1 Chapter 16 When demands are unknown, expected values are the keys for deciding how much to order...

1

Chapter 16

When demands are unknown, expected values are the keys for deciding how much to order and how often.

Inventory Decisions

with Uncertain Factors

2

Inventory Decisions with Uncertain Factors

Two basic inventory decisions are evaluated: Single-period inventory—e.g., newspapers.

Probability distribution is for period’s demand. Multi-stage inventory—e.g., birthday cards.

Probability distribution is for lead-time demand. There are two demand probability distributions:

Deterministic (tabular). Continuous (normal curve).

There are two analytical approaches: Tabular: maximizing expected payoff Model: marginal analysis or EOQ.

Two cases are modeled: Backordering. Lost sales.

3

Making an Inventory Decision:Maximizing Expected Payoff

Problem: A drugstore stocks Fortunes.They sell for $3 and cost $2.10. Unsold copies are returned for $.70 credit. There are four levels of demand possible. Using profit as payoff, the following applies.

DemandEvent

Proba-bility

ACTS

Q = 20 Q = 21 Q = 22 Q = 23D = 20 .2 $18.00 $16.60 $15.20 $13.80D = 21 .4 8.00 18.90 17.50 16.10D = 22 .3 8.00 18.90 19.80 18.40D = 23 .1 8.00 18.90 19.80 20.70

4

Making an Inventory Decision:Maximizing Expected Payoff

Solution: The owner does not consider stocking less than the minimum demand or more than the maximum. (Why?)

The expected payoffs are computed for each possible order quantity:

Q = 20 Q = 21 Q = 22 Q = 23$18.00 $18.44 $17.90 $16.79

maximum According to the Bayes decision rule,

stocking 21 magazines is optimal. If the probabilities were long-run frequencies,

then doing so would maximize long-run profit. Maximizing expected payoff is assumed proper.

5

The Single-Period Model:The Newsvendor Problem

The payoff table approach can be cumber-some with many levels of demand.

The same result is achieved with a marginal analysis model. The decision variable is

Q = Order Quantity The model minimizes total expected cost

for the period, using parameters: c = Unit procurement costhE = Additional cost of each item held at

end of inventory cycle pS = Penalty for each item short pR = Selling price

The event variable is uncertain demand D.

6


The shortage penalty here applies regardless of duration of stockout.

Sales will equal D if demand falls at or below Q and Q if sales are greater.

If D < Q, there are Q D leftovers, each costing: hE + c

If D > Q, there are D Q shortages, each costing: pS + pR c

The objective is to minimize total expected cost:

where is the expected demand.

TEC(Q) = ]Pr[]Pr[0

dDQdcppdDdQchcQd

RS

Q

dE

7


This is the expression for optimal order quantity:

Problem: A newsvendor sells Wall Street Journals. She loses pS = $.02 in future profits each time a customer wants to buy a paper when out of stock. They sell for pR = $.23 and cost c = $.20. Unsold copies cost hE = $.01 to dispose. Demands between 21 and 30 are equally likely. How many should she stock?

Solution: The expected demand is = 25 copies.

Q* is the smallest possible demand such that

chcppcpp

*QDERS

RS

]Pr[

8


The following ratio is computed:

Each demand level has probability .1. The smallest cumulative probability exceeding this is .20, corresponding to 22 papers. Thus, Q* = 22.

The above is sensitive to the parameter levels. Raising pS to $.04 will increase Q* to 23. Raising pS to $.10 will increase Q* to 24.

192

2001202302202302

......

...chcpp

cpp

ERS

RS

9

Continuous Demand Distribution:Christmas Tree Problem

When demand is continuous the marginal analysis involves areas under normal curve.

Problem: Demand for noble firs is approximately normally distributed with = 2,000 and = 500. Trees sell for pR = $9 and cost c = $3. Loss of goodwill is pS = $1 per tree out of stock. Disposal cost is hE = $.50 per tree. How many trees should be stocked?Solution: The following applies:

This normal curve area corresponds to z = .43, and the demand at or beyond this determines Q*.Q* = + z = 2,000 + .43(500) = 2,215 trees

6667

350391391

..chcpp

cpp

ERS

RS

10

Continuous Demand Distribution:Christmas Tree Problem

The following is used in computing the total expected cost:

The above uses the expected shortage:

where L(x) is the tabled loss function.

QQ

LQ

QQ

LQB

][ QBcppQBQchc*QTEC RSE

11

Multiperiod Inventory Policies

When demand is uncertain, multiperiod inventory might look like this over time.

