Cover Some Realistic Situations Which Relax One Or More Of The Eoq Assumptions

04/09/23 Special Inventory Models CJA10.14.1

• Cover some realistic situations which relax one or more of the EOQ assumptions– Non-Instantaneous Replenishment– Quantity Discounts– One-Period Decisions

Operations Management

Special Inventory Models


• Item used or sold as they are completed, without waiting for a full lot to be completed

• Usual case is where production rate, p, exceeds the demand rate, d, so there is a buildup rate of (p – d) units during time when both production and demand occur.

• Both p and d expressed in same time interval.


Non-Instantaneous Replenishment


• Recall…TC = annual holding costs plus annual setup costs

=

Ave. Inv. Level =

• Now...

Avg. Inv. Level =


Economic Lot Size (ELS)

2

)0(min)(max levelQlevel

2

)0(min)(max max levelIlevel


• Buildup at a rate of ( p-d) (units/day) during production phase until a lot size of Q is produced

• Buildup continues for Q / p days (units/units/day)

• Imax = units per day, ( p-d),

x number of days (Q / p )

where: p = production rated = demand rateQ = lot size




Total Cost = Annual holding costs + annual ordering costs

Setting up the total cost equation, where D is the annual demand:



2

max SQ

DH

ITC


• Differentiation of this equation with respect to Q,setting the result equal to zero, and solving for Q results in the

Economic Production Lot Size, ELS:

• Since p > d, the second term is greater than 1,

– so the ELS is __________ than the EOQ



H

DS2 ELS


Demand = 30 barrels/day Setup cost = $200Production rate = 190 barrels/day Annual holding cost = $0.21/barrelAnnual demand = 10,500 barrels Plant operates 350 days/year

ELS = pp - d

2DSH


Example




Example

SELS

DH

p

dpELSTC

2


TBOELS = (350 days/year)ELSD



Example


Production time = ELSp



Example


• Quantity discounts are price incentives to purchase large quantities

• Price break is the minimum purchase quantity to get a certain discount price

• The item’s price is no longer fixed so there arethree relevant cost components

– annual purchase costs in addition to annual holding costs and annual ordering (setup) costs


Quantity Discounts

PDSQ

DH

QTC

2


• There are cost curves for each price level

• The feasible total cost begins at the top curve,

then drops down, curve by curve,

at the price breaks.

• The EOQs do not necessarily produce the

best (“minimum total annual cost”) lot size.

Quantity Discounts

Feasible Price-Quantity Combinations


C for P = $4.00

C for P = $3.50

C for P = $3.00

Total annualcost, $

Purchase quantity, Q

0 100 200

(a) Total cost curves with purchased materials added

Purchase Discounts

Total Cost Curves

Firstprice break

Second price break


Step 1:

Beginning with the lowest price, calculate the EOQ for each price level until a feasible EOQ is found.

– it is feasible if the quantity lies in the range corresponding to its price.

As subsequent prices are larger than the previous one,the holding cost, H, (H = i·P ) gets larger.

Since H is in the denominator of the EOQ formula,

the EOQ gets smaller.

Purchase Discounts

Solution Procedure


Annual demand = 936 unitsOrdering cost = $100.00Holding cost = 25% of unit price

Order Quantity Price per Unit

0 - 249 $60.00250 - 499 $59.00500 or more $58.00

Purchase Discounts

Example

Pi

SDEOQ

2

00.58$


$55,000

$56,000

$57,000

$58,000

$59,000

$60,000

$61,000

$62,000

0 250 500 750

Order Quantity, Q

Tota

l Ann

ual C

ost,

$

Purchase Discounts

Example

Price = $60.00

Price = $59.00

Price = $58.00


Step 2:

If the first feasible EOQ found is for the lowest price level, this quantity is the best lot size.

Otherwise, calculate the total cost for the first feasible EOQ and for the most economical, feasible order quantity at each lower price level.

The quantity with the lowest total cost is optimal.

Purchase Discounts

Solution Procedure


$55,000

$56,000

$57,000

$58,000

$59,000

$60,000

$61,000

$62,000

0 250 500 750

Order Quantity, Q

Tota

l Ann

ual C

ost,

$

Purchase Discounts

Example

Price = $60.00

Price = $59.00

Price = $58.00


Annual demand = 936 unitsOrdering cost = $100.00Holding cost = 25% of unit price

Order Quantity Price per Unit

0 - 249 $60.00250 - 499 $59.00500 or more $58.00

Purchase Discounts

Example

DPSQ

DPi

QTCQ

2


• The best purchase quantity is 250 units, which does not correspond to the deepest discount price.

• This is not always true - EOQ is affected by:

– small discounts, quantity break points,

– large holding cost, and

– small demand.

• Small lot sizes may be better even though the price is not the lowest

Quantity Discounts

Example

04/09/23 Special Inventory Models

• Single-Period Inventory Model

– One time purchasing decision (Example: vendor selling t-shirts at a

football game)

– Seeks to balance the costs of inventory overstock and under stock

• Multi-Period Inventory Models

– Fixed-Order Quantity Models

> Event triggered (Example: running out of stock)

– Fixed-Time Period Models

> Time triggered (Example: Monthly sales call by sales representative)


Inventory Models


• Problem for seasonal and high fashion goods.

• Only allowed to order one time.

• Short selling seasons and long lead times prohibit the possibility of placing a second order.

• A balance between ordering enough to meet demand and not having any left over at the end of the season.

• Sometimes referred to as the ”Newsvendor” problem


One-Period Decisions


• List different demand levels and probabilities

• Develop a payoff table, where each new row represents a different order quantity and each column represents a different demand.


Selecting the Purchase Quantity


• The payoff is:

where: p = profit per unit sold during the season l = loss per unit disposed of after the seasonQ = purchase quantityD = demand level



demand) exceedsquantity (purchase if

markup) fullat sold units (all if Payoff

DQ

DQ

DQlpD

pQ


• Calculate the expected payoff of each Q. For a specific Q, first multiply each payoff by its demand probability, and then add the products.

• Choose the order quantity Q with the highest expected payoff.




For one item, p = $10 and l= $5. The probability distribution for the season’s demand is:

Demand Demand(D) Probability10 0.220 0.330 0.340 0.150 0.1


Example


Complete the following payoff matrix, as well as the column on the right showing expected payoff.

D ExpectedQ 10 20 30 40 50 Payoff--- (.2) (.3) (.3) (.1) (.1) ---

10 $100 $100 $100 $100 $100 $10020 50 200 200 200 200 17030 0 ____ 300 ____ 300 ____40 –50 100 250 400 400 17550 –100 50 200 350 500 140


Example


Payoff if Q = 30 and D = 20:

Payoff if Q = 30 and D = 40:

Expected payoff if Q = 30:

What is the best choice for Q?


Example

Technology

Cover Some Realistic Situations Which Relax One Or More Of The Eoq Assumptions