Managing Flow Variability: Safety Inventory The Magnitude of Shortages (Out of Stock) The Newsvendor Ardavan Asef-Vaziri, Oct 2011 1 Managing Flow Variability: Safety Inventory Optimal Service Level: The Newsvendor Problem How do we choose what level of service a firm should offer? Cost of ordering too much: holding cost, salvage Cost of ordering too little: loss of sale, low service level The decision maker balances the expected costs of ordering too much with the expected costs of ordering too little to determine the optimal order quantity. The Newsvendor Ardavan Asef-Vaziri, Oct 2011 2 Managing Flow Variability: Safety Inventory News Vendor Model; Assumptions Demand is random

Distribution of demand is known No initial inventory Set-up cost is zero Single period Zero lead time Linear costs Purchasing (production) Salvage value Revenue Goodwill The Newsvendor Ardavan Asef-Vaziri, Oct 2011 3 Managing Flow Variability: Safety Inventory Optimal Service Level: The Newsvendor Problem Cost =1800, Sales Price = 2500, Salvage Price = 1700 Underage Cost = 2500-1800 = 700, Overage Cost = 1800-1700 = 100 Demand 100 110 120 130 140 150 160 170 180

190 200 Probability of Demand 0.02 0.05 0.08 0.09 0.11 0.16 0.20 0.15 0.08 0.05 0.01 What is probability of demand to be equal to 130? 0.09 0.35 What is probability of demand to be less than or equal to 140? 0.02+0.05+0.08+0.09+0.11= What is probability of demand to be greater than or equal to 140? 1-0.35+0.11= 0.76 What is probability of demand to be equal to 133? 0 P(R Q ) = 1-P(R Q)+P(R = Q) The Newsvendor R is quantity of demand Q is the quantity ordered Ardavan Asef-Vaziri, Oct 2011 4 Managing Flow Variability: Safety Inventory Optimal Service Level: The Newsvendor Problem Demand 100

101 102 103 104 105 106 107 108 109 Probability of Demand 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 Demand 110 111 112 113 114 115 116 117 118 119 Probability of Demand 0.005 0.005 0.005

0.005 0.005 0.005 0.005 0.005 0.005 0.005 What is probability of demand to be equal to 116? 0.005 What is probability of demand to be less than or equal to 116? 0.02+0.035 = 0.055 What is probability of demand to be greater than or equal to 116? 1-0.055+0.005 = 0.95 What is probability of demand to be equal to 113.3? 0 The Newsvendor Ardavan Asef-Vaziri, Oct 2011 5 Managing Flow Variability: Safety Inventory Optimal Service Level: The Newsvendor Problem Average Demand 100 110 120 130 140 150 160 170 180 190 200 The Newsvendor

Probability of Demand 0.02 0.05 0.08 0.09 0.11 0.16 0.20 0.15 0.08 0.05 0.01 What is probability of demand to be equal to 130? 0 What is probability of demand to be less than or equal to 145? 0.02+0.05+0.08+0.09+0.11 = 0.35 What is probability of demand to be greater than or equal to 145? 1-0.35 = 0.65 P(R Q) = 1-P(R Q) Ardavan Asef-Vaziri, Oct 2011 6 Managing Flow Variability: Safety Inventory Compute the Average Demand N Average Demand Qi P( R Qi ) i 1 Average Demand = +1000.02 +1100.05+1200.08 +1300.09+1400.11 +1500.16 +1600.20 +1700.15 +1800.08 +1900.05+2000.01

Average Demand = 151.6 Qi 100 110 120 130 140 150 160 170 180 190 200 P( R =Q i ) 0.02 0.05 0.08 0.09 0.11 0.16 0.20 0.15 0.08 0.05 0.01 How many units should I have to sell 151.6 units (on average)? 200 How many units do I sell (on average) if I have 100 units? 100 The Newsvendor Ardavan Asef-Vaziri, Oct 2011 7 Managing Flow Variability: Safety Inventory

Deamand (Qi) 100 110 120 130 140 150 160 170 180 190 200 Porbability Prob (R Qi) 0.02 1.00 0.05 0.98 0.08 0.93 0.09