12

Multiperiod Inventory Policies

The multiperiod decisions involve two variables: Order quantity Q Reorder point r

The following parameters apply: A = mean annual demand rate k = ordering cost c = unit procurement cost pS = cost of short item (no matter how long) h = annual holding cost per dollar value = mean lead-time demand

13

Multiperiod Inventory Policies: Discrete Lead-Time Demand

The following is used to compute the expected shortage per inventory cycle:

The following is used to compute the total annual expected cost:

rd

L dDrdrB Pr

(With Backordering)

rBQA

prQ

hckQA

Q,rTEC S

2

14


Solution Algorithm. Calculate the starting order quantity:

Determine the reorder point r*:

Determine optimal order quantity:

r* is smallest level such that

(with backordering) Ap

hcQ*rD

S

1Pr

hcAk

Q2

1

hc

rBpkA*Q S 2

15


Recompute r* after getting Q*, and vice versa, until one of them stops changing.

Problem: Annual demand for printer cartridges costing c = $1.50 is A = 1,500. Ordering cost is k = $5 and holding cost is $.12 per dollar per year. Shortage cost is pS = $.12, no matter how long. Lead-time demand has the following distribution.

Find the optimal inventory policy.

16


Solution: The starting order quantity is:

Using the above, we compute:

The smallest cumulative lead-time demand probability > .93 is .95, corresponding to 7 cartridges. Thus, r* = 7 cartridges. We compute:B(7) = (8–7)(.03) + (9–7)(.01) + (10–7)(.01)=.08 and the optimal order quantity is:

28950112

5500121

..,

Q

9350015

289151211 .

,...

AphcQ

S

29050112

0850550012 ..

..,*Q

17


Solution (continued): Substituting the above into the expression used for finding r* the same value as before is found. r* does not change, and the optimal inventory policy is:

r* = 7 Q* = 290 The Lost Sales Case:

There is a new parameter: pR = selling price The condition for reorder point changes to:

r* is smallest level such that

(lost sales) AcpphcQ

Acpp*rD

RS

RS

Pr

18


The Lost Sales Case (continued): The optimal order quantity expression is:

(lost sales)

hc

rBcppkA*Q RS 2

19

Multiperiod Inventory Policies: Continuous Lead-Time Demand

The formulas and algorithms for the continuous case are the same, except for the expected shortage:

where and are the parameters of the normal lead-time demand distribution and L(x) is the tabled losss function.

rr

Lr

rr

LrB

20

Inventory Spreadsheet Templates

Payoff Table

Newsvendor

Christmas Tree

Multiperiod Discrete Backordering

Multiperiod Discrete Lost Sales

Multiperiod Normal Backordering

Multiperiod Normal Lost Sales

21

Payoff Table(Figure 16-1)

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101112131415161718

A B C D E F

PROBLEM: Fortune Magazine

Act 1 Act 2 Act 3 Act 4Events Probability Q = 20 Q = 21 Q = 22 Q = 23

1 D = 20 0.2 $18.00 $16.60 $15.20 $13.802 D = 21 0.4 $18.00 $18.90 $17.50 $16.103 D = 22 0.3 $18.00 $18.90 $19.80 $18.404 D = 23 0.1 $18.00 $18.90 $19.80 $20.70

Act 1 Act 2 Act 3 Act 4Q = 20 Q = 21 Q = 22 Q = 23

Expected Payoff $18.00 $18.44 $17.96 $16.79

PAYOFF TABLE EVALUATION

Problem Data

Act Summary

1. Enter problem name in B3.

1. Enter problem name in B3.

2. Enter data in B9:F12 and labels in A9:A12 and C8:F8.

2. Enter data in B9:F12 and labels in A9:A12 and C8:F8.

3. If more events or acts are required, expand the table by inserting additional rows and/or columns. Make sure the formulas in the Act Summary table include all the rows of the expanded table.

3. If more events or acts are required, expand the table by inserting additional rows and/or columns. Make sure the formulas in the Act Summary table include all the rows of the expanded table.

4. Expected payoffs4. Expected payoffs

18C

=SUMPRODUCT($B$9:$B$12,C9:C12)

Copy cell C18 over to D18:F18.

Copy cell C18 over to D18:F18.

