Inventory Models Uncertain Demand: The Newsvendor Model Background: expected value A fruit seller example Undamaged mango Damaged mango Profit $4 $1 Probability 80% 20% What is the expected profit for a stock of 100 mangoes ? 0.8 x 100 ($4) + 0.2 x 100 x ($1) = 320 + 20 = $340 random variable: ai probability: pi Expected value = a1 p1 + a2 p2 + + ak pk = i = 1,,k aipi

Probabilistic models: Flower seller example Wedding bouquets: Selling price: $50 (if sold on same day), $ 0 (if not sold on that day) Cost = $35 number of bouquets probability 3 4 5 6 7 8 9 0.05 0.12 0.20 0.24

0.17 0.14 0.08 How many bouquets should he make each morning to maximize the expected profit? Probabilistic models: Flower seller example.. number of bouquets probability 3 4 5 6 7 8 9 0.05

0.12 0.20 0.24 0.17 0.14 0.08 CASE 1: Make 3 bouquets probability( demand 3) = 1 Exp. Profit = 3x50 3x35 = $45 CASE 2: Make 4 bouquets if demand = 3, then revenue = 3x $50 = $150 if demand = 4 or more, then revenue = 4x $50 = $200 prob = 0.05 prob = 0.95 Exp. Profit = 150x0.05 + 200x0.95 4x35 = $57.5 Probabilistic models: Flower seller example Compute expected profit for each case number of bouquets

probability Expected profit 3 4 5 6 7 8 9 0.05 0.12 0.20 0.24 0.17 0.14 0.08

45 57.5 64 60.5 45 21 -10 Making 5 bouquets will maximize expected profit. Probabilistic models: definitions number of bouquets probability 3 4 5 6

7 8 9 0.05 0.12 0.20 0.24 0.17 0.14 0.08 Discrete random variable Probability (sum of all likelihoods = 1) Continuous random variable: Example, height of people in a city -4 -3140

-2 150 -1 160 0 170 1 180 2 190 3 200 4 Probability density function (area under curve = integral over entire range = 1) Probabilistic models: normal distribution function Standard normal distribution curve: mean = 0, std dev. = 1 P( a x b) = ab f(x) dx -4 -3

-2 -1 0 a 1 2 3 4 b Property: normally distributed random variable x, mean = , standard deviation = , Corresponding standard random variable: z = (x )/ z is normally distributed, with a = 0 and = 1. The Newsvendor Model Assumptions: - Plan for single period inventory level - Demand is unknown

- p(y) = probability( demand = y), known - Zero setup (ordering) cost Example: Mrs. Kandells Christmas Tree Shop Order for Christmas trees must be placed in Sept Cost per tree: $25 Price per tree: $55 before Dec 25 $15 after Dec 25 If she orders too few, the unit shortage cost is cu = 55 25 = $30 If she orders too many, the unit overage cost is co = 25 15 = $10 Past Data Sales 22 24 26 28 30

32 34 36 Probability .05 .10 .15 .20 .20 .15 .10 .05 How many trees should she order? Stockout and Markdown Risks 1. Mrs. Kandell has only one chance to order until the sales begin: no information to revise the forecast; after the sales start: too late to order more.

2. She has to decide an order quantity Q now D total demand before Christmas F(x) the demand distribution, D > Q stockout, at a cost of: cu (D Q)+ = cu max{D Q, 0} D < Q overstock, at a cost of co (QD)+ = co max{Q D, 0} Key elements of the model 1. Uncertain demand 2. One chance to order (long) before demand 3. ( order > demand OR order < demand) COST Model development Stockout cost = cu max{D Q, 0} Overstock cost = co max{Q D, 0} Total cost = G(Q) = cu (D Q)+ + co (Q D)+ Expected cost, E( G(Q) ) = E(cu (D Q)+ + co (Q D)+) = cu E(D Q)+ + co E(Q D)+ Q [cu ( x Q) co (Q x) ]P( x) [cu ( x Q) ]P ( x) [co (Q x) ]P ( x) x 0

x Q x 0 Model Development: generalization Suppose Demand a continuous variable ++ good approximation when number of possibilities is high -- difficult to generate probabilities, but ++ probability distribution can be guessed Q E (G (Q)) [cu ( x Q) ]P( x) [co (Q x) ]P( x) x Q x 0 Q g (Q) E ( G (Q)) c x 0

0 (Q x) P( x) dx c x Q u ( x Q) P( x) dx Model solution Q g (Q) E ( G (Q)) c x 0 0 (Q x) P( x) dx c

u ( x Q) P( x) dx x Q d g (Q) 0 Minimize g(Q) dQ Q d c0 (Q x) P( x) dx cu ( x Q) P( x) dx 0 dQ x 0 x Q g(Q) is a convex function: it has a unique minimum when g(Q) is at minimum value, F(Q) = cu/(cu + co) The Critical Ratio Solution to the Newsvendor problem: c dg (Q) 0 F (Q*) u dQ

c0 cu = cu /(co + cu ) is called the critical ratio relative importance of stockout cost vs. markdown cost Mrs. Kandells Problem, solved: cu = 55 25 = $30 Past Data D 22 Probability 0.05 F (D ) 0.05 co = 25 15 = $10 24 0.1 0.15 26 0.15 0.3 28 0.2 0.5 = cu /(co + cu ) = 30/(30 + 10) = 0.75

NOTE: 30 0.2 0.7 32 0.15 0.85 34 0.1 0.95 36 0.05 1 optimum 31 E(D) = 22x 0.05 + 24 x 0.1 + + 36 x 0.05 = 29 Newsvendor model: effect of critical ratio D 22 Probability 0.05 F (D ) 0.05 24 0.1

0.15 26 0.15 0.3 28 0.2 0.5 30 0.2 0.7 32 0.15 0.85 34 0.1 0.95 36 0.05 1 = cu /(co + cu ) = 30/(30 + 10) = 0.75 optimum: 31

overstock cost less significant order more overstock cost dominates order less Summary When demand is uncertain, we minimize expected costs newsvendor model: single period, with over- and under-stock costs Critical ratio determines the optimum order point Critical ratio affects the direction and magnitude of order quantity Concluding remarks on inventory control Inventory costs lead to success/failure of a company Example: Dell Inc. Dell's direct model enables us to keep low component inventories that enable us to give customers immediate savings when component prices are reduced, ... Because of our inventory management, Dell is able to offer some of the newest technologies at low prices while our competitors struggle to sell off older products. Drive to reduce inventory costs was main motivation for Supply Chain Management next: Quality Control