0.85 0.11 0.76 0.16 0.65 0.20 0.49 0.15 0.29 0.08 0.14 0.05 0.06 0.01 0.01 Suppose I have ordered 140 units. On average, how many of them are sold? In other words, what is the expected value of the number of sold units? When I can sell all 140 units? I can sell all 140 units if R 140 Prob(R 140) = 0.76 The expected number of units sold for this part- is (0.76)(140) = 106.4 Also, there is 0.02 probability that I sell 100 units 2 units Also, there is 0.05 probability that I sell 110 units5.5 Also, there is 0.08 probability that I sell 120 units 9.6 Also, there is 0.09 probability that I sell 130 units 11.7 106.4 + 2 + 5.5 + 9.6 + 11.7 = 135.2 The Newsvendor

Ardavan Asef-Vaziri, Oct 2011 8 Managing Flow Variability: Safety Inventory Deamand (Qi) 100 110 120 130 140 150 160 170 180 190 200 Porbability Prob (R Qi) 0.02 1.00 0.05 0.98

0.08 0.93 0.09 0.85 0.11 0.76 0.16 0.65 0.20 0.49 0.15 0.29 0.08 0.14 0.05 0.06 0.01 0.01 Suppose I have ordered 140 units. On average, how many of them are salvaged? In other words, what is the expected value of the number of salvaged units? 0.02 probability that I sell 100 units. In that case 40 units are salvaged 0.02(40) = .8 0.05 probability to sell 110 30 salvaged 0.05(30)= 1.5 0.08 probability to sell 120 20 salvaged 0.08(20) = 1.6 0.09 probability to sell 130 10 salvaged 0.09(10) =0.9 0.8 + 1.5 + 1.6 + 0.9 = 4.8 Total number Sold

135.2 @ 700 = 94640 Total number Salvaged 4.8 @ -100 = -480 Expected Profit = 94640 480 = 94,160 The Newsvendor Ardavan Asef-Vaziri, Oct 2011 9 Managing Flow Variability: Safety Inventory Cumulative Probabilities Qi 100 110 120 130 140 150 160 170 180 190 200 Probabilities P(R =Qi) P(R

0.09 0.15 0.85 0.11 0.24 0.76 0.16 0.35 0.65 0.2 0.51 0.49 0.15 0.71 0.29 0.08 0.86 0.14 0.05 0.94 0.06 0.01 0.99 0.01 The Newsvendor Ardavan Asef-Vaziri, Oct 2011 10 Managing Flow Variability: Safety Inventory Number of Units Sold, Salvaged Qi 100 110

120 130 140 150 160 170 180 190 200 Probabilities P(R =Qi) P(R

0.14 0.05 0.94 0.06 0.01 0.99 0.01 The Newsvendor Units Sold Salvage 100 0 109.8 0.2 119.1 0.9 127.6 2.4 135.2 4.8 141.7 8.3 146.6 13.4 149.5 20.5 150.9 29.1 151.5 38.5 151.6 48.4 Ardavan Asef-Vaziri, Oct 2011

[email protected] [email protected] 11 Managing Flow Variability: Safety Inventory Total Revenue for Different Ordering Policies Qi 100 110 120 130 140 150 160 170 180 190 200 Probabilities P(R =Qi) P(R

0.24 0.76 0.16 0.35 0.65 0.2 0.51 0.49 0.15 0.71 0.29 0.08 0.86 0.14 0.05 0.94 0.06 0.01 0.99 0.01 The Newsvendor Units Sold Salvaged 100 0 109.8 0.2 119.1 0.9 127.6 2.4 135.2 4.8 141.7 8.3

146.6 13.4 149.5 20.5 150.9 29.1 151.5 38.5 151.6 48.4 Ardavan Asef-Vaziri, Oct 2011 Sold 70000 76860 83370 89320 94640 99190 102620 104650 105630 106050 106120 Revenue Salvaged 0 20 90 240 480 830 1340 2050 2910 3850

