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### Payoff Table(Figure 16-1)

### Newsvendor Problem (Figure 16-3)

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

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.

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.

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.

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 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 demandD.

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 - Dleftovers, each costing:
- hE + c
- If D > Q, there are D -Qshortages, each costing:
- pS + pR- c
- The objective is to minimize total expected cost:

where m is the expected demand.

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 m= 25 copies.

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 pSto $.04 will increase Q* to 23. Raising pSto $.10 will increase Q* to 24.

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:

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

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.

Multiperiod Inventory Policies

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

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

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:

Multiperiod Inventory Policies: Discrete Lead-Time Demand

- Solution Algorithm.
- Calculate the starting order quantity:
- Determine the reorder point r*:
- Determine optimal order quantity:

Multiperiod Inventory Policies: Discrete Lead-Time Demand

- Recompute r* after getting Q*, andvice 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.

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:

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:

Multiperiod Inventory Policies: Discrete Lead-Time Demand

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

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.

Inventory Spreadsheet Templates

- Payoff Table
- Newsvendor
- Christmas Tree
- Multiperiod Discrete Backordering
- Multiperiod Discrete Lost Sales
- Multiperiod Normal Backordering
- Multiperiod Normal Lost Sales

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

1. Enter problem name in B3.

Copy cell C18 over to D18:F18.

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.

1. Enter the problem name in C3.

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.

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

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

in cell G18.

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

Christmas Tree Problem (Figure 16-6)

2. Enter the problem parameters in G6:G11.

1. Enter the problem name in C3.

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

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

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.

Christmas Tree Formulas

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

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.

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.

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 G6:G11.

6. Iteration 1 results are here

4. Enter the demands and probabilities in

C23:D42.

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.

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*), PD>Q*.

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.

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.

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).

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).

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

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.

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.

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.

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 G6:G12.

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*.

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.

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.

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).

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 G6:G13.

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*.

Multiperiod Normal Lost SalesSummaryTo 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.

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.

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