22

Newsvendor Problem (Figure 16-3)

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8

910111213141516171819202122232425262728293031

A B C D E F G

PROBLEM: Wall Street Journal

Parameter Values:Cost per Item Procured: c = 0.20Additional Cost for Each Leftover Item Held: hE = 0.01

Penalty for Each Item Short: pS = 0.02

Selling Price per Unit: pR = 0.23Number of demands for probability distribution = 11

Optimal Values:Optimal Order Quantity: Q* = 47Expected Demand: mu = 49.5Total Expected Cost: TEC(Q*) = $10.07Expected Shortages: B(Q*) = 2.66Probability of Shortage: P[D>Q*] = 0.80

Cumulative Number ofDemand Probability Probability shortages

45 0.05 0.05 0.046 0.06 0.11 0.047 0.09 0.20 0.048 0.12 0.32 1.049 0.17 0.49 2.050 0.20 0.69 3.051 0.12 0.81 4.052 0.08 0.89 5.053 0.06 0.95 6.054 0.04 0.99 7.055 0.01 1.00 8.0

SINGLE PERIOD INVENTORY MODEL -- NEWSVENDOR PROBLEM

1. Enter the problem name in C3.


2. Enter the problem parameters in G6:G10.


3. Enter the demands and probabilities in C21:D40.

3. Enter the demands and probabilities in C21:D40.

4. If the number of demands for probability distribution is greater than 20 add the appropriate number of rows and copy the formulas in columns E and F down for these rows.


5. To calculate the expected profit, enter =SUMPRODUCT(C21:C40,D21:D40)*G9-G15 in cell G18.

5. To calculate the expected profit, enter =SUMPRODUCT(C21:C40,D21:D40)*G9-G15 in cell G18.

6. Optimal values: Q*, mu, TEC(Q*), B(Q*), PD>Q*.


23

Newsvendor Formulas

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8

910111213141516171819202122232425262728293031

A B C D E F G

PROBLEM: Wall Street Journal

Parameter Values:Cost per Item Procured: c = 0.20Additional Cost for Each Leftover Item Held: hE = 0.01


Selling Price per Unit: pR = 0.23Number of demands for probability distribution = 11

Optimal Values:Optimal Order Quantity: Q* = 47Expected Demand: mu = 49.5Total Expected Cost: TEC(Q*) = $10.07Expected Shortages: B(Q*) = 2.66Probability of Shortage: P[D>Q*] = 0.80

Cumulative Number ofDemand Probability Probability shortages

45 0.05 0.05 0.046 0.06 0.11 0.047 0.09 0.20 0.048 0.12 0.32 1.049 0.17 0.49 2.050 0.20 0.69 3.051 0.12 0.81 4.052 0.08 0.89 5.053 0.06 0.95 6.054 0.04 0.99 7.055 0.01 1.00 8.0

SINGLE PERIOD INVENTORY MODEL -- NEWSVENDOR PROBLEM1314151617

G

=INDEX(C21:C40,MATCH(F13LOOKUP((G8+G9-G6)/(G8+G9+G7),E21:E40,C21:C40),C21:C40)+1)=SUMPRODUCT(C21:C40,D21:D40)=G6*G14+(G7+G6)*(G13-G14+G16)+(G8+G9-G6)*G16=SUMPRODUCT(D21:D40,F21:F40)=1-VLOOKUP(G13,C21:F40,3)

21

22

E F

=IF(ROW(C21)-20<=$G$10,D21,"")

=IF(ROW(C21)-20<=$G$10,IF($G$13>C2

1,0,C21-$G$13),"")

=IF(ROW(C22)-20<=$G$10,D22+E21,"")

=IF(ROW(C22)-20<=$G$10,IF($G$13>C2

2,0,C22-$G$13),"")

24

Christmas Tree Problem (Figure 16-6)

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10

1112131415161718

A B C D E F G

PROBLEM: Noble Fir

Parameter Values:Mean of Demand Distribution: mu = 2000Stand. Deviation of Demand Distribution: sigma = 500Cost per Item Procured: c = 3.00Additional Cost for Each Leftover Item Held: hE = 0.50


Selling Price per Unit: pR = 9.00

Optimal Values:Optimal Order Quantity: Q* = 2215Expected Demand: mu = 2000Total Expected Cost: TEC(Q*) = $7,910.35Expected Shortages: B(Q*) = 110.15Probability of Shortage: P[D>Q*] = 0.33

SINGLE PERIOD INVENTORY MODEL - CHRISTMAS TREE PROBLEM



The Normal Loss Table L(D) is on the next worksheet. It is used in the spreadsheet calculations.

The Normal Loss Table L(D) is on the next worksheet. It is used in the spreadsheet calculations.





25

The Normal Loss Table L(D)

The Normal Loss Table L(D) is used the calculations in the Christmas Tree template.

The Normal Loss Table L(D) is used the calculations in the Christmas Tree template.