4840 Total 70000 76840 83280 89080 94160 98360 101280 102600 102720 102200 101280 12 Managing Flow Variability: Safety Inventory Denim Wholesaler; Marginal Analysis The demand for denim is: 1000 with probability 0.10 2000 with probability 0.15 3000 with probability 0.15 4000 with probability 0.20 Unit Revenue (p ) = 30 Unit purchase cost (c )= 10 Salvage value (v )= 5 Goodwill cost (g )= 0 5000 with probability 0.15 6000 with probability 0.15 7000 with probability 0.10 How much should we order? The Newsvendor Ardavan Asef-Vaziri, Oct 2011

13 Managing Flow Variability: Safety Inventory Marginal Analysis Marginal analysis: What is the value of an additional unit ordered? Suppose the wholesaler purchases 1000 units What is the value of having the 1001st unit? Marginal Cost: The retailer must salvage the additional unit and losses $5 (10 5). P(R 1000) = 0.1 Expected Marginal Cost = 0.1(5) = 0.5 The Newsvendor Ardavan Asef-Vaziri, Oct 2011 14 Managing Flow Variability: Safety Inventory Marginal Analysis Marginal Profit: The retailer makes and extra profit of $20 (30 10) P(R > 1000) = 0.9 Expected Marginal Profit= 0.9(20) = 18 MP MC Expected Value = 18-0.5 = 17.5 By purchasing an additional unit, the expected profit increases by $17.5 The retailer should purchase at least 1,001 units. The Newsvendor Ardavan Asef-Vaziri, Oct 2011 15

Managing Flow Variability: Safety Inventory Marginal Analysis Should he purchase 1,002 units? Marginal Cost: $5 salvage P(R 1001) = 0.1 Expected Marginal Cost = 0.5 Marginal Profit: $20 profit P(R >1002) = 0.9 18 Expected Marginal Profit = 18 Expected Value = 18-0.5 = 17.5 Assuming that the initial purchasing quantity is between 1000 and 2000, then by purchasing an additional unit exactly the same savings will be achieved. Conclusion: Wholesaler should purchase at least 2000 units. The Newsvendor Ardavan Asef-Vaziri, Oct 2011 16 Managing Flow Variability: Safety Inventory Marginal Analysis Marginal analysis: What is the value of an additional unit ordered? Suppose the retailer purchases 2000 units What is the value of having the 2001st unit? Marginal Cost: The retailer must salvage the additional unit and losses $5 (10 5). P(R 2000) = 0.25 Expected Marginal Cost = 0.25(5) = 1.25 The Newsvendor Ardavan Asef-Vaziri, Oct 2011 17 Managing Flow Variability: Safety Inventory

Marginal Analysis Marginal Profit: The retailer makes and extra profit of $20 (30 10) P(R > 2000) = 0.75 Expected Marginal Profit= 0.75(20) = 15 MP MC Expected Value = 15-1.25 = 13.75 By purchasing an additional unit, the expected profit increases by $13.75 The retailer should purchase at least 2,001 units. The Newsvendor Ardavan Asef-Vaziri, Oct 2011 18 Managing Flow Variability: Safety Inventory Marginal Analysis Should he purchase 2,002 units? Marginal Cost: $5 salvage P(R 2001) = 0.25 Expected Marginal Cost = 1.25 Marginal Profit: $20 profit P(R >2002) = 0.75 Expected Marginal Profit = 15 Expected Value = 15-1.25 = 13.75 Assuming that the initial purchasing quantity is between 2000 and 3000, then by purchasing an additional unit exactly the same savings will be achieved. Conclusion: Wholesaler should purchase at least 3000 units. The Newsvendor Ardavan Asef-Vaziri, Oct 2011 19 Managing Flow Variability: Safety Inventory

Marginal Analysis Why does the marginal value of an additional unit decrease, as the purchasing quantity increases? Expected cost of an additional unit increases Expected savings of an additional unit decreases Cumulative Expected Expected Expected Demand Probability Probability Marginal Cost Marginal Profit Marginal Value 1000 0.10 0.1 0.50 18 17.50 2000 0.15 0.25 1.25 15 13.75 3000 0.15 0.40 2.00 12 10.00 4000 0.20 0.60 3.00 8 5.00 5000 0.15 0.75 3.75

