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
1
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:
2
 Backordering.
 Lost sales.
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.
3
ACTS
Demand ProbaEvent bility Q = 20 Q = 21 Q = 22 Q = 23
D = 20
.2
$18.00 $16.60 $15.20 $13.80
D = 21
.4
18.00 18.90 17.50 16.10
D = 22
.3
18.00 18.90 19.80 18.40
D = 23
.1
18.00 18.90 19.80 20.70
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.
4
 If the probabilities were long-run frequencies,
then doing so would maximize long-run profit.
 Maximizing expected payoff is assumed proper.
The Single-Period Model:
The Newsvendor Problem
 The payoff table approach can be cumbersome 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 cost
hE = 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.
5
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 + p R - c
 The objective is to minimize total expected cost:
TEC(Q) =
Q
cm   hE  c Q - d  Pr[ D  d ]    pS  pR - c d - Q  Pr[ D  d ]
d 0
6
d Q
where m is the expected demand.
The Single-Period Model:
The Newsvendor Problem
 This is the expression for optimal order quantity:
Q* is the smallest possible demand such that
pS  p R - c
Pr[ D  Q* ] 
 pS  pR - c   hE  c 
 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 m = 25 copies.
7
The Single-Period Model:
The Newsvendor Problem
The following ratio is computed:
pS  p R - c
.02  .23 - .20

 .192
 pS  pR - c   hE  c  .02  .23 - .20  .01  .20
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.
8
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 m = 2,000 and s = 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:
pS  p R - c
1 9 - 3

 .6667
 pS  pR - c   hE  c  1  9 - 3  .50  3
This normal curve area corresponds to z = .43, and
the demand at or beyond this determines Q*.
9
Q* = m + zs = 2,000 + .43(500) = 2,215 trees
Continuous Demand Distribution:
Christmas Tree Problem
 The following is used in computing the total
expected cost:
TEC Q *   cm  hE  c [Q - m  BQ ] pS  pR - c  BQ 
 The above uses the expected shortage:
 Q - m 
Qm
s L s 
BQ   
m
-Q

 m - Q  s L
 Qm
 s 

where L(x) is the tabled loss function.
10
Multiperiod Inventory Policies
 When demand is uncertain, multiperiod
inventory might look like this over time.
11
Multiperiod Inventory Policies
 The multiperiod decisions involve two
variables:
 Order quantity Q
 Reorder point r
 The following parameters apply:






12
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
m = mean lead-time demand
Multiperiod Inventory Policies:
Discrete Lead-Time Demand
 The following is used to compute the
expected shortage per inventory cycle:
Br    d - r PrL D  d 
d r
 The following is used to compute the total
annual expected cost:
(With Backordering)
Q
 A
 A


TEC r ,Q    k  hc   r - m   pS   Br 
2

Q
Q
13
Multiperiod Inventory Policies:
Discrete Lead-Time Demand
 Solution Algorithm.
 Calculate the starting order quantity:
2 Ak
Q1 
hc
 Determine the reorder point r*:
r* is smallest level such that
hcQ
PrD  r *   1 (with backordering)
pS A
 Determine optimal order quantity:
14
2 Ak  pS Br 
Q* 
hc
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.
15
 Find the optimal inventory policy.
Multiperiod Inventory Policies:
Discrete Lead-Time Demand
16
 Solution: The starting order quantity is:
21,5005
Q1 
 289
.121.50 
Using the above, we compute:
hcQ
.12.15289
1 1 .93
pS A
.51,500
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:
21,5005  .50.08
Q* 
 290
.121.50 
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
 pS  p R - c  A
PrD  r *  
hcQ   pS  pR - c A
17
(lost sales)
Multiperiod Inventory Policies:
Discrete Lead-Time Demand
 The Lost Sales Case (continued):
 The optimal order quantity expression is:
2 Ak   pS  pR - c Br 
Q* 
hc
(lost sales)
18
Multiperiod Inventory Policies:
Continuous Lead-Time Demand
 The formulas and algorithms for the
continuous case are the same, except for the
expected shortage:
 r - m
rm
s L s 
B r   
m
-r

 m - r  s L
 rm
 s 

 where m and s are the parameters of the
normal lead-time demand distribution and
L(x) is the tabled losss function.
19
Inventory Spreadsheet Templates
 Payoff Table
 Newsvendor
 Christmas Tree
 Multiperiod Discrete Backordering
 Multiperiod Discrete Lost Sales
 Multiperiod Normal Backordering
 Multiperiod Normal Lost Sales
20
Payoff Table
(Figure 16-1)
1. Enter problem
name in B3.
A
B
C
D
2. Enter data in
B9:F12 and labels in
A9:A12 and C8:F8.
