1. Inventory Management Part Deux: The Newsvendor Problem
3/21/2016

2. My objectives in teaching this class:
Give you a set of analytical tools that you can actually use in future jobs
Develop an ability to identify the tradeoffs inherent in all decisions
Make you better at Microsoft Excel

3. Certain & Constant Demand
E[O/P]Q
Certain & Constant Demand

4. E[O/P]Q Decision: How much to order/produce at a time Tradeoff:
Fixed setup vs. variable holding costs
Heroic Assumptions:
Constant demand
All costs constant & precisely known
No randomness anywhere

5. Newsvendor Model Decision: How much to order at a time Tradeoff:
Different costs for too few/too many
Heroic Assumptions:
Random but statistically well-defined demand
All costs constant & precisely known
No randomness anywhere

7. Newsvendor Model
We must decide TODAY how many papers to buy for resale TOMORROW. We don’t know exactly how many people will come tomorrow, but we do know the demand DISTRIBUTION (i.e. historical demand). We know the purchase cost, selling price, and salvage value of each paper.

8. Demand Demand~Uniform(0,100) J F M A S O N D 1 91 62 5 76 94 27 3 85J F M A S O N D 1 91 62 5 76 94 27 3 85 15 17 2 63 12 98 37 67 84 60 19 49 79 89 43 86 88 35 4 10 18 22 56 58 30 42 32 93 95 57 16 80 46 7 20 55 75 36 6 90 74 54 23 11 44 83 33 61 28 53 29 8 87 99 65 9 50 81 51 24 52 25 13 59 47 39 14 66 40 41 82 38 77 97 92 31 26 72 34 69 96 78 68
100 71 21 48 73 45 64
Demand~Uniform(0,100)

9. Demand
Demand~Uniform(0,100)

10. Economic Factors
Cost to buy each paper is $0.30. Selling price of each paper is $1.20. At the end of the day, unsold papers are worthless (i.e., they have no salvage value)

11. Economic Factors
Cost = $0.30 Selling price = $1.20 What if we order one paper too many? What if we order one paper too few?

12. Objective
Maximize expected profit

13. Aside: Expected Value
For every possible outcome, multiply the value of that outcome by the probability of that outcome, and add them all up. 𝐸𝑉= 𝑖 ∈ 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑝𝑟𝑜 𝑏 𝑖 ×𝑣𝑎𝑙𝑢 𝑒 𝑖

14. Aside: Expected Value
“I will give you $10 for sure.” 𝐸𝑉=100%×$10=$10

15. Aside: Expected Value
“Flip a coin. If it’s heads, I’ll give you $10, but if it is tails, you don’t get anything.” 𝐸𝑉= 50%×$ %×$0 =$5

16. Aside: Expected Value
“Roll one die. If it is a six, I’ll give you $10, otherwise, you don’t get anything.” 𝐸𝑉= 1 6 ×$ ×$0 =$1.67

17. Aside: Expected Value
“Roll one die. If you roll a one, I’ll give you $1; if you roll a two, I’ll give you $2; …” 𝐸𝑉= 1 6 ×$ ×$ ×$ ×$ ×$ ×$6 =$3.50

19. Review of Setting
Cost = $0.30 Selling price = $1.20 Demand ~ Uniform (0,100) Average Demand = ? Underage Cost = ? Overage Cost = ?

20. Review of Setting
Cost = $0.30 Selling price = $1.20 Demand ~ Uniform (0,100) Average Demand = 50 Underage Cost = $0.90 Overage Cost = $0.30

21. Marginal Analysis
Average demand is 50 papers. Let the “default” action be to order 50 papers. Now, would we be better off by ordering 51, instead?
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22. Marginal Analysis
P(51st item is not sold) = P(Demand ≤ 50) P(51st item IS sold) = P(Demand > 50) E[Marginal Cost] = P(Demand ≤ 50) × Overage Cost E[Marginal Profit] = P(Demand > 50) × Underage Cost E[Marginal Cost] = P(Demand ≤ 50) × $0.30 E[Marginal Profit] = P(Demand > 50) × $0.90
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23. Marginal Analysis
Thus, we should increase our order from 50 to 51 IF … E[Marginal Cost] < E[Marginal Profit] P(Demand ≤ 50) × $0.30 < P(Demand > 50) × $ %× $0.30 < 50% × $0.90 $0.15 < $0.45
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24. Marginal Analysis In general, we increase or order while …
E[Marginal Cost] < E[Marginal Profit]
And we stop increasing once
E Marginal Cost = E[Marginal Profit]
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25. Marginal Analysis In other words, we stop at an optimal 𝑞 ∗ when …
E Marginal Cost = E[Marginal Profit]
𝑃 Demand ≤ 𝑞 ∗ × 𝑐 𝑜 = 𝑃(Demand > 𝑞 ∗ ) × 𝑐 𝑢
𝑃 Demand ≤ 𝑞 ∗ × 𝑐 𝑜 = 1−𝑃 Demand≤ 𝑞 ∗ × 𝑐 𝑢
𝑃 Demand ≤ 𝑞 ∗ = 𝑐 𝑢 𝑐 𝑜 + 𝑐 𝑢
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26. Unit Profit Margin or “Contribution”
Critical Fractile
Back to our original example, we have …
𝑃 Demand ≤ 𝑞 ∗ = 𝑐 𝑢 𝑐 𝑜 + 𝑐 𝑢 = $0.90 $0.30+$0.90 =0.75
If there is no salvage value in the case of overage and no chance for emergency restock in the case of shortage, then …
𝑃 Demand ≤ 𝑞 ∗ = 𝑐 𝑢 𝑐 𝑜 + 𝑐 𝑢 = 𝑝−𝑐 𝑝 = $0.90 $1.20 =0.75
Unit Profit Margin or “Contribution”

27. Critical Fractile
𝑃 Demand ≤ 𝑞 ∗ =0.75 What does this “critical fractile” really mean?
75%
25%
𝑞 ∗ =75

28. Critical Fractile
𝑃 Demand ≤ 𝑞 ∗ =0.75 What does this “critical fractile” really mean?
75%
25%
𝑞 ∗ =66

29. Critical Fractile
𝑃 Demand ≤ 𝑞 ∗ =0.75 What does this “critical fractile” really mean?
75%
25%
𝑞 ∗ =57

30. Solving for Critical Fractile
𝑃 Demand ≤ 𝑞 ∗ =0.75
But how do we solve for 𝑞 ∗ ?
1. Using math: Inverse Cumulative Demand Function
𝑞 ∗ = 𝐹 −1 𝑐 𝑢 𝑐 𝑜 + 𝑐 𝑢
2. Using simple Excel functions

31. Solving for Critical Fractile
𝐶𝐹= 𝑐 𝑢 𝑐 𝑜 + 𝑐 𝑢 If demand ~ Uniform(min, max) … 𝑞 ∗ = CF * (max – min) + min If demand ~ Normal(µ, σ2) … 𝑞 ∗ = NORM.INV(CF, µ, σ)