Inventory Management Subject to Uncertain Demand ISYE 3104 - Fall 2012 nar Keskinocak, 2007 nar Keskinocak, ı © Esma Gel, Gel, P © Esma Inventory Control Subject to Uncertain Demand In the presence of uncertain demand, the objective is to minimize the expected cost or to maximize the expected profit Two types of inventory control models Fixed time period - Periodic review One period (Newsvendor model) Multiple periods Fixed order quantity - Continuous review (Q,R) models 1 Types of Inventory Control Policies Fixed order quantity policies The order quantity is always the same but the time between the orders will vary depending on demand and the current inventory levels Inventory levels are continuously monitored and an order is placed whenever the inventory level drops below a prespecified reorder point. Continuous review policy Types of Inventory Control Policies Fixed time period policies The time between orders is constant, but the quantity ordered each time varies with demand and the current level of inventory Inventory is reviewed and replenished in given time intervals, such as a week or month (i.e., review period) Depending on the current inventory level an order size is determined to (possibly) increase the inventory level up-to a prespecified level (i.e., order- up-to level) Periodic review policy 2 Inventory Control - Demand Variability Constant/Stationary Variable/Non-Stationary Economic Order Aggregate Planning – Quantity (EOQ) – Planning for capacity Tradeoff between levels given a forecast fixed cost and holding Materials Requirements cost Deterministic Planning (MRP) Uncertainty Lot size/Reorder point (Q,R) or (s,S) models – Very difficult problem! Tradeoff between fixed cost, holding cost, and Stochastic shortage cost Newsvendor – single period READ THE APPENDIX ON PROBABILITY REVIEW 3 Newsvendor Models ISYE 3104 – Fall 2012 nar Keskinocak, 2007 nar Keskinocak, ı © Esma Gel, Gel, P © Esma Example - INFORMS INFORMS (The Institute for Operations Research and Management Science, www.informs.org) will hold its annual meeting in Washington D.C. in 2008. Six months before the meeting begins, INFORMS must decide how many rooms should be reserved at the conference hotel. At this time, rooms can be reserved at a cost of $50 per room. It is estimated that the demand for rooms is normally distributed with mean 5000 and standard deviation 2000. If the number of rooms required exceeds the number of rooms reserved, extra rooms will have to be found at neighboring hotels at a cost of $80 per room. The inconvenience of staying at another hotel is estimated at $10. How many rooms should be reserved to minimize the expected cost? 4 Example – Fashion Bags The buyer for What-a-Markup Fashion Bags must decide on the quantity of a high-priced woman’s handbag to procure in Italy for the following Christmas Season. The unit cost of the handbag to the store is $28.50 and the handbag will sell for $150. A discount firm purchases any handbags not sold by the end of the season for $20. In addition, the store accountants estimate that there is cost of $0.40 for each dollar tied up in inventory at the end of the season (after all sales have been made). Newsvendor model - Properties One-time decision Current decisions only impact the “next period” but not future periods Retail: fashion/seasonal items One-time events 5 Tradeoff in inventory decisions Demand Supply Shortage Supply < Demand Lost sales / Lost profit Excess inventory Supply > Demand Inventory cost Supply Demand Newsvendor model - Properties One-time decision Current decisions only impact the “next period” but not future periods Retail: fashion/seasonal items One-time events Relevant costs Co: Unit cost of excess inventory (cost of overage) Cu: Unit cost of shortage (cost of underage) Demand D is a random variable with Probability density function (pdf), f(x) cumulative distribution function (cdf), F(x) Objective: Choose the ordering quantity Q (before you know the demand) to minimize Total expected overage and underage costs 6 Newsvendor model – Finding the optimal order quantity First, write the cost function: G(Q, D) :total overage underage cost (if Q is ordered and demand is D) cu (D Q) if D Q G(Q, D) : co (Q D) if D Q G(Q) expected overage underage cost if Q is ordered G(Q) E[G(Q, D] G(Q, x) f (x)dx Critical Ratio 0 Q c c (Q D) f (x)dx c (D Q) f (x)dx F(Q) u o u 0 Q cu co Derivation of the Critical Ratio - 1 G(Q) expected overage underage cost if Q is ordered Q G(Q) c (Q D) f (x)dx c (D Q) f (x)dx c G c G o u o 1 u 2 0 Q To take the derivative of G(Q), use Leibniz's Rule d f2 ( y) f2 ( y) h(x, y) h(x, y)dx dx h( f (y), y). f ' (y) h( f (y), y). f ' (y) dy y 2 2 1 1 f1 ( y) f1 ( y) In G1: y Q f2 (y) Q f1(y) 0 h(x,Q) (Q x) f (x) 7 Derivation of the Critical Ratio - 2 Q G (Q) (Q D) f (x)dx In G1: y Q f2 (y) Q f1(y) 0 1 0 h(x,Q) (Q x) f (x) To take the derivative of G(Q), use Leibniz's Rule d f2 ( y) f2 ( y) h(x, y) h(x, y)dx dx h( f (y), y). f ' (y) h( f (y), y). f ' (y) dy y 2 2 1 1 f1 ( y) f1 ( y) h(x, y) [(Q D) f (x)] f (x) y Q ' h( f2 (y), y) h(Q,Q) (Q Q) f (x) 0 f2 (y) 1 ' h( f1(y), y) h(0,Q) (Q 0) f (x) Qf (x) f1 (y) 0 d Q d G (Q) f (x)dx Similarly, G (Q) f (x)dx 1 2 dQ 0 dQ Q Derivation of the Critical Ratio - 3 d Q G(Q) c f (x)dx c f (x)dx c F(Q) c (1 F(Q)) o u o u dQ 0 Q (co cu )F(Q) cu 0 c F(Q) u cu co Recall: f (x)dx 1 0 Q f (x)dx F(Q) P(D Q) f (x)dx 1 F(Q) P(D Q) 0 Q Also need to check if G(Q) is convex. Second derivative is (co+cu)f(Q)≥0 8 Example - INFORMS D = number of rooms actually required Q = number of rooms reserved What are Co and Cu ? Example - INFORMS D = number of rooms actually required Q = number of rooms reserved DQ: cost = 50Q DQ: cost = 50Q+80(D-Q)+10(D-Q) =90D-40Q Cost of Cost of Cost of reserving reserving inconvenience extra rooms in other hotels 9 Example - INFORMS D = number of rooms actually required Q = number of rooms reserved Co DQ: cost = 50Q Cu DQ: cost = 50Q+80(D-Q)+10(D-Q)=90D-40Q Cost of Cost of Cost of reserving reserving inconvenience extra rooms in other hotels cu 40 4 F(Q) What is Q? cu co 40 50 9 Example - INFORMS Recall: D Normal(5000,2000) We have lookup tables for Standard Normal distribution ( =0, =1) Convert to Standard Normal! 4 F(Q) P(D Q) 0.444 9 D Q P(D Q) P(D Q ) P P(Z z) (z) Standard normal Zz Find z from the lookup table Q = z + 10 Example - INFORMS Recall: D Normal(5000,2000) We have lookup tables for Standard Normal distribution ( =0, =1) Convert to Standard Normal! 4 F(Q) P(D Q) 0.444 9 D Q P(D Q) P(D Q ) P P(Z z) (z) Safety Standard normal Zzstock Expected demand Find z from the lookup table Q = z + Standard Normal Distribution - 1 z 2 e 2 Density function (z) 2 Symmetric around the mean Area under the entire curve 1 P(Z ) 11 Standard Normal Distribution - 2 z 2 e 2 Density function (z) We will use ɸ(z) and F(z) 2 interchangeably Symmetric around the mean Area under the entire curve 1 P(Z ) F(z)=P(Z≤z) z Standard Normal Distribution - 2 z 2 e 2 Density function (z) We will use ɸ(z) and F(z) 2 interchangeably Symmetric around the mean Area under the entire curve 1 P(Z ) F(z)=P(Z≤z) F(-z)=P(Z≤-z)=1-F(z) -z z 12 Example - INFORMS P(Z z) (z) Area under the curve 0.444 z -0.14 Q* z 2000(0.14) 5000 Q* 4720 z = - 0.14 Example - INFORMS Why is Q*<5000, i.e., less than the expected demand? Expected total cost Expected cost of underage Expected cost of overage Q*=4720 13 Example - INFORMS What if co=150? Expected c 40 total cost F(Q) u cu co 40 150 0.21 z 0.805 Expected Q* z Expected cost of (2000)(0.805) 5000 cost of overage 3390 underage Q*=3390 Example - INFORMS What if co=150? Expected c 40 total cost F(Q) u c c 40 150 u o C Q* 0.21 z 0.805 o Expected Cu Q* Q* z Expected cost of (2000)(0.805) 5000 cost of overage 3390 underage Q*=3390 14 When do we have Q*=Expected demand? Q* z If Q* then z 0 !!! P(Z z) (z) Area under the curve 0.5 z 0 For Q* we need cu F(Q) 0.5, i.e., cu co!!! cu co z = 0 Optimal quantity versus expected demand cu cu co F(Q) 0.5 z 0 Q* cu co If shortages are more costly, order more than expected demand cu cu co F(Q) 0.5 z 0 Q* cu co If excess inventory is more costly order less than expected demand cu cu co F(Q) 0.5 z 0 Q* cu co If shortages and excess inventory cost the same, order expected demand Note: Assuming the demand distribution is symmetric around its mean 15 The impact of standard deviation What happens to the optimal quantity as the standard deviation increases? The impact of standard deviation What happens to the optimal quantity as the standard deviation increases? (Assuming the demand distribution is symmetric around its mean) cu co z 0 Q* z increases in If shortages are more costly, Q* increases in std.