E
PAYOFF TABLE EVALUATION
1
2
3
PROBLEM: Fortune Magazine
4
5
Problem Data
C
6
18 =SUMPRODUCT($B$9:$B$12,C9:C12)
7
Act 1
Act 2
Act 3
8
Events
Probability Q = 20
Q = 21
Q = 22
9
1 D = 20
0.2
$18.00
$16.60
$15.20
10
2 D = 21
0.4
$18.00
$18.90
$17.50
11
3 D = 22
0.3
$18.00
$18.90
$19.80
12
4 D = 23
0.1
$18.00
$18.90
$19.80
13
4. Expected payoffs
14
Act Summary
15
16
Act 1
Act 2
Act 3
17
Q = 20
Q = 21
Q = 22
18 Expected Payoff
$18.00
$18.44
$17.96
F
Copy cell C18
over to D18:F18.
21
Act 4
Q = 23
$13.80
$16.10
$18.40
$20.70
Act 4
Q = 23
$16.79
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.
1. Enter the
problem name
in C3.
6. Optimal
values: Q*, mu,
TEC(Q*), B(Q*),
PD>Q*.
3. Enter the
demands and
probabilities in
C21:D40.
22
Newsvendor
Problem (Figure 16-3)
A
B
C
D
E
F
G
1
SINGLE PERIOD INVENTORY MODEL -- NEWSVENDOR PROBLEM
2
3 PROBLEM:
Wall Street Journal
4
5
Parameter Values:
6
Cost per Item Procured: c =
0.20
Additional Cost for Each Leftover Item Held: hE =
7
0.01
Penalty for Each Item Short: pS =
8
0.02
Selling Price per Unit: pR =
9
0.23
10
Number of demands for probability distribution =
11
11
12
Optimal Values:
13
Optimal Order Quantity: Q* =
47
14
Expected Demand: mu =
49.5
15
Total Expected Cost: TEC(Q*) =
$10.07
16
Expected Shortages: B(Q*) =
2.66
17
Probability of Shortage: P[D>Q*] =
0.80
18
19
Cumulative Number of
20
Demand Probability Probability
shortages
21
45
0.05
0.05
0.0
22
46
0.06
0.11
0.0
23
47
0.09
0.20
0.0
24
48
0.12
0.32
1.0
25
49
0.17
0.49
2.0
26
50
0.20
0.69
3.0
27
51
0.12
0.81
4.0
28
52
0.08
0.89
5.0
29
53
0.06
0.95
6.0
30
54
0.04
0.99
7.0
31
55
0.01
1.00
8.0
2. Enter the
problem
parameters in
G6:G10.
4. If the number
of demands for
probability
distribution is
greater than 20
add the
appropriate
number of rows
and copy the
formulas in
columns E and F
down for these
rows.
5. To calculate
the expected
profit, enter
=SUMPRODUCT(
C21:C40,D21:D4
0)*G9-G15
in cell G18.
Newsvendor Formulas
G
13
14
15
16
17
E
F
=IF(ROW(C21)=INDEX(C21:C40,MATCH(F13LOOKUP((G8+G9A
B
C
D
E
F
G
=IF(ROW(C21)20<=$G$10,IF($G$13>C2
G6)/(G8+G9+G7),E21:E40,C21:C40),C21:C40)+1)
1
SINGLE PERIOD INVENTORY
-- NEWSVENDOR PROBLEM
20<=$G$10,D21,"")
1,0,C21-$G$13),"")
21 MODEL
=SUMPRODUCT(C21:C40,D21:D40)
2
=G6*G14+(G7+G6)*(G13-G14+G16)+(G8+G9-G6)*G16
=IF(ROW(C22)3 PROBLEM:
Wall Street Journal
=SUMPRODUCT(D21:D40,F21:F40)
=IF(ROW(C22)20<=$G$10,IF($G$13>C2
4
2,0,C22-$G$13),"")
22 20<=$G$10,D22+E21,"")
=1-VLOOKUP(G13,C21:F40,3)
5
Parameter Values:
6
Cost per Item Procured: c =
0.20
Additional Cost for Each Leftover Item Held: hE =
7
0.01
Penalty for Each Item Short: pS =
8
0.02
Selling Price per Unit: pR =
9
0.23
10
Number of demands for probability distribution =
11
11
12
Optimal Values:
13
Optimal Order Quantity: Q* =
47
14
Expected Demand: mu =
49.5
15
Total Expected Cost: TEC(Q*) =
$10.07
16
Expected Shortages: B(Q*) =
2.66
17
Probability of Shortage: P[D>Q*] =
0.80
18
19
Cumulative Number of
20
Demand Probability Probability
shortages
21
45
0.05
0.05
0.0
22
46
0.06
0.11
0.0
23
23
47
0.09
0.20
0.0
1. Enter the
problem name
in C3.
Christmas Tree
Problem (Figure 16-6)
2. Enter the
problem
parameters in
G6:G11.
A
B
C
D
E
F
G
SINGLE PERIOD INVENTORY MODEL - CHRISTMAS TREE PROBLEM
3. Optimal
values: Q*, mu,
TEC(Q*), B(Q*),
PD>Q*.
24
1
2
3 PROBLEM:
Noble Fir
4
5
Parameter Values:
6
Mean of Demand Distribution: mu =
7
Stand. Deviation of Demand Distribution: sigma =
8
Cost per Item Procured: c =
Additional Cost for Each Leftover Item Held: hE =
9
Penalty for Each Item Short: pS =
10
Selling Price per Unit: pR =
11
12
13
Optimal Values:
14
Optimal Order Quantity: Q* =
2215
15
Expected Demand: mu =
2000
16
Total Expected Cost: TEC(Q*) =
$7,910.35
17
Expected Shortages: B(Q*) =
110.15
18
Probability of Shortage: P[D>Q*] =
0.33
The Normal Loss Table L(D) is on the next worksheet. It
is used in the spreadsheet calculations.
