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Transcript of 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 Single-Period Model:The Newsvendor Problem
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
The Single-Period Model:The Newsvendor Problem
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 Single-Period Model:The Newsvendor Problem
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.
LQ
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
Multiperiod Inventory Policies: Discrete Lead-Time Demand
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
Multiperiod Inventory Policies: Discrete Lead-Time Demand
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
Multiperiod Inventory Policies: Discrete Lead-Time Demand
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
Multiperiod Inventory Policies: Discrete Lead-Time Demand
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
Multiperiod Inventory Policies: Discrete Lead-Time Demand
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)
123456789
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)
1234567
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.
1. Enter the problem name in C3.
2. Enter the problem parameters in G6:G10.
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.
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*.
6. Optimal values: Q*, mu, TEC(Q*), B(Q*), PD>Q*.
23
Newsvendor Formulas
1234567
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*) = $10.07Expected Shortages: B(Q*) = 2.66Probability of Shortage: P[D>Q*] = 0.80
Cumulative Number ofDemand Probability Probability shortages
45 0.05 0.05 0.046 0.06 0.11 0.047 0.09 0.20 0.048 0.12 0.32 1.049 0.17 0.49 2.050 0.20 0.69 3.051 0.12 0.81 4.052 0.08 0.89 5.053 0.06 0.95 6.054 0.04 0.99 7.055 0.01 1.00 8.0
SINGLE PERIOD INVENTORY MODEL -- NEWSVENDOR PROBLEM1314151617
G
=INDEX(C21:C40,MATCH(F13LOOKUP((G8+G9-G6)/(G8+G9+G7),E21:E40,C21:C40),C21:C40)+1)=SUMPRODUCT(C21:C40,D21:D40)=G6*G14+(G7+G6)*(G13-G14+G16)+(G8+G9-G6)*G16=SUMPRODUCT(D21:D40,F21:F40)=1-VLOOKUP(G13,C21:F40,3)
21
22
E F
=IF(ROW(C21)-20<=$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)
123456789
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
Penalty for Each Item Short: pS = 1.00
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
1. Enter the problem name in C3.
1. Enter the problem name in C3.
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.
2. Enter the problem parameters in G6:G11.
2. Enter the problem parameters in G6:G11.
3. Optimal values: Q*, mu, TEC(Q*), B(Q*), PD>Q*.
3. Optimal values: Q*, mu, TEC(Q*), B(Q*), PD>Q*.
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
123456789
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
Penalty for Each Item Short: pS = 1.00
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
Multiperiod Discrete Backordering
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
Multiperiod Discrete Backordering
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.
3. Enter the problem parameters in G6:G11.
3. Enter the problem parameters in G6:G11.
6. Iteration 1 results are here
6. Iteration 1 results are here
5. 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. 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.
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).
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*.
4. Optimal values: Q*, r*, mu, TEC(Q*), B(Q*), PD>Q*.
123456789
10
111213141516171819202122
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 = 0.5
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
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 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
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
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.
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.
Notice the answers do not change after the second iteration.
32
Multiperiod Discrete BackorderingIteration 1 Formulas
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
14
1516
171819
G=SQRT((2*G7*G6)/(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)
33
123456789
10
111213141516171819202122
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 = 0.5
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
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 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)
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.
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.
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)
123456789
10
1112131415161718192021222324252627282930313233
A B C D E F GMULTI-PERIOD EOQ MODEL (Lost Sales) - DISCRETE LEAD-TIME DEMAND
PROBLEM: Compact Disks
Parameter ValuesFixed Cost per Order: k = 20Annual Demand Rate: A = 5000Unit cost of Procuring an Item: c = 0.45Annual Holding Cost per Dollar Value: h = 0.12Shortage Cost per Unit: pS = 0.1
Shortage Cost per Unit: pR = 0.9Number of demands for probability distribution = 10
Optimal Values:Optimal Order Quantity: Q* = 1935Optimal Reorder Point: r* = 160Expected Lead-Time Demand: mu = 123Total Expected Cost: TEC(Q*) = 106.48$ Expected Shortage: B(r*) = 0.40Probability of Shortage: P[D>r*] = 0.03
Cumulative Number ofDemand Probability Probability Shortages
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.
2. To quickly find the optimal solution skip to the last iteration by clicking on tab 10 (shown here).
2. To quickly find the optimal solution skip to the last iteration by clicking on tab 10 (shown here).
3. Optimal values: Q*, r*, mu, TEC(Q*), B(Q*), PD>Q*.
3. Optimal values: Q*, r*, mu, TEC(Q*), B(Q*), PD>Q*.
36
Multiperiod Discrete Lost SalesSummary
123456789
10
11
12131415
16171819202122232425
A B C D E F GMULTI-PERIOD EOQ MODEL (Backordering) - DISCRETE LEAD-TIME DEMAND
PROBLEM: Compact Discs
Parameter ValuesFixed Cost per Order: k = 20Annual Demand Rate: A = 5000Unit cost of Procuring an Item: c = 0.45Annual Holding Cost per Dollar Value: h = 0.12Shortage Cost per Unit: pS = 0.1
Shortage Cost per Unit: pR = 0.9
Number of demands for probability distribution = 10
Iteration, i Qi ri B(ri) TEC(Qi,ri)
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$
To quickly find the optimal solution click on the Summary tab. It provides a summary of all the 10 iterations.
