© 2007 Pearson Education Special Inventory Models Supplement D.

© 2007 Pearson Education

Special Inventory Models

Supplement DSupplement D


Special Inventory Models

Three common situations require relaxation of one or more of the assumptions on which the EOQ model is based.

Noninstantaneous Replenishment occurs when production is not instantaneous and inventory is replenished gradually, rather than in lots.

Quantity Discounts occur when the unit cost of purchased materials is reduced for larger order quantities.

One-Period Decisions: Retailers and manufacturers of fashion goods often face situations in which demand is uncertain and occurs during just one period or season.


If an item is being produced internally rather than purchased, finished units may be used or sold as soon as they are completed, without waiting until a full lot is completed.

Production rate, p, exceeds the demand rate, d. Cycle inventory accumulates faster than demand

occursa buildup of p – d units occurs per time period,

continuing until the lot size, Q, has been produced.

Noninstantaneous Replenishment


Production quantityProduction quantity

Demand during Demand during production intervalproduction interval

Maximum inventoryMaximum inventory

Production Production and demandand demand

Demand Demand onlyonly

TBOTBO

On

-han

d i

nve

nto

ryO

n-h

and

in

ven

tory QQ

TimeTime

IImaxmax

p – d



Cycle inventory is no longer Q/2, as it was with the basic EOQ method; instead, it is the maximum cycle inventory (Imax / 2)


C = ( ) + (S)DQ

Q p – d2 p

D = annual demandd = daily demandp = production rateS = setup costsQ = ELS

Imax = (p – d) = Q( )Qp

p – dp

Total annual cost (C) = Annual holding cost + annual ordering or setup cost


Economic production lot size (ELS) is the optimal lot size in a situation in which replenishment is not instantaneous.

Economic Lot Size (ELS)

ELS =p

p – d2DS

H

D = annual demandd = daily demandp = production rateS = setup costsH = annual unit holding cost


Finding the ELSFinding the ELSExample D.1Example D.1

The manager of a chemical plant must determine the following for a particular chemical:

1. Determine the economic production lot size (ELS).2. Determine the total annual setup and inventory

holding costs. 3. Determine the TBO, or cycle length, for the ELS.4. Determine the production time per lot.

• What are the advantages of reducing the setup time by 10 percent?

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


ELSELS = = pp

pp – – dd22DSDS

HH

Finding the ELS for the Finding the ELS for the Example D.1Example D.1 chemical chemical


ELSELS = = 190190190190 – – 3030

2(2(10,50010,500)()($200$200))$0.21$0.21

ELSELS = 4873.4 barrels = 4873.4 barrels

D = annual demandd = daily demandp = production rateS = setup costsH = unit holding costQ = ELS


Finding the Finding the Total Annual Cost Total Annual Cost



CC = = ( ( ))((HH) + () + (SS))DDQQ

Q Q pp – – d2 2 pp

CC = = ( ( )) ($0.21) + ( ($0.21) + ($200$200))10,50010,5004873.44873.4

4873.4 4873.4 190190 – – 30 2 2 190190

CC = $430.91 + $430.91 = $430.91 + $430.91 CC = $861.82 = $861.82

Example D.1Example D.1


Finding the TBO Finding the TBO



TBOTBOELSELS = (350 days/year) = (350 days/year)ELSELS

DD

TBOTBOELSELS = (350 days/year) = (350 days/year)4873.44873.410,50010,500

TBOTBOELSELS = 162.4, or 162 days = 162.4, or 162 days



Finding the Finding the Production Time per Lot Production Time per Lot



Production time = Production time = ELSELS

pp

Production time = Production time = 4873.44873.4

190190

Production time = 25.6, or 26 daysProduction time = 25.6, or 26 days



Advantage of Reducing Advantage of Reducing Setup TimeSetup Time

OM Explorer Solver for the Economic Production Lot Size Showing the effect of a 10 Percent Reduction in setup cost.

$180 vs original $200


Application D.1

38.1555

3560

60

000,2

000,100080,1022

dp

p

H

DSELS

or 1555 engines


Application D.1continued


Quantity Discounts

Quantity discounts, which are price incentives to purchase large quantities, create pressure to maintain a large inventory.

