Supply Chain Management

Lecture 20

Outline

• Today– Chapter 11

• Sections 1, 2, 3, 7, 8– Skipping 11.2 “Evaluating Safety Inventory Given Desired Fill rate”

• Thursday– Homework 4 due before class– Chapter 12

• Sections 12.1, 12.2 up to and including Example 12.2, 12.3

• Friday– Homework 5 online

• Due Thursday April 8 before class

Staples Visit

• Date– Friday April 2, 10:30am – 2:30pm

• (11:30am – 1:30pm on site, the rest in transit)• Free transportation

• Location– Staples fulfillment Center, Brighton, CO

• What– Lunch and Learn

Email: Ann.Richmond@colorado.edu

Summary

• Inