorder continues Since customers do not like to wait, there is a

shortage penalty per item on the waiting list, p. The penalty p may either be given directly, or

imputed from a service level, L, where L is the proportion of demand met on time or alternatively the probability of providing an item from inventory (rather than from the waiting list). It is straight forward to impute p from L. (pp. 600-01).

The total relevant cost equals the ordering cost + holding cost + shortage cost.

Model 3: Inventory with Production In this model the firm produces the item for its own use.

The firm wishes to optimize the quantity to be produced and used in every cycle, Q and correspondingly the optimal length of the production and use cycles. (T1 and T respectively,).

First the firm sets up the machines, produces the product and uses it. (production rate B must be greater than use rate A.). Since the production is on-going, inventory is gradually built up to an optimal maximal level.

Once the maximum inventory level is reached the firm stops production. For the rest of the use cycle, the firm only uses the already produced inventory in stock. The process is repeated.

Model 3, Inventory with Production, Continued In this model set up costs replaces ordering

cost. Mathematically, it has the same formula, The quantity Q stands for the quantity to use

and produce in the cycle. Average inventory is half the maximum

inventory, as in Model 1. However it is lesser than half of Q, the quantity used in a cycle, as the inventory is only gradually built up..

Total Relevant Cost equals Set Up Cost + Holding Costs.

An interesting feature of the Deterministic Models

In the Regular EOQ Model, at optimum,

Ordering Cost = Holding Cost

In other models, similar observation is made:

In the Inventory with Back Order Case

Ordering Cost = Holding Cost + Shortage Cost

In the inventory with Production case,

Set Up Cost = Holding Cost

Models with Stochastic Demand: The Newsvendor Problem In this case we are dealing with

stochastic demand and given short cycles.

Unlike the deterministic models of chapter 15 we are not determining the cycle’s length There is some given cycle and we are provided with the holding cost of left over item at the end of the cycle.

Properties of the Newsvendor Problem Continued.

The firm faces possibilities of either shortage and surplus penalties

We wish to find Q, the optimal order quantity to be stocked at the beginning of a cycle, expected surplus and surplus cost, and expected shortage and Shortage costs, and total expected costs.

Inputs for Model 4: An Inventory Model with Discrete Demand DistributionC Procurement Cost hE Additional cost of each item held at the end of

the inventory cycle.

pS Penalty for each item short (loss of goodwill)

pR Selling Price

Discrete Distribution of the quantity demand for the item in a cycle.

Per Unit Penalties for left over Item and for Shortage The penalty for a left over item hE +C, the

surplus penalty per item, is the item cost c and the residual cost associated with the left over item. If the left over item has some residual value hE will be negative and would reduce the surplus penalty per item.

Shortage penalty per item, pS + (pR –C) . It has two components. The first is the good will loss, pS. The second is the opportunity loss encountered. If the firm had the item it could have made a profit pR –C by selling it.

The Critical Share and Finding the optimal Order Quantity

The critical share is the share of the per item shortage penalty in the sum of the per units shortage and surplus penalties,

)()( chcpp

cpp

ERS

RS

The critical share formula

The critical share is,

pS +(pR –c)

____________________________

pS +(pR –c) + (hE+c)

The critical share and optimal Q

Marginal considerations, based on calculus led to the result that the optimal quantity is to be this where the cumulative probability of the demand equal the critical share.

In the discrete case we look at the level of the cumulative distribution just surpass the critical share. Mean demand for a cycle is labeled μ.

Total Expected Costs

Total expected cost is equal to the expected procurement cost + the expected shortage cost + the expected surplus cost.

Expected shortage cost equals the penalty per unit of shortage times the expected shortage, B(Q).

Expected surplus cost equals the penalty per unit of surplus times the expected surplus.

Model 5. Inventory with Normally Distributed Demand In this case the mean and standard deviation of

the demand are known and the determination of the ordered quantity, Q, based on the critical share may be very precisely determined. The rest of the inputs are the same as in Model 4.

The following computations of the expected shortage and surplus using the Normal Loss Function L(x), and the computations of the associated expected costs depend on the whether Q is smaller or larger than the mean demand μ. See page 630.