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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 ModelsModel 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 firms

total cost.

- Determining the economic order quantity uniquely

determines the cycles 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 items 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 lists 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 cycles 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 Distribution

- C 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,

The critical share formula

- The critical share is,
- pS (pR c)
- ____________________________
- pS (pR c) (hEc)

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.

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