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

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 inventorye.g., newspapers.
• Probability distribution is for periods demand.
• Multi-stage inventorye.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 DecisionMaximizing 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 ACTS ACTS ACTS
DemandEvent Proba-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
4
Making an Inventory DecisionMaximizing 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 ModelThe 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 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.

6
The Single-Period ModelThe 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 lt Q, there are Q - D leftovers, each
  costing
• hE c
• If D gt Q, there are D - Q shortages, each
  costing
• pS pR - c
• The objective is to minimize total expected cost
• where m is the expected demand.

TEC(Q)
7
The Single-Period ModelThe 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 m 25 copies.

Q is the smallest possible demand such that
8
The Single-Period ModelThe 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.

9
Continuous Demand DistributionChristmas 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
• This normal curve area corresponds to z .43,
  and the demand at or beyond this determines Q.
• Q m zs 2,000 .43(500) 2,215 trees

10
Continuous Demand DistributionChristmas 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.

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
• m 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

(With Backordering)
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)

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 gt .93 is .95, corresponding to 7
  cartridges. Thus, r 7 cartridges. We
  compute
• B(7) (87)(.03) (97)(.01) (107)(.01).08
  and the optimal order quantity is

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)
18
Multiperiod Inventory Policies Discrete
Lead-Time Demand

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

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

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)
2. Enter data in B9F12 and labels in A9A12 and
C8F8.
1. Enter problem name in B3.
Copy cell C18 over to D18F18.
4. Expected payoffs
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.
22
Newsvendor Problem (Figure 16-3)
1. Enter the problem name in C3.
2. Enter the problem parameters in G6G10.
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.
6. Optimal values Q, mu, TEC(Q), B(Q),
P?DgtQ?.
5. To calculate the expected profit, enter
SUMPRODUCT(C21C40,D21D40)G9-G15 in cell G18.
3. Enter the demands and probabilities in C21D40.
23
Newsvendor Formulas
24
Christmas Tree Problem (Figure 16-6)
2. Enter the problem parameters in G6G11.
1. Enter the problem name in C3.
3. Optimal values Q, mu, TEC(Q), B(Q),
P?DgtQ?.
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)
Note that many rows have been hidden because the
entire table is too big to show on one page.
The Normal Loss Table L(D) is used the
calculations in the Christmas Tree template.
26
Christmas Tree Formulas
L(D)!A2B501 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.
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.
29
Multiperiod Discrete BackorderingIteration 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 G6G11.
6. Iteration 1 results are here
4. Enter the demands and probabilities
in C23D42.
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.
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?DgtQ?.
31
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
33
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.
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)
1. Start with worksheet 1 (tab 1) and enter the
problem name in B3, the problem parameters in
G6G12, and the demands and probabilities
in C24D43.
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?DgtQ?.
36
Multiperiod Discrete Lost SalesSummary
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.
37
Multiperiod Discrete Lost SalesIteration 1
Formulas
38
Multiperiod Discrete Lost SalesIteration 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.
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)
1. Start with worksheet 1 (tab 1) and enter the
problem name in B3 and the problem parameters in
G6G12.
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? DgtQ?.
41
Multiperiod Normal 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 third
iteration.
42
Multiperiod Normal BackorderingIteration 1
Formulas
43
Multiperiod Normal BackorderingIteration 10
Formulas
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)
1. Start with worksheet 1 (tab 1) and enter the
problem name in B3 and the problem parameters in
G6G13.
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? DgtQ?.
46
Multiperiod Normal Lost SalesSummary
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.
47
Multiperiod Normal Lost SalesIteration 1 Formulas
48
Multiperiod Normal Lost SalesIteration 10
Formulas
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.