1

23456

102103104105106202203204205206497498499500501

A BD L(D)

0.00 0.39890.01 0.39400.02 0.38900.03 0.38410.04 0.37931.00 0.083321.01 0.081741.02 0.080191.03 0.078661.04 0.077162.00 8.491E-032.01 8.266E-032.02 8.046E-032.03 7.832E-032.04 7.623E-034.95 6.982E-084.96 6.620E-084.97 6.276E-084.98 5.950E-084.99 5.640E-08

Note that many rows have been hidden because the entire table is too big to show on one page.

Note that many rows have been hidden because the entire table is too big to show on one page.

26

Christmas Tree Formulas

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10

1112131415161718

A B C D E F G

PROBLEM: Noble Fir

Parameter Values:Mean of Demand Distribution: mu = 2000Stand. Deviation of Demand Distribution: sigma = 500Cost per Item Procured: c = 3.00Additional Cost for Each Leftover Item Held: hE = 0.50


Selling Price per Unit: pR = 9.00

Optimal Values:Optimal Order Quantity: Q* = 2215Expected Demand: mu = 2000Total Expected Cost: TEC(Q*) = $7,910.35Expected Shortages: B(Q*) = 110.15Probability of Shortage: P[D>Q*] = 0.33

SINGLE PERIOD INVENTORY MODEL - CHRISTMAS TREE PROBLEM

141516

1718

F=NORMINV((G10+G11-G8)/(G10+G11+G9),G6,G7)=G6=G8*F15+(G9+G8)*(F14-F15+F17)+(G10+G11-G8)*F17

=IF(F14<G6,G6-F14+G7*VLOOKUP((G6-F14)/G7,'L(D)'!A2:B501,2),G7*VLOOKUP((F14-G6)/G7,'L(D)'!A2:B501,2))=1-NORMDIST(F14,G6,G7,TRUE)

‘L(D)’!A2:B501 refers to the normal loss table L(D) table located on the L(D) worksheet

‘L(D)’!A2:B501 refers to the normal loss table L(D) table located on the L(D) worksheet

27


The solution to multiperiod models with discrete lead-time demand and backordering is based on the newsvendor spreadsheet. It varies in two respects:

some formulas are a little different (described in Appendix 16-1)

it contains many worksheets because of the iterative nature of the solution process.

Ten iterations are done in this spreadsheet. This is sufficient for all problems in the book and will solve most other multiperiod, discrete, backordering models. However, addition iterations can be added whenever necessary.

Ten iterations are done in this spreadsheet. This is sufficient for all problems in the book and will solve most other multiperiod, discrete, backordering models. However, addition iterations can be added whenever necessary.

28


Each of the ten worksheets appear as tabs in the spreadsheet, numbered 1, 2, 3, . . . , 10. The problem data is entered in worksheet 1 (tab 1). Intermediate solution results for iteration 1 are on tab 1, the results for iteration 2 are on tab 2, and so forth up to the results for iteration 10 which appear on tab 10. An optimal solution is obtained when the results converge and do not vary with increasing iterations. Normally, an optimal solution is obtained after 2 or 3 iterations.

A summary worksheet is provided after the iterations. It summarizes the intermediate results of all the iterations.

A summary worksheet is provided after the iterations. It summarizes the intermediate results of all the iterations.

29

Multiperiod Discrete BackorderingIteration 1

4. Enter the demands and probabilities inC23:D42.

4. Enter the demands and probabilities inC23:D42.

12345678910111213141516171819202122

2324252627282930313233

A B C D E F GMULTI-PERIOD EOQ MODEL (Backordering) - DISCRETE LEAD-TIME DEMAND

PROBLEM: Printer Cartridges

Parameter ValuesFixed Cost per Order: k = 5Annual Demand Rate: A = 1500Unit cost of Procuring an Item: c = 1.5Annual Holding Cost per Dollar Value: h = 0.12Shortage Cost per Unit: pS = 0.5Number of demands for probability distribution = 11

Optimal Values:Optimal Order Quantity: Q* = 288.68Optimal Reorder Point: r* = 7Expected Lead-Time Demand: mu = 4Total Expected Cost: TEC(Q*) = 52.7094$ Expected Shortage: B(r*) = 0.08Probability of Shortage: P[D>r*] = 0.05

Cumulative Number ofDemand Probability Probability Shortages

0 0.01 0.01 01 0.07 0.08 02 0.16 0.24 03 0.20 0.44 04 0.19 0.63 05 0.16 0.79 06 0.10 0.89 07 0.06 0.95 08 0.03 0.98 19 0.01 0.99 210 0.01 1.00 3