5 1.25 6000 0.15 0.90 4.50 2 -2.50 7000 0.10 1.00 5.00 0 -5.00 The Newsvendor Ardavan Asef-Vaziri, Oct 2011 20 Managing Flow Variability: Safety Inventory Marginal Analysis What is the optimal purchasing quantity? Answer: Choose the quantity that makes marginal value: zero Marginal value 17.5 13.75 10 5 1.3 -2.5 Quantity 1000 2000

3000 4000 5000 6000 7000 8000 -5 The Newsvendor Ardavan Asef-Vaziri, Oct 2011 21 Managing Flow Variability: Safety Inventory Additional Example On consecutive Sundays, Mac, the owner of your local newsstand, purchases a number of copies of The Computer Journal. He pays 25 cents for each copy and sells each for 75 cents. Copies he has not sold during the week can be returned to his supplier for 10 cents each. The supplier is able to salvage the paper for printing future issues. Mac has kept careful records of the demand each week for the journal. The observed demand during the past weeks has the following distribution: Qi 4 5 6 7 8 9 10 11 12 13 P(R=Qi) 0.04 0.06 0.16 0.18 0.2 0.1 0.1 0.08 0.04 0.04

The Newsvendor Ardavan Asef-Vaziri, Oct 2011 22 Managing Flow Variability: Safety Inventory Additional Example Qi 4 5 6 7 8 P(R=Qi) 0.04 0.06 0.16 0.18 0.2 9 0.1 10 0.1 11 12 13 0.08 0.04 0.04 a) How many units are sold if we have ordered 7 units There is 0.18 + 0.20 + 0.10 + 0.10 + 0.08 + 0.04 + 0.04 = 0.74 There is 0.74 probability that demand is greater than or equal to 7. There is 0.16 probability that demand is equal to 6. There is 0.06 probability that demand is equal to 5. There is 0.04 probability that demand is equal to 4. The expected number of units sold is 0.74(7) + 0.16 (6) + 0.06 (5) + 0.04 (4) = 6.6 The Newsvendor Ardavan Asef-Vaziri, Oct 2011

23 Managing Flow Variability: Safety Inventory Additional Example Qi 4 5 6 7 8 P(R=Qi) 0.04 0.06 0.16 0.18 0.2 9 0.1 10 0.1 11 12 13 0.08 0.04 0.04 b) How many units are salvaged? 7-6.6 = 0.4. Alternatively, we can compute it directly There is 0.74 probability that we salvage 7 7 = 0 units There is 0.16 probability that we salvage 7- 6 = 1 units There is 0.06 probability that we salvage 7- 5 = 2 units There is 0.04 probability that we salvage 7-4 = 3 units The expected number of units salvaged is 0.74(0) + 0.16 (1) + 0.06 (2) + 0.04 (3) = 0.4 and 7-0.4 = 6.6 sold The Newsvendor Ardavan Asef-Vaziri, Oct 2011 24 Managing Flow Variability: Safety Inventory

Additional Example c) Compute the total profit if we order 7 units. Out of 7 units, 6.6 sold, 0.4 salvaged. P = 75, c= 25, v=10. Profit per unit sold = 75-25 = 50 Cost per unit salvaged = 25-10 = 15 Total Profit = 6.6(50) + 0.4(15) = 330 - 6 = 324 The Newsvendor Ardavan Asef-Vaziri, Oct 2011 25 Managing Flow Variability: Safety Inventory Additional Example Qi 4 5 6 7 8 P(R=Qi) 0.04 0.06 0.16 0.18 0.2 9 0.1 10 0.1 11 12 13 0.08 0.04 0.04 d) Compute the expected Marginal profit of ordering the 8th unit. MP = 75-25 = 50 P(R 8) = 0.2 + 0.1 + 0.1 + 0.08 + 0.04 + 0.04 = 0.56 Expected Marginal profit = 0.56(50) = 28

e) Compute the expected Marginal cost of ordering the 8th unit. MC = 25 10 = 15 P(R 7) = 1-0.56 = 0.44 Expected Marginal cost = 0.44(15) = 6.6 The Newsvendor Ardavan Asef-Vaziri, Oct 2011 26 Managing Flow Variability: Safety Inventory Another Example Swell Productions (The Retailer) is sponsoring an outdoor conclave for owners of collectible and classic Fords. The concession stand in the T-Bird area will sell clothing such as official Thunderbird racing jerseys. The following table shows the probability of jerseys sales quantities. Probability Demand 0.05 100 0.10 200 0.30 300 0.20 400 0.20 500 0.15 600