For any per-unit price level, P, the total cost is:

Total annual cost = Annual holding cost + Annual ordering or setup cost + Annual cost of materials

CC = ( = (HH) + () + (SS) + ) + PDPDQQ22

DDQQ

D = annual demandS = setup costsP = per-unit price levelH = unit holding costQ = ELS

© 2007 Pearson EducationTotal cost curves with Total cost curves with

purchased materials addedpurchased materials added

Quantity Discounts

EOQs and price break quantitiesEOQs and price break quantities

CC for for PP = $4.00 = $4.00CC for for PP = $3.50 = $3.50CC for for PP = $3.00 = $3.00

PDPD for forPP = $4.00 = $4.00 PDPD for for

PP = $3.50 = $3.50 PDPD for forPP = $3.00 = $3.00

EOQ EOQ 4.004.00

EOQ EOQ 3.503.50

EOQ EOQ 3.003.00

First First price price breakbreak

Second Second price price breakbreak

To

tal c

ost

(d

olla

rs)

To

tal c

ost

(d

olla

rs)

To

tal c

ost

(d

olla

rs)

To

tal c

ost

(d

olla

rs)

Purchase quantity (Purchase quantity (QQ))00 100100 200200 300300

Purchase quantity (Purchase quantity (QQ))00 100100 200200 300300

First price 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 it lies in the range corresponding to its

price.

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 larger price break quantity at each lower price level. The quantity with the lowest total cost is optimal.

Finding Q with Quantity Discounts


Annual demand (D) = 936 unitsOrdering cost (S) = $45

Holding cost (H) = 25% of unit price

Order Quantity Price per Unit

0 – 299 $60.00300 – 499 $58.80500 or more $57.00

A supplier for St. LeRoy Hospital has introduced quantity discounts to encourage larger order quantities of a special catheter. The price schedule is:

Example D.2

EOQ EOQ 57.0057.00 = =22DSDS

HH2(936)(45)2(936)(45)0.25(0.25(57.0057.00))

== = 77 units= 77 units

Step 1: Start with lowest price level:


Example D.2 continued

Annual demand (D) = 936 unitsOrdering cost (S) = $45

Holding cost (H) = 25% of unit price

Order Quantity Price per Unit

0 – 299 $60.00300 – 499 $58.80500 or more $57.00

EOQ EOQ 57.0057.00 = =22DSDS

HH2(936)(45)2(936)(45)0.25(0.25(57.0057.00))


EOQ EOQ 58.8058.80 = =22DSDS

HH2(936)(45)2(936)(45)0.25(0.25(58.8058.80))


EOQ EOQ 60.0060.00 = =22DSDS

HH2(936)(45)2(936)(45)0.25(0.25(60.0060.00))


This quantity is feasible because it lies in the range corresponding to its price.

Not feasible

Not feasible

Feasible


Step 2: The first feasible EOQ of 75 does not correspond to the lowest price level. Hence, we must compare its total cost with the price break quantities (300 and 500 units) at the lower price levels ($58.80 and $57.00):

Example D.2 continued

CC = ( = (HH) + () + (SS) + ) + PPDDQQ22

DDQQ

CC7575 = [( = [(0.25)($60.000.25)($60.00)] + ($45) + $60.00()] + ($45) + $60.00(936936))757522

9369367575

CC7575 = $57,284 = $57,284

CC300300 = [(0.25)($58.80)] + ($45) + $58.80(936) = [(0.25)($58.80)] + ($45) + $58.80(936)300300

22936936300300

= $57,382= $57,382

CC500500 = [(0.25)($57.00)] + ($45) + $57.00(936) = [(0.25)($57.00)] + ($45) + $57.00(936)500500

22936936500500

= $56,999= $56,999

The best purchase quantity is 500 units, which qualifies for the deepest discount.

© 2007 Pearson Education© 2007 Pearson Education

Decision Point:If the price per unit for the range of 300 to 499 units is reduced to $58.00, the best decision is to order 300 catheters, as shown below. This shows that the decision is sensitive to the price schedule. A reduction of slightly more than 1 percent is enough to make the difference in this example.