1. Start with worksheet 1 (tab 1). It gives the results of the first iteration.

1. Start with worksheet 1 (tab 1). It gives the results of the first iteration.

2. Enter the problem name in B3.

2. Enter the problem name in B3.



6. Iteration 1 results are here

6. Iteration 1 results are here



30

Multiperiod Discrete Backordering(Figure 16-8)

1. Tab 2 gives the results of the second iteration, tab 3 the results of the 3rd iteration, etc.

1. Tab 2 gives the results of the second iteration, tab 3 the results of the 3rd iteration, etc.

2. The optimal solution occurs when the answers do not change from iteration to iteration.

2. The optimal solution occurs when the answers do not change from iteration to iteration.

3. To quickly find the optimal solution skip to the last iteration by clicking on tab 10 (shown here).


4. Optimal values: Q*, r*, mu, TEC(Q*), B(Q*), PD>Q*.


123456789

10

111213141516171819202122

2324252627282930313233



Parameter ValuesFixed Cost per Order: k = 5Annual Demand Rate: A = 1500Unit cost of Procuring an Item: c = 1.5Annual Holding Cost per Dollar Value: h = 0.12Shortage Cost per Unit: pS = 0.5

Number of demands for probability distribution = 11

Optimal Values:Optimal Order Quantity: Q* = 290Optimal Reorder Point: r* = 7Expected Lead-Time Demand: mu = 4Total Expected Cost: TEC(Q*) = 52.71$ Expected Shortage: B(r*) = 0.08Probability of Shortage: P[D>r*] = 0.05


0 0.01 0.01 01 0.07 0.08 02 0.16 0.24 03 0.20 0.44 04 0.19 0.63 05 0.16 0.79 06 0.10 0.89 07 0.06 0.95 08 0.03 0.98 19 0.01 0.99 2

10 0.01 1.00 3

14

1516

171819

G=SQRT((2*G7*(G6+'9'!G18*G10))/(G9*G8))

=INDEX(C23:C42,MATCH(LOOKUP(1-((G9*G8*G14)/(G10*G7)),E23:E42,C23:C42),C23:C42,0)+1)=SUMPRODUCT(C23:C42,D23:D42)

=(G7/G14)*G6+G9*G8*(G14/2+G15-G16)+G10*(G7/G14)*G18=SUMPRODUCT(D23:D42,F23:F42)=1-VLOOKUP(G15,C23:E42,3)

31

12345678910111213

14151617181920212223




Iteration, i Qi ri B(ri) TEC(Qi,ri)

1 289 7 0.08 52.71$ 2 290 7 0.08 52.71$ 3 290 7 0.08 52.71$ 4 290 7 0.08 52.71$ 5 290 7 0.08 52.71$ 6 290 7 0.08 52.71$ 7 290 7 0.08 52.71$ 8 290 7 0.08 52.71$ 9 290 7 0.08 52.71$ 10 290 7 0.08 52.71$

Multiperiod Discrete BackorderingSummary

To quickly find the optimal solution click on the Summary tab. It provides a summary of all the 10 iterations.


Notice the answers do not change after the second iteration.


32

Multiperiod Discrete BackorderingIteration 1 Formulas

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2324252627282930313233




Optimal Values:Optimal Order Quantity: Q* = 288.68Optimal Reorder Point: r* = 7Expected Lead-Time Demand: mu = 4Total Expected Cost: TEC(Q*) = 52.7094$ Expected Shortage: B(r*) = 0.08Probability of Shortage: P[D>r*] = 0.05


0 0.01 0.01 01 0.07 0.08 02 0.16 0.24 03 0.20 0.44 04 0.19 0.63 05 0.16 0.79 06 0.10 0.89 07 0.06 0.95 08 0.03 0.98 19 0.01 0.99 210 0.01 1.00 3

14

1516

171819

G=SQRT((2*G7*G6)/(G9*G8))



33

123456789

10

111213141516171819202122

2324252627282930313233







0 0.01 0.01 01 0.07 0.08 02 0.16 0.24 03 0.20 0.44 04 0.19 0.63 05 0.16 0.79 06 0.10 0.89 07 0.06 0.95 08 0.03 0.98 19 0.01 0.99 2

10 0.01 1.00 3

14

1516

171819

G=SQRT((2*G7*(G6+'9'!G18*G10))/(G9*G8))



Multiperiod Discrete BackorderingIteration 10 Formulas

Only one formula changes on the iteration 2 - 10 worksheets, in cell G14. The formula in this cell always refers back the the previous iteration. For example, the worksheet shown here is for iteration 10 so the formula in cell G14 refers back to iteration 9.