2000
500
3.00
0.50
1.00
9.00
The Normal Loss Table L(D)
The Normal
Loss Table L(D)
is used the
calculations in
the Christmas
Tree template.
25
1
2
3
4
5
6
102
103
104
105
106
202
203
204
205
206
497
498
499
500
501
A
D
0.00
0.01
0.02
0.03
0.04
1.00
1.01
1.02
1.03
1.04
2.00
2.01
2.02
2.03
2.04
4.95
4.96
4.97
4.98
4.99
B
L(D)
0.3989
0.3940
0.3890
0.3841
0.3793
0.08332
0.08174
0.08019
0.07866
0.07716
8.491E-03
8.266E-03
8.046E-03
7.832E-03
7.623E-03
6.982E-08
6.620E-08
6.276E-08
5.950E-08
5.640E-08
Note that many
rows have been
hidden because
the entire table
is too big to
show on one
page.
Christmas Tree Formulas
A
B
C
D
E
F
G
SINGLE PERIOD INVENTORY MODEL - CHRISTMAS TREE PROBLEM
‘L(D)’!A2:B501
refers to the
normal loss
table L(D) table
located on the
L(D) worksheet
26
1
2
3 PROBLEM:
Noble Fir
4
5
Parameter Values:
6
Mean of Demand Distribution: mu =
7
Stand. Deviation of Demand Distribution: sigma =
8
Cost per Item Procured: c =
Additional Cost for Each Leftover Item Held: hE =
9
Penalty for Each Item Short: pS =
10
Selling Price per Unit: pR =
11
12
13
Optimal Values:
14
Optimal Order Quantity: Q* =
2215
15
Expected Demand: mu =
2000
16
Total Expected Cost: TEC(Q*) =
$7,910.35
17
Expected Shortages: B(Q*) =
110.15
18
Probability of Shortage: P[D>Q*] =
0.33
F
14 =NORMINV((G10+G11-G8)/(G10+G11+G9),G6,G7)
15 =G6
16 =G8*F15+(G9+G8)*(F14-F15+F17)+(G10+G11-G8)*F17
=IF(F14<G6,G6-F14+G7*VLOOKUP((G6F14)/G7,'L(D)'!A2:B501,2),G7*VLOOKUP((F1417 G6)/G7,'L(D)'!A2:B501,2))
18 =1-NORMDIST(F14,G6,G7,TRUE)
2000
500
3.00
0.50
1.00
9.00
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.
27
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.
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.
28
Multiperiod Discrete Backordering
Iteration 1
1. Start with
worksheet 1 (tab
1). It gives the
results of the
first iteration.
2. Enter the problem
name in B3.
3. Enter the problem
parameters in
G6:G11.
6. Iteration 1
results are here
4. Enter the demands
and probabilities in
C23:D42.
A
B
C
D
E
F
G
1 MULTI-PERIOD EOQ MODEL (Backordering) - DISCRETE LEAD-TIME DEMAND
2
3 PROBLEM:
Printer Cartridges
4
5
Parameter Values
6
Fixed Cost per Order: k =
5
7
Annual Demand Rate: A =
1500
8
Unit cost of Procuring an Item: c =
1.5
9
Annual Holding Cost per Dollar Value: h =
0.12
Shortage Cost per Unit: pS =
10
0.5
11
Number of demands for probability distribution =
11
12
13
Optimal Values:
14
Optimal Order Quantity: Q* =
288.68
15
Optimal Reorder Point: r* =
7
16
Expected Lead-Time Demand: mu =
4
17
Total Expected Cost: TEC(Q*) =
$ 52.7094
18
Expected Shortage: B(r*) =
0.08
19
Probability of Shortage: P[D>r*] =
0.05
20
21
Cumulative
Number of
22
Demand Probability
Probability
Shortages
23
0
0.01
0.01
0
24
1
0.07
0.08
0
25
2
0.16
0.24
0
26
3
0.20
0.44
0
27
4
0.19
0.63
0
28
5
0.16
0.79
0
29
6
0.10
0.89
0
30
7
0.06
0.95
0
31
8
0.03
0.98
1
32
9
0.01
0.99
2
33
10
0.01
1.00
3
5. If the number of demands for probability distribution is greater than 20 add the
29
appropriate number of rows and copy the formulas in columns E and F down for these rows.
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.
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*),
PD>Q*.