Application D.2


Application D.2Solution


One-Period Decisions

This type of situation is often called the newsboy problem. If the newspaper seller does not buy enough newspapers to resell on the street corner, sales opportunities are lost. If the seller buys too many newspapers, the overage cannot be sold because nobody wants yesterday’s newspaper.

1. List the different levels of demand that are possible, along with the estimated probability of each.

2. Develop a payoff table that shows the profit for each purchase quantity, Q, at each assumed demand level.

3. Calculate the expected payoff for each Q (or row in the payoff table) by using the expected value decision rule.

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


The payoff for a given quantity-demand combination depends on whether all units are sold at the regular profit margin, which results in two possible cases.

1. If demand is high enough (Q < D) then all of the cases are sold at the full profit margin, p, during the regular season.

Payoff = (Profit per unit)(Purchase quantity) = pQ

2. If the purchase quantity exceeds the eventual demand (Q > D), only D units are sold at the full profit margin, and the remaining units purchased must be disposed of at a loss, l, after the season.

One-Period Decisions

Payoff = (Profit per unit during season) (Demand) – (Loss per unit) (Amount disposed of after season) = pD – l(Q – D)


10 $100 $100 $100 $100 $100 10020 50 200 200 200 200 17030 0 150 300 300 300 19540 –50 100 250 400 400 17550 –100 50 200 350 500 140

Q 10 20 30 40 50 Expected Payoff

Demand 10 20 30 40 50

Demand Probability 0.2 0.3 0.3 0.1 0.1

Example D.3A gift museum shop sells a Christmas ornament at a $10 profit per unit during the holiday season, but it takes a $5 loss per unit after the season is over. The following is the discrete probability distribution for the season’s demand:

Expected payoff if Q = 30: 0(0.2)+(150(0.3)+300(0.3+0.1+0.1) = $195Payoff if Q = 30 and D = 40: pD = 10(30) = $300Payoff if Q = 30 and D = 20: pD – l(Q – D)=10(20) – 5(30 – 20) = $150


Example D.3 OM Explorer Solution


Solved Problem 1

For Peachy Keen, Inc., the average demand for mohair sweaters is 100 per week. The production facility has the capacity to sew 400 sweaters per week. Setup cost is $351. The value of finished goods inventory is $40 per sweater. The annual per-unit inventory holding cost is 20 percent of the item’s value.

a. What is the economic production lot size (ELS)?

b. What is the average time between orders (TBO)?

c. What is the minimum total of the annual holding cost and setup cost?


Solved Problem 1

ELS = ELS = pp

pp – – dd22DDSS

HH

ELS = ELS = 400400(400 – 100)(400 – 100)

22(100)(52)(100)(52)($351)($351)0.20($40)0.20($40)

= 780 sweaters= 780 sweaters

TBOTBOELSELS = =ELSELS

DD780780

5,2005,200= 0.15 year or 7.8 weeks= 0.15 year or 7.8 weeks==

CC = = ( ( ))((HH) + () + (SS))DDQQ

Q pQ p – – d2 p 2 p

CC = = ( ( )) ( (0.20 x $400.20 x $40) + ($351)) + ($351)5,2005,200780780

780 400 – 780 400 – 100 2 400 400

= 2,340/year + $2,340/year= 2,340/year + $2,340/year = $4,680/year= $4,680/year

D = 5,200p = 400

d = 100S = $351

H = 20% of $40

a.

c. b.


Solved Problem 3

For Swell Productions, a concession stand will sell poodle skirts and other souvenirs of the 1950s a one-time event. Skirts are purchased for $40 each and are sold during for $75 each.

Unsold skirts can be returned for a refund of $30 each. Sales depend on the weather, attendance, and other variables.

The following table shows the probability of various sales quantities. How many skirts should be ordered?


Solved Problem 3

Probabilities0.05 0.11 0.34 0.34 0.11 0.05

The highest expected payoff occurs when 400 skirts are ordered.

Date post:	01-Apr-2015
Category:	Documents
Upload:	ashton-coles
View:	234 times
Download:	2 times

© 2007 Pearson Education Special Inventory Models Supplement D.

Documents