The term ‘9’!G18 means the value of G18 (expected number of shortages) from iteration 9.


14G

=SQRT((2*G7*(G6+'9'!G18*G10))/(G9*G8))

34

Multiperiod Discrete Lost Sales

The solution to multiperiod models with discrete lead-time demand and lost sales is based on the backordering case just described. It varies only in that some formulas are different (described in Appendix 16-1).

35

Multiperiod Discrete Lost Sales(Figure 16-9)

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10

1112131415161718192021222324252627282930313233

A B C D E F GMULTI-PERIOD EOQ MODEL (Lost Sales) - DISCRETE LEAD-TIME DEMAND

PROBLEM: Compact Disks


Shortage Cost per Unit: pR = 0.9Number of demands for probability distribution = 10



90 0.05 0.05 0100 0.12 0.17 0110 0.17 0.34 0120 0.22 0.56 0130 0.19 0.75 0140 0.14 0.89 0150 0.05 0.94 0160 0.03 0.97 0170 0.02 0.99 10180 0.01 1.00 20

1. Start with worksheet 1 (tab 1) and enter the problem name in B3, the problem parameters in G6:G12, and the demands and probabilities inC24:D43.

1. Start with worksheet 1 (tab 1) and enter the problem name in B3, the problem parameters in G6:G12, and the demands and probabilities inC24:D43.





36

Multiperiod Discrete Lost SalesSummary

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10

11

12131415

16171819202122232425


PROBLEM: Compact Discs


Shortage Cost per Unit: pR = 0.9



1 1925 160 0.40 106.48$ 2 1935 160 0.40 106.48$ 3 1935 160 0.40 106.48$ 4 1935 160 0.40 106.48$ 5 1935 160 0.40 106.48$ 6 1935 160 0.40 106.48$ 7 1935 160 0.40 106.48$ 8 1935 160 0.40 106.48$ 9 1935 160 0.40 106.48$

10 1935 160 0.40 106.48$





37

Multiperiod Discrete Lost SalesIteration 1 Formulas

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10

11

121314151617181920212223

24252627282930313233


PROBLEM: Compact Discs


Shortage Cost per Unit: pR = 0.9




90 0.05 0.05 0100 0.12 0.17 0110 0.17 0.34 0120 0.22 0.56 0130 0.19 0.75 0140 0.14 0.89 0150 0.05 0.94 0160 0.03 0.97 0170 0.02 0.99 10180 0.01 1.00 20

15

1617

181920

G=SQRT((2*G7*G6)/(G9*G8))

=INDEX(C24:C43,MATCH(LOOKUP((G10+G11-G8)*G7/(G9*G8*G15+(G10+G11-G8)*G7),E24:E43,C24:C43),C24:C43,0)+1)=SUMPRODUCT(C24:C43,D24:D43)

=((G7*(1-G19/G15))/G15)*G6+G9*G8*(G15/2+G16-G17+G19)+(G10+G11-G8)*(G7*(1-G19/G15)/G15)*G19=SUMPRODUCT(D24:D43,F24:F43)=1-VLOOKUP(G16,C24:E43,3)

38

Multiperiod Discrete Lost SalesIteration 10 Formulas

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10

1112131415161718192021222324252627282930313233


PROBLEM: Compact Disks


Shortage Cost per Unit: pR = 0.9Number of demands for probability distribution = 10



90 0.05 0.05 0100 0.12 0.17 0110 0.17 0.34 0120 0.22 0.56 0130 0.19 0.75 0140 0.14 0.89 0150 0.05 0.94 0160 0.03 0.97 0170 0.02 0.99 10180 0.01 1.00 20

15G

=SQRT((2*G7*(G6+'9'!G19*(G10+G11-G8))/(G9*G8)))





39

Multiperiod Normal Backordering

The solution to multiperiod models with normal lead-time demand and backordering is a variation of the Christmas Tree template and it incorporates features from the multiperiod, discrete leadtime template. The formulas are described in Appendix 16-1.