A) Compute the average demand (units that can be sold) for Swell Productions jerseys. 0.05 100 0.1 0.3 0.2 0.2 0.15 1 The Newsvendor Ardavan Asef-Vaziri, Oct 2011 200 300 400 500 600 385 27 Managing Flow Variability: Safety Inventory Another Example B) Given the average demand you have obtained in the previous part. How many units should Swell Productions order to be able to have the average number of units sold equal to the average demand. 600 C) Supposed Swell Productions has ordered 400 units. Compute the marginal profit of ordering one more unit. 40(0.35) = 14 The Newsvendor

Ardavan Asef-Vaziri, Oct 2011 28 Managing Flow Variability: Safety Inventory Another Example D) Supposed Swell Productions has ordered 400 units. Compute the marginal cost of ordering one more unit. 20(0.65) = 13 E) Suppose your computations indicates that it is at Swell Productions benefit to order 401 units (this may or may not the correct answer). How many units should Swell Productions order? 500 The Newsvendor Ardavan Asef-Vaziri, Oct 2011 29 Managing Flow Variability: Safety Inventory Another Example F) Suppose Swell Productions has ordered 500 units. Compute the expected value of the number of units salvaged. 0.05(400)+0.1(300) +0.3 (200) + 0.2 (100) = 20+30+60+20 = 130 G) Suppose Swell Productions orders 500 jerseys. Compute the expected number of jerseys that can be sold. 0.05 0.1 0.3 0.2 0.35

500-130= 370 100 200 300 400 500 370 The Newsvendor Ardavan Asef-Vaziri, Oct 2011 30 Managing Flow Variability: Safety Inventory Another Example H) Supposed Swell Productions has ordered 500 units. Compute the expected value of Swell Productions total net profit. -130(20) + 370(40) = -2600+ 14800 = 12200 I) At what purchasing price (current purchasing price is $40 and current salvage value is 20) will you order 600 units? MC=0.85 (c-20) = 0.85c -17 MP = 0.15(80-c) = 12- 0.15c 12-0.15c> 0.85c-17 29> c The Newsvendor Ardavan Asef-Vaziri, Oct 2011 31 Managing Flow Variability: Safety Inventory Another Example It really does not make sense to say the break-even point is where 385(40) = 215X. Because X and 40 are not independent.

To correct the above statement, one may say: the break-even point is where 385(80-c) = 215(c-20) One may try to solve the above equation. But it does not make sense because in that case and under the price of c, we will make 0 profit if we order 600. While we already make $12200, by ordering 500. Therefore no matter what c value comes out of the above equation, it makes our profit equal 0. Why we should order 600 for 0 profit compared to 500 with $12200 The Newsvendor Ardavan Asef-Vaziri, Oct 2011 32 Managing Flow Variability: Safety Inventory Another Example A correct second procedure to solve the problem is to say; we order 600 if its total expected revenue is greater than ordering 500. Ordering 500 # of units sold is 370 and salvaged 130 Ordering 600 # of units sold is 385 and salvaged 215 For sale we get (80-c) for salvage we pay (c-20) Therefore, total revenue of 600 must be greater than that of 500 385(80-c) 215(c-20) > 370(80-c) 130(c-20) By solving this equation we will get 29>c The Newsvendor Ardavan Asef-Vaziri, Oct 2011 33 Managing Flow Variability: Safety Inventory Another Example Indeed we could have also said that by ordering 600 we sell 15 units more and salvage 85 units more. And the sale revenue must be greater than salvaged marginal cost

15(80-c) > 85(c-20) 29>c The Newsvendor Ardavan Asef-Vaziri, Oct 2011 34