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.
Notice the answers do not change after the second iteration.
37
Multiperiod Discrete Lost SalesIteration 1 Formulas
123456789
10
11
121314151617181920212223
24252627282930313233
A B C D E F GMULTI-PERIOD EOQ MODEL (Lost Sales) - DISCRETE LEAD-TIME DEMAND
PROBLEM: Compact Discs
Parameter ValuesFixed Cost per Order: k = 20Annual Demand Rate: A = 5000Unit cost of Procuring an Item: c = 0.45Annual Holding Cost per Dollar Value: h = 0.12Shortage Cost per Unit: pS = 0.1
Shortage Cost per Unit: pR = 0.9
Number of demands for probability distribution = 10
Optimal Values:Optimal Order Quantity: Q* = 1925Optimal Reorder Point: r* = 160Expected Lead-Time Demand: mu = 123Total Expected Cost: TEC(Q*) = 106.48$ Expected Shortage: B(r*) = 0.40Probability of Shortage: P[D>r*] = 0.03
Cumulative Number ofDemand Probability Probability Shortages
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
123456789
10
1112131415161718192021222324252627282930313233
A B C D E F GMULTI-PERIOD EOQ MODEL (Lost Sales) - DISCRETE LEAD-TIME DEMAND
PROBLEM: Compact Disks
Parameter ValuesFixed Cost per Order: k = 20Annual Demand Rate: A = 5000Unit cost of Procuring an Item: c = 0.45Annual Holding Cost per Dollar Value: h = 0.12Shortage Cost per Unit: pS = 0.1
Shortage Cost per Unit: pR = 0.9Number of demands for probability distribution = 10
Optimal Values:Optimal Order Quantity: Q* = 1935Optimal Reorder Point: r* = 160Expected Lead-Time Demand: mu = 123Total Expected Cost: TEC(Q*) = 106.48$ Expected Shortage: B(r*) = 0.40Probability of Shortage: P[D>r*] = 0.03
Cumulative Number ofDemand Probability Probability Shortages
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)))
Only one formula changes on the iteration 2 - 10 worksheets, in cell G15. 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 G15 refers back to iteration 9.
Only one formula changes on the iteration 2 - 10 worksheets, in cell G15. 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 G15 refers back to iteration 9.
The term ‘9’!G19 means the value of G19 (expected number of shortages) from iteration 9.
The term ‘9’!G19 means the value of G19 (expected number of shortages) from iteration 9.
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)
123456789
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.
2. To quickly find the optimal solution skip to the last iteration by clicking on tab 10 (shown here).
2. To quickly find the optimal solution skip to the last iteration by clicking on tab 10 (shown here).
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*.
41
Multiperiod Normal BackorderingSummary
123456789
101112
131415
16171819202122232425
A B C D E F GMULTI-PERIOD EOQ MODEL (Backordering) - NORMAL LEAD-TIME DEMAND
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
Iteration, i Qi ri B(ri) TEC(Qi,ri)
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$
To quickly find the optimal solution click on the Summary tab. It provides a summary of all the 10 iterations.
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 third iteration.
Notice the answers do not change after the third iteration.
42
Multiperiod Normal BackorderingIteration 1 Formulas
123456789
101112
1314151617181920
A B C D E F GMULTI-PERIOD EOQ MODEL (Backordering) - NORMAL LEAD-TIME DEMAND
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* = 1,000,000 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
123456789
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 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.
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.
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)
123456789
10111213
1415161718192021
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
1. Start with worksheet 1 (tab 1) and enter the problem name in B3 and the problem parameters in G6:G13.
1. Start with worksheet 1 (tab 1) and enter the problem name in B3 and the problem parameters in G6:G13.
2. To quickly find the optimal solution skip to the last iteration by clicking on tab 10 (shown here).
2. To quickly find the optimal solution skip to the last iteration by clicking on tab 10 (shown here).
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
123456789
101112
13141516171819202122232425
A B C D E F GMULTI-PERIOD EOQ MODEL (Lost Sales) - NORMAL LEAD-TIME DEMAND
PROBLEM: Roger's Sentinel Station
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$
To quickly find the optimal solution click on the Summary tab. It provides a summary of all the 10 iterations.
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.
Notice the answers do not change after the second iteration.
47
Multiperiod Normal Lost SalesIteration 1 Formulas
123456789
10111213
1415161718192021
A B C D E F GMULTI-PERIOD EOQ MODEL (Lost Sales) - NORMAL LEAD-TIME DEMAND
PROBLEM: Roger's Sentinel Station
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
1718
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)
48
Multiperiod Normal Lost SalesIteration 10 Formulas
123456789
10111213
1415161718192021
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
Only one formula changes on the iteration 2 - 10 worksheets, in cell F16. 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 F16 refers back to iteration 9.
Only one formula changes on the iteration 2 - 10 worksheets, in cell F16. 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 F16 refers back to iteration 9.
The term ‘9’!F20 means the value of F20 (expected number of shortages) from iteration 9.
The term ‘9’!F20 means the value of F20 (expected number of shortages) from iteration 9.
16F
=SQRT((2*G9*(G8+'9'!F20*(G12+G13-G10))/(G11*G10)))