40

Multiperiod Normal Backordering(Figure 16-10)

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101112

1314151617181920

A B C D E F G

PROBLEM: Unleaded Gas at Oil Refinery

Parameter Values:Mean of Demand Distribution: mu = 200,000Stand. Deviation of Demand Distribution: sigma = 20,000Fixed Cost per Order: k = 1,000Annual Demand Rate: A = 40,000,000Unit Cost of Procuring an Item: c = 0.40Annual Holding Cost per Dollar Value: h = 0.20Shortage Cost per Unit: pS = 0.05

Optimal Values:Optimal Order Quantity: Q* = 1,008,256Optimal Reorder Point: r* = 234,937Expected Demand: mu = 200,000Total Expected Cost: TEC(Q*) = 83,455.46Expected Shortages: B(r*) = 331.60Probability of Shortage: P[D>r*] = 0.04

MULTI-PERIOD EOQ MODEL (Backordering) - NORMAL LEAD-TIME DEMAND1. Start with worksheet 1 (tab 1) and enter the problem name in B3 and the problem parameters in G6:G12.

1. Start with worksheet 1 (tab 1) and enter the problem name in B3 and the problem parameters in G6:G12.



3. Optimal values: Q*, r*, mu, TEC(Q*), B(Q*), P D>Q*.3. Optimal values: Q*, r*, mu, TEC(Q*), B(Q*), P D>Q*.

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Multiperiod Normal BackorderingSummary

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A B C D E F GMULTI-PERIOD EOQ MODEL (Backordering) - NORMAL LEAD-TIME DEMAND




1 1,000,000 235,014 323 83,447.90$ 2 1,008,053 234,939 332 83,455.61$ 3 1,008,256 234,937 332 83,455.46$ 4 1,008,256 234,937 332 83,455.46$ 5 1,008,256 234,937 332 83,455.46$ 6 1,008,256 234,937 332 83,455.46$ 7 1,008,256 234,937 332 83,455.46$ 8 1,008,256 234,937 332 83,455.46$ 9 1,008,256 234,937 332 83,455.46$

10 1,008,256 234,937 332 83,455.46$



Notice the answers do not change after the third iteration.

Notice the answers do not change after the third iteration.

42

Multiperiod Normal BackorderingIteration 1 Formulas

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A B C D E F GMULTI-PERIOD EOQ MODEL (Backordering) - NORMAL LEAD-TIME DEMAND



Optimal Values:Optimal Order Quantity: Q* = 1,000,000 Optimal Reorder Point: r* = 235,014 Expected Demand: mu = 200,000 Total Expected Cost: TEC(Q*) = 83,447.90$ Expected Shortages: B(r*) = 323.40Probability of Shortage: P[D>r*] = 0.04

151617

18

1920

F=SQRT((2*G9*G8)/(G11*G10))=NORMINV(1-((G11*G10*F15)/(G12*G9)),G6,G7)=G6

=(G9/F15)*G8+G11*G10*(F15/2+F16-G6)+G12*(G9/F15)*F19

=IF(F16<F17,F17-F16+G7*VLOOKUP((F17-F16)/G7,'L(D)'!A2:B501,2),G7*VLOOKUP((F16-F17)/G7,'L(D)'!A2:B501,2))=1-NORMDIST(F16,G6,G7,TRUE)

43

Multiperiod Normal BackorderingIteration 10 Formulas

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A B C D E F G



Optimal Values:Optimal Order Quantity: Q* = 1,008,256Optimal Reorder Point: r* = 234,937Expected Demand: mu = 200,000Total Expected Cost: TEC(Q*) = 83,455.46Expected Shortages: B(r*) = 331.60Probability of Shortage: P[D>r*] = 0.04

MULTI-PERIOD EOQ MODEL (Backordering) - NORMAL LEAD-TIME DEMAND

15F

=SQRT((2*G9*(G8+G12*'9'!F19))/(G11*G10))

Only one formula changes on the iteration 2 - 10 worksheets, in cell F15. The formula in this cell always refers back the the previous iteration. For example, the worksheet shown here is for iteration 10 so the formula in cell F15 refers back to iteration 9.


The term ‘9’!F19 means the value of F19 (expected number of shortages) from iteration 9.


44

Multiperiod Normal Lost Sales

The solution to multiperiod models with normal lead-time demand and lost sales is based on the backordering case just described. It varies only in that some formulas are different (described in Appendix 16-1).