30
A
B
C
D
E
F
G
1 MULTI-PERIOD EOQ MODEL (Backordering) - DISCRETE LEAD-TIME DEMAND
2
3 PROBLEM:
Printer Cartridges
4
5
Parameter Values
6
Fixed Cost per Order: k =
5
7
Annual Demand Rate: A =
1500
8
Unit cost of Procuring an Item: c =
1.5
9
Annual Holding Cost per Dollar Value: h =
0.12
Shortage Cost per Unit: pS =
10
0.5
11
Number of demands for probability distribution =
11
12
13
Optimal Values:
14 =SQRT((2
14
Optimal Order Quantity: Q* =
290
=INDEX(C
15
Optimal Reorder Point: r* =
7
((G9*G8*G
16
Expected Lead-Time Demand: mu =
4 15 ),C23:C42,
17
Total Expected Cost: TEC(Q*) =
$ 52.71 16 =SUMPRO
18
Expected Shortage: B(r*) =
0.08
=(G7/G14)
19
Probability of Shortage: P[D>r*] =
0.05
G16)+G10
17
20
18
=SUMPRO
21
Cumulative
Number of
19
=1-VLOOK
22
Demand Probability
Probability
Shortages
23
0
0.01
0.01
0
24
1
0.07
0.08
0
25
2
0.16
0.24
0
26
3
0.20
0.44
0
27
4
0.19
0.63
0
28
5
0.16
0.79
0
29
6
0.10
0.89
0
30
7
0.06
0.95
0
31
8
0.03
0.98
1
32
9
0.01
0.99
2
33
10
0.01
1.00
3
Multiperiod Discrete Backordering
Summary
A
B
C
D
E
F
G
1 MULTI-PERIOD EOQ MODEL (Backordering) - DISCRETE LEAD-TIME DEMAND
2
3 PROBLEM: Printer Cartridges
4
5
Parameter Values
To quickly find the
6
Fixed Cost per Order: k =
5
optimal solution click on7
Annual Demand Rate: A =
1500
8
Unit
cost
of
Procuring
an
Item:
c
=
1.5
the Summary tab. It
9
Annual Holding Cost per Dollar Value: h =
0.12
provides a summary of all
Shortage Cost per Unit: pS =
10
0.5
the 10 iterations.
11
Number of demands for probability distribution =
11
12
Qi
ri
B(ri) TEC(Qi,ri)
13
Iteration, i
31
14
15
16
17
18
19
20
21
22
23
1
2
3
4
5
6
7
8
9
10
289
290
290
290
290
290
290
290
290
290
7
7
7
7
7
7
7
7
7
7
0.08
0.08
0.08
0.08
0.08
0.08
0.08
0.08
0.08
0.08
$
$
$
$
$
$
$
$
$
$
52.71
52.71
52.71
52.71
52.71
52.71
52.71
52.71
52.71
52.71
Notice the
answers do not
change after the
second iteration.
Multiperiod Discrete Backordering
Iteration 1 Formulas
G
14 =SQRT((2*G7*G6)/(G9*G8))
=INDEX(C23:C42,MATCH(LOOKUP(1((G9*G8*G14)/(G10*G7)),E23:E42,C23:
15 C42),C23:C42,0)+1)
16 =SUMPRODUCT(C23:C42,D23:D42)
=(G7/G14)*G6+G9*G8*(G14/2+G1517 G16)+G10*(G7/G14)*G18
18 =SUMPRODUCT(D23:D42,F23:F42)
19 =1-VLOOKUP(G15,C23:E42,3)
32
A
B
C
D
E
F
G
1 MULTI-PERIOD EOQ MODEL (Backordering) - DISCRETE LEAD-TIME DEMAND
2
3 PROBLEM:
Printer Cartridges
4
5
Parameter Values
6
Fixed Cost per Order: k =
5
7
Annual Demand Rate: A =
1500
8
Unit cost of Procuring an Item: c =
1.5
9
Annual Holding Cost per Dollar Value: h =
0.12
Shortage Cost per Unit: pS =
10
0.5
11
Number of demands for probability distribution =
11
12
13
Optimal Values:
14
Optimal Order Quantity: Q* =
288.68
15
Optimal Reorder Point: r* =
7
16
Expected Lead-Time Demand: mu =
4
17
Total Expected Cost: TEC(Q*) =
$ 52.7094
18
Expected Shortage: B(r*) =
0.08
19
Probability of Shortage: P[D>r*] =
0.05
20
21
Cumulative
Number of
22
Demand Probability
Probability
Shortages
23
0
0.01
0.01
0
24
1
0.07
0.08
0
25
2
0.16
0.24
0
26
3
0.20
0.44
0
27
4
0.19
0.63
0
28
5
0.16
0.79
0
29
6
0.10
0.89
0
30
7
0.06
0.95
0
31
8
0.03
0.98
1
32
9
0.01
0.99
2
33
10
0.01
1.00
3
Multiperiod Discrete Backordering
Iteration 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.
