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

 DQ: cost = 50Q DQ: 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  DQ: cost = 50Q Cu DQ: 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. dev.

cu  co  z  0  Q*  z   decreases in  If excess inventory is more costly , Q* decreases in std. dev.

cu  co  z  0  Q*   does not change in  If shortages and excess inventory cost the same, order expected demand regardless of standard deviation

16 Example - INFORMS

cu=40 < co=50 z=-0.14 cu=70 > co=50 z=0.21

Q* Q*

Standard deviation Standard deviation

Example – Fashion Bags Input

 Unit cost c = $28.50

 Selling price p = $150

 Salvage value s = $20

 Cost of inventory = $0.40 for each dollar tied up in inventory at the end of the season

17 Example – Fashion Bags Input

 Unit cost c = $28.50

 Selling price p = $150

 Salvage value s = $20

 Cost of inventory = $0.40 for each dollar tied up in inventory at the end of the season

Computed input:

 Holding cost per bag h =

 cu =

 co = c F(Q)  u  cu  co

Example – Fashion Bags Input

 Unit cost c = $28.50

 Selling price p = $150

 Salvage value s = $20

 Cost of inventory = $0.40 for each dollar tied up in inventory at the end of the season

Computed input:

 Holding cost per bag h = (0.40)(28.50) = $11.4

 cu = p – c = $121.5

 co = c – s + h = $19.9 c 121.5 F(Q)  u   0.86 cu  co 121.5 19.9

18 Example – Fashion Bags – Normally distributed demand

Demand ~ Normal( 150,  20)

F(Q)  0.86  z 1.08 Area under the curve= Critical ratio = 0.86

 Q*  z    (20)(1.08) 150 Q*172

z = 1.08

Example – Fashion Bags – Uniformly distributed demand

 Demand Uniform between 50 and 250  =150

 Same expected demand as in Normal distribution

1 0.86  (Q*50)  200 Uniform density Area under the curve = Q*  222 Critical ratio = 0.86

1/200

Q 50 250

Q*= 222

19 Example – Fashion Bags

 Even though both the Normal and the Uniform distributions have the same mean (=150), why did we get different quantities?

 Normal distribution  Q*=172

 Uniform distribution  Q*=222

Example – Fashion Bags

 Even though both the Normal and the Uniform distributions have the same mean (=150), why did we get different quantities?

 Normal distribution  Q*=172

 Uniform distribution  Q*=222

Because of the variance (equivalently, standard deviation) and the shape of the distribution !!!

 Normal  =20

 Uniform  =57.7 Normal

Uniform

20