45

Multiperiod Normal Lost Sales(Figure 16-11)

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A B C D E F G

PROBLEM: Roger's Sentinel Station

Parameter Values:Mean of Demand Distribution: mu = 1,000Stand. Deviation of Demand Distribution: sigma = 50Fixed Cost per Order: k = 100Annual Demand Rate: A = 500,000Unit Cost of Procuring an Item: c = 1.48Annual Holding Cost per Dollar Value: h = 0.19Shortage Cost per Unit: pS = 0.25Selling Price per Unit: pR = 1.75

Optimal Values:Optimal Order Quantity: Q* = 18,876Optimal Reorder Point: r* = 1,103Expected Demand: mu = 1,000Total Expected Cost: TEC(Q*) = 5,336.87Expected Shortages: B(r*) = 0.37Probability of Shortage: P[D>r*] = 0.02

MULTI-PERIOD EOQ MODEL (Lost Sales) - NORMAL LEAD-TIME DEMAND





3. Optimal values: Q*, r*, mu, TEC(Q*), B(Q*), P D>Q*.3. Optimal values: Q*, r*, mu, TEC(Q*), B(Q*), P D>Q*.

46

Multiperiod Normal Lost SalesSummary

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A B C D E F GMULTI-PERIOD EOQ MODEL (Lost Sales) - NORMAL LEAD-TIME DEMAND


Parameter Values:Mean of Demand Distribution: mu = 1,000 Stand. Deviation of Demand Distribution: sigma = 50 Fixed Cost per Order: k = 100 Annual Demand Rate: A = 500,000 Unit Cost of Procuring an Item: c = 1.48Annual Holding Cost per Dollar Value: h = 0.19Shortage Cost per Unit: pS = 0.25

Iteration, i Qi ri B(ri) TEC(Qi,ri)1 18,858 1,103 0.37 5,336.87$ 2 18,876 1,103 0.37 5,336.87$ 3 18,876 1,103 0.37 5,336.87$ 4 18,876 1,103 0.37 5,336.87$ 5 18,876 1,103 0.37 5,336.87$ 6 18,876 1,103 0.37 5,336.87$ 7 18,876 1,103 0.37 5,336.87$ 8 18,876 1,103 0.37 5,336.87$ 9 18,876 1,103 0.37 5,336.87$

10 18,876 1,103 0.37 5,336.87$





47

Multiperiod Normal Lost SalesIteration 1 Formulas

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A B C D E F GMULTI-PERIOD EOQ MODEL (Lost Sales) - NORMAL LEAD-TIME DEMAND


Parameter Values:Mean of Demand Distribution: mu = 1,000 Stand. Deviation of Demand Distribution: sigma = 50 Fixed Cost per Order: k = 100 Annual Demand Rate: A = 500,000 Unit Cost of Procuring an Item: c = 1.48Annual Holding Cost per Dollar Value: h = 0.19Shortage Cost per Unit: pS = 0.25Selling Price per Unit: pR = 1.75

Optimal Values:Optimal Order Quantity: Q* = 18,858 Optimal Reorder Point: r* = 1,103 Expected Demand: mu = 1,000 Total Expected Cost: TEC(Q*) = 5,336.87 Expected Shortages: B(r*) = 0.37Probability of Shortage: P[D>r*] = 0.02

16

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19

2021

F=SQRT((2*G9*G8)/(G11*G10))

=NORMINV((G12+G13-G10)*G9/(G11*G10*F16+(G12+G13-G10)*G9),G6,G7)=G6

=(G9*(1-F20/F16))/F16*G8+G11*G10*(F16/2+F17-F18+F20)+(G12+G13-G10)*(G9*(1-F20/F16)/F16)*F20

=IF(F17<F18,F18-F17+G7*VLOOKUP((F18-F17)/G7,'L(D)'!A2:B501,2),G7*VLOOKUP((F17-F18)/G7,'L(D)'!A2:B501,2))=1-NORMDIST(F17,G6,G7,TRUE)

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Multiperiod Normal Lost SalesIteration 10 Formulas

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A B C D E F G


Parameter Values:Mean of Demand Distribution: mu = 1,000Stand. Deviation of Demand Distribution: sigma = 50Fixed Cost per Order: k = 100Annual Demand Rate: A = 500,000Unit Cost of Procuring an Item: c = 1.48Annual Holding Cost per Dollar Value: h = 0.19Shortage Cost per Unit: pS = 0.25Selling Price per Unit: pR = 1.75

Optimal Values:Optimal Order Quantity: Q* = 18,876Optimal Reorder Point: r* = 1,103Expected Demand: mu = 1,000Total Expected Cost: TEC(Q*) = 5,336.87Expected Shortages: B(r*) = 0.37Probability of Shortage: P[D>r*] = 0.02

MULTI-PERIOD EOQ MODEL (Lost Sales) - NORMAL LEAD-TIME DEMAND





16F

=SQRT((2*G9*(G8+'9'!F20*(G12+G13-G10))/(G11*G10)))

1 Chapter 16 When demands are unknown, expected values are the keys for deciding how much to order...

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