A
B
C
D
E
F
G
1 MULTI-PERIOD EOQ MODEL (Backordering) - DISCRETE LEAD-TIME DEMAND
2
3 PROBLEM:
Printer Cartridges
4
5
Parameter Values
6
Fixed Cost per Order: k =
5
7
Annual Demand Rate: A =
1500
8
Unit cost of Procuring an Item: c =
1.5
9
Annual Holding Cost per Dollar Value: h =
0.12
Shortage Cost per Unit: pS =
10
0.5
11
Number of demands for probability distribution =
11
12
13
Optimal Values:
14 =SQ
14
Optimal Order Quantity: Q* =
290
=IN
15
Optimal Reorder Point: r* =
7
((G9
16
Expected Lead-Time Demand: mu =
4 15 ),C2
17
Total Expected Cost: TEC(Q*) =
$ 52.71 16 =SU
18
Expected Shortage: B(r*) =
0.08
=(G
19
Probability of Shortage: P[D>r*] =
0.05
G16
17
20
18
=SU
21
Cumulative
Number of
19
=1-V
22
Demand Probability
Probability
Shortages
23
0
0.01
0.01
0
24
1
0.07
0.08
0
25
2
0.16
0.24
0
26
3
0.20
0.44
0
27
4
0.19
0.63
0
28
5
0.16
0.79
0
29
6
0.10
0.89
0
30
7
0.06
0.95
0
31
8
0.03
0.98
1
32
9
0.01
0.99
2
33
10
0.01
1.00
3
G
14 =SQRT((2*G7*(G6+'9'!G18*G10))/(G9*G8))
The term ‘9’!G18
means the value
of G18 (expected
number of
shortages) from
iteration 9.
33
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).
34
Multiperiod Discrete Lost Sales
(Figure 16-9)
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 in
C24:D43.
2. To quickly find
the optimal solution
skip to the last
iteration by clicking
on tab 10 (shown
here).
35
A
B
C
D
E
F
G
1 MULTI-PERIOD EOQ MODEL (Lost Sales) - DISCRETE LEAD-TIME DEMAND
2
3 PROBLEM:
Compact Disks
4
5
Parameter Values
6
Fixed Cost per Order: k =
20
7
Annual Demand Rate: A =
5000
8
Unit cost of Procuring an Item: c =
0.45
9
Annual Holding Cost per Dollar Value: h =
0.12
Shortage Cost per Unit: pS =
10
0.1
Shortage Cost per Unit: pR =
11
0.9
12
Number of demands for probability distribution =
10
13
14
Optimal Values:
15
Optimal Order Quantity: Q* =
1935
16
Optimal Reorder Point: r* =
160
17
Expected Lead-Time Demand: mu =
123
18
Total Expected Cost: TEC(Q*) =
$ 106.48
19
Expected Shortage: B(r*) =
0.40
20
Probability of Shortage: P[D>r*] =
0.03
21
22
Cumulative
Number of
23
Demand Probability
Probability
Shortages
24
90
0.05
0.05
0
25
100
0.12
0.17
0
3. Optimal
26
110
0.17
0.34
0
27
120
0.22
0.56
0
values: Q*, r*,
28
130
0.19
0.75
0
mu, TEC(Q*),
29
140
0.14
0.89
0
30
150
0.05
0.94
0
B(Q*), PD>Q*.
31
160
0.03
0.97
0
32
170
0.02
0.99
10
33
180
0.01
1.00
20
Multiperiod Discrete Lost Sales
Summary
A
B
C
D
E
F
G
1 MULTI-PERIOD EOQ MODEL (Backordering) - DISCRETE LEAD-TIME DEMAND
2
3 PROBLEM: Compact Discs
4
5
Parameter Values
6
Fixed Cost per Order: k =
20
7
Annual Demand Rate: A =
5000
8
Unit
cost
of
Procuring
an
Item:
c
=
0.45
on
9
Annual Holding Cost per Dollar Value: h =
0.12
Shortage Cost per Unit: pS =
10
0.1
To quickly find the
optimal solution click
the Summary tab. It
provides a summary of all
11
the 10 iterations.
12
36
Shortage Cost per Unit: pR =
Number of demands for probability distribution =
13
14
15
Iteration, i
Qi
ri
B(ri)
TEC(Qi,ri)
16
17
18
19
20
21
22
23
24
25
1
2
3
4
5
6
7
8
9
10
1925
1935
1935
1935
1935
1935
1935
1935
1935
1935
160
160
160
160
160
160
160
160
160
160
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
$
$
$
$
$
$
$
$
$
$
106.48
106.48
106.48
106.48
106.48
106.48
106.48
106.48
106.48
106.48
0.9
10
Notice the
answers do not
change after the
second iteration.
Multiperiod Discrete Lost Sales
Iteration 1 Formulas
G
15 =SQRT((2*G7*G6)/(G9*G8))
=INDEX(C24:C43,MATCH(LOOKUP((G10+G1
1-G8)*G7/(G9*G8*G15+(G10+G1116 G8)*G7),E24:E43,C24:C43),C24:C43,0)+1)
17 =SUMPRODUCT(C24:C43,D24:D43)
=((G7*(1G19/G15))/G15)*G6+G9*G8*(G15/2+G16G17+G19)+(G10+G11-G8)*(G7*(118 G19/G15)/G15)*G19
19 =SUMPRODUCT(D24:D43,F24:F43)
20 =1-VLOOKUP(G16,C24:E43,3)
37
A
B
C
D
E
F
1 MULTI-PERIOD EOQ MODEL (Lost Sales) - DISCRETE LEAD-TIME DEMAND
2
3 PROBLEM:
Compact Discs
4
5
Parameter Values
6
Fixed Cost per Order: k =
7
Annual Demand Rate: A =
8
Unit cost of Procuring an Item: c =
9
Annual Holding Cost per Dollar Value: h =
Shortage Cost per Unit: pS =
10
Shortage Cost per Unit: pR =
11
12
Number of demands for probability distribution =
13
14
Optimal Values:
15
Optimal Order Quantity: Q* =
16
Optimal Reorder Point: r* =
17
Expected Lead-Time Demand: mu =
18
Total Expected Cost: TEC(Q*) =
$
19
Expected Shortage: B(r*) =
20
Probability of Shortage: P[D>r*] =
21
22
Cumulative
Number of
23
Demand Probability
Probability
Shortages
24
90
0.05
0.05
0
25
100
0.12
0.17
0
26
110
0.17
0.34
0
27
120
0.22
0.56
0
28
130
0.19
0.75
0
29
140
0.14
0.89
0
30
150
0.05
0.94
0
31
160
0.03
0.97
0
32
170
0.02
0.99
10
33
180
0.01
1.00
20
G
20
5000
0.45
0.12
0.1
0.9
10
1925
160
123
106.48
0.40
0.03
Multiperiod Discrete Lost Sales
Iteration 10 Formulas
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.
A
B
C
D
E
F
1 MULTI-PERIOD EOQ MODEL (Lost Sales) - DISCRETE LEAD-TIME DEMAND
2
3 PROBLEM:
Compact Disks
4
5
Parameter Values
6
Fixed Cost per Order: k =
7
Annual Demand Rate: A =
8
Unit cost of Procuring an Item: c =
9
Annual Holding Cost per Dollar Value: h =
Shortage Cost per Unit: pS =
10
Shortage Cost per Unit: pR =
11
12
Number of demands for probability distribution =
13
14
Optimal Values:
15
Optimal Order Quantity: Q* =
16
Optimal Reorder Point: r* =
G
17
Expected Lead-Time Demand: mu =
15 =SQRT((2*G7*(G6+'9'!G19*(G10+G11-G8))/(G9*G8)))
18
Total Expected Cost: TEC(Q*) =
$
19
Expected Shortage: B(r*) =
20
Probability of Shortage: P[D>r*] =
The term ‘9’!G19
21
22
Cumulative
Number of
means the value
23
Demand Probability
Probability
Shortages
of G19 (expected
24
90
0.05
0.05
0
25
100
0.12
0.17
0
number of
26
110
0.17
0.34
0
shortages) from
27
120
0.22
0.56
0
28
130
0.19
0.75
0
iteration 9.
29
140
0.14
0.89
0
30
150
0.05
0.94
0
31
160
0.03
0.97
0
38
32
170
0.02
0.99
10
33
180
0.01
1.00
20
G
20
5000
0.45
0.12
0.1
0.9
10
1935
160
123
106.48
0.40
0.03
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.
39
Multiperiod Normal Backordering
(Figure 16-10)
A
1. Start with
worksheet 1
(tab 1) and
enter the
problem
name in B3
and the
problem
parameters in
G6:G12.
B
C
D
E
F
G
MULTI-PERIOD EOQ MODEL (Backordering) - NORMAL LEAD-TIME DEMAND
1
2
3 PROBLEM: Unleaded Gas at Oil Refinery
4
5
Parameter Values:
6
Mean of Demand Distribution: mu =
7
Stand. Deviation of Demand Distribution: sigma =
8
Fixed Cost per Order: k =
9
Annual Demand Rate: A =
10
Unit Cost of Procuring an Item: c =
11
Annual Holding Cost per Dollar Value: h =
Shortage Cost per Unit: pS =
12
13
14
Optimal Values:
2. To quickly find15the
Optimal Order Quantity: Q* =
optimal solution 16
skip
Optimal Reorder Point: r* =
17
Expected Demand: mu =
to the last iteration
1810
Total Expected Cost: TEC(Q*) =
by clicking on tab
Expected Shortages: B(r*) =
(shown here). 19
20
Probability of Shortage: P[D>r*] =
40
200,000
20,000
1,000
40,000,000
0.40
0.20
0.05
1,008,256
234,937
200,000
83,455.46
331.60
0.04
3. Optimal values: Q*, r*, mu, TEC(Q*), B(Q*), P D>Q*.
Multiperiod Normal Backordering
Summary
A
B
C
D
E
F
1 MULTI-PERIOD EOQ MODEL (Backordering) - NORMAL LEAD-TIME DEMAND
2
3 PROBLEM: Unleaded Gas at Oil Refinery
4
5
Parameter Values:
Mean of Demand Distribution: mu =
To quickly find the6
7
Stand. Deviation of Demand Distribution: sigma =
optimal solution click
on
8
Fixed Cost per Order: k =
the Summary tab. 9It
Annual Demand Rate: A =
provides a summary
10 of all
Unit Cost of Procuring an Item: c =
Annual Holding Cost per Dollar Value: h =
the 10 iterations. 11
Shortage Cost per Unit: pS =
12
13
14
Qi
ri
B(ri)
TEC(Qi,ri)
15 Iteration, i
41
16
17
18
19
20
21
22
23
24
25
1
2
3
4
5
6
7
8
9
10
1,000,000
1,008,053
1,008,256
1,008,256
1,008,256
1,008,256
1,008,256
1,008,256
1,008,256
1,008,256
235,014
234,939
234,937
234,937
234,937
234,937
234,937
234,937
234,937
234,937
323
332
332
332
332
332
332
332
332
332
$
$
$
$
$
$
$
$
$
$
83,447.90
83,455.61
83,455.46
83,455.46
83,455.46
83,455.46
83,455.46
83,455.46
83,455.46
83,455.46
G
200,000
20,000
1,000
40,000,000
0.40
0.20
0.05
Notice the
answers do not
change after the
third iteration.
Multiperiod Normal Backordering
Iteration 1 Formulas
15
16
17
18
A
B
C
D
E
F
1 MULTI-PERIOD EOQ MODEL (Backordering) - NORMAL LEAD-TIME DEMAND
2
3 PROBLEM: Unleaded Gas at Oil Refinery
4
5
Parameter Values:
6
Mean of Demand Distribution: mu =
7
Stand. Deviation of Demand Distribution: sigma =
8
Fixed Cost per Order: k =
9
Annual Demand Rate: A =
10
Unit Cost of Procuring an Item: c =
11
Annual Holding Cost per Dollar Value: h =
Shortage Cost per Unit: pS =
12
13
14
Optimal Values:
15 F
Optimal Order Quantity: Q* =
1,000,000
16
Optimal Reorder Point: r* =
235,014
=SQRT((2*G9*G8)/(G11*G10))
=NORMINV(1-((G11*G10*F15)/(G12*G9)),G6,G7)
17
Expected Demand: mu =
200,000
=G6
18
Total Expected Cost: TEC(Q*) =
$
83,447.90
=(G9/F15)*G8+G11*G10*(F15/2+F1619
Expected Shortages: B(r*) =
323.40
G6)+G12*(G9/F15)*F19
20
Probability of Shortage: P[D>r*] =
0.04
=IF(F16<F17,F17-F16+G7*VLOOKUP((F17F16)/G7,'L(D)'!A2:B501,2),G7*VLOOKUP((F1619 F17)/G7,'L(D)'!A2:B501,2))
20 =1-NORMDIST(F16,G6,G7,TRUE)
42
G
200,000
20,000
1,000
40,000,000
0.40
0.20
0.05
Multiperiod Normal Backordering
Iteration 10 Formulas
Only one formula changes on
the iteration 2 - 10
worksheets, in cell F15. AThe
B
C
D
E
F
G
formula in this cell
1 alwaysMULTI-PERIOD EOQ MODEL (Backordering) - NORMAL LEAD-TIME DEMAND
2 previous
refers back the the
3 PROBLEM:
iteration. For example,
the Unleaded Gas at Oil Refinery
4
worksheet shown5 here is for Parameter Values:
iteration 10 so the6 formula in
Mean of Demand Distribution: mu =
200,000
cell F15 refers back
7 to
Stand. Deviation of Demand Distribution: sigma =
20,000
8
Fixed Cost per Order: k =
1,000
iteration 9.
The term
‘9’!F19
means the
value of F19
(expected
number of
shortages)
from iteration
9.
43
9
10
11
12
13
14
15
16
17
18
19
20
Annual Demand Rate: A =
Unit Cost of Procuring an Item: c =
Annual Holding Cost per Dollar Value: h =
Shortage Cost per Unit: pS =
Optimal Values:
Optimal Order Quantity: Q* =
Optimal Reorder Point: r* =
Expected Demand: mu =
Total Expected Cost: TEC(Q*) =
Expected Shortages: B(r*) =
Probability of Shortage: P[D>r*] =
F
15 =SQRT((2*G9*(G8+G12*'9'!F19))/(G11*G10))
40,000,000
0.40
0.20
0.05
1,008,256
234,937
200,000
83,455.46
331.60
0.04
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).
44
Multiperiod Normal Lost Sales
(Figure 16-11)
A
B
C
D
E
F
MULTI-PERIOD EOQ MODEL (Lost Sales) - NORMAL LEAD-TIME DEMAND
1
2
1. Start with
3 PROBLEM: Roger's Sentinel Station
worksheet 1
4
(tab 1) and
5
Parameter Values:
6
Mean of Demand Distribution: mu =
enter the
7
Stand. Deviation of Demand Distribution: sigma =
problem
8
Fixed Cost per Order: k =
name in B3
9
Annual Demand Rate: A =
and the
10
Unit Cost of Procuring an Item: c =
problem
11
Annual Holding Cost per Dollar Value: h =
Shortage Cost per Unit: pS =
12
parameters in
Selling Price per Unit: pR =
13
G6:G13.
14
15
Optimal Values:
2. To quickly find
16
Optimal Order Quantity: Q* =
the optimal solution
17
Optimal Reorder Point: r* =
Expected Demand: mu =
skip to the last18
19
Total Expected Cost: TEC(Q*) =
iteration by clicking
20
Expected Shortages: B(r*) =
on tab 10 (shown
21
Probability of Shortage: P[D>r*] =
here).
45
G
1,000
50
100
500,000
1.48
0.19
0.25
1.75
18,876
1,103
1,000
5,336.87
0.37
0.02
3. Optimal values: Q*, r*, mu, TEC(Q*), B(Q*), P D>Q*.
Multiperiod Normal Lost Sales
Summary
A
B
C
D
E
F
G
1 MULTI-PERIOD EOQ MODEL (Lost Sales) - NORMAL LEAD-TIME DEMAND
2
3 PROBLEM: Roger's Sentinel Station
4
5
Parameter Values:
6
Mean of Demand Distribution: mu =
1,000
To quickly find the 7
Stand. Deviation of Demand Distribution: sigma =
50
8 on
Fixed Cost per Order: k =
100
optimal solution click
9
Annual Demand Rate: A =
500,000
the Summary tab. It
10
Unit Cost of Procuring an Item: c =
1.48
provides a summary
of
all
11
Annual Holding Cost per Dollar Value: h =
0.19
Shortage
Cost
per
Unit:
p
=
12
0.25
S
the 10 iterations.
13
14
15 Iteration, i
Qi
ri
B(ri)
TEC(Qi,ri)
16
1
18,858
1,103
0.37 $ 5,336.87
17
2
18,876
1,103
0.37 $ 5,336.87
18
3
18,876
1,103
0.37 $ 5,336.87
Notice the
19
4
18,876
1,103
0.37 $ 5,336.87
answers do not
20
5
18,876
1,103
0.37 $ 5,336.87
change after the
21
6
18,876
1,103
0.37 $ 5,336.87
22
7
18,876
1,103
0.37 $ 5,336.87
second iteration.
23
8
18,876
1,103
0.37 $ 5,336.87
24
9
18,876
1,103
0.37 $ 5,336.87
25
10
18,876
1,103
0.37 $ 5,336.87
46
Multiperiod Normal Lost Sales
Iteration 1 Formulas
16
17
18
19
A
B
C
D
E
F
1 MULTI-PERIOD EOQ MODEL (Lost Sales) - NORMAL LEAD-TIME DEMAND
2
3 PROBLEM: Roger's Sentinel Station
4
5
Parameter Values:
6
Mean of Demand Distribution: mu =
7
Stand. Deviation of Demand Distribution: sigma =
8
Fixed Cost per Order: k =
9
Annual Demand Rate: A =
10
Unit Cost of Procuring an Item: c =
11
Annual Holding Cost per Dollar Value: h =
F
Shortage Cost per Unit: pS =
12
=SQRT((2*G9*G8)/(G11*G10))
Selling Price per Unit: pR =
13
=NORMINV((G12+G1314
G10)*G9/(G11*G10*F16+(G12+G13Optimal Values:
G10)*G9),G6,G7) 15
16
Optimal Order Quantity: Q* =
18,858
=G6
17
Optimal Reorder Point: r* =
1,103
=(G9*(118
Expected Demand: mu =
1,000
F20/F16))/F16*G8+G11*G10*(F16/2+F1719
Total Expected Cost: TEC(Q*) =
5,336.87
F18+F20)+(G12+G13-G10)*(G9*(1Expected Shortages: B(r*) =
0.37
F20/F16)/F16)*F20 20
21
Probability of Shortage: P[D>r*] =
0.02
=IF(F17<F18,F18-F17+G7*VLOOKUP((F18-
F17)/G7,'L(D)'!A2:B501,2),G7*VLOOKUP((F1720 F18)/G7,'L(D)'!A2:B501,2))
21 =1-NORMDIST(F17,G6,G7,TRUE)
47
G
1,000
50
100
500,000
1.48
0.19
0.25
1.75
Multiperiod Normal Lost Sales
Iteration 10 Formulas
Only one formula changes on
A
B
C
D
E
F
G
the iteration 2 - 10
1 F16. The
MULTI-PERIOD EOQ MODEL (Lost Sales) - NORMAL LEAD-TIME DEMAND
worksheets, in cell
2
formula in this cell
always
3 PROBLEM: Roger's Sentinel Station
refers back the the
4 previous
iteration. For example,
the
5
Parameter Values:
Mean of Demand Distribution: mu =
1,000
worksheet shown6here is for
Stand. Deviation of Demand Distribution: sigma =
50
iteration 10 so the7 formula in
8
Fixed Cost per Order: k =
100
cell F16 refers back
to
9
Annual Demand Rate: A =
500,000
iteration 9.
10
Unit Cost of Procuring an Item: c =
1.48
The term
‘9’!F20
means the
value of F20
(expected
number of
shortages)
from iteration
9.
48
11
12
13
14
15
16
17
18
19
20
21
Annual Holding Cost per Dollar Value: h =
Shortage Cost per Unit: pS =
Selling Price per Unit: pR =
Optimal Values:
Optimal Order Quantity: Q* =
Optimal Reorder Point: r* =
Expected Demand: mu =
Total Expected Cost: TEC(Q*) =
Expected Shortages: B(r*) =
Probability of Shortage: P[D>r*] =
0.19
0.25
1.75
18,876
1,103
1,000
5,336.87
0.37
0.02
F
16 =SQRT((2*G9*(G8+'9'!F20*(G12+G13-G10))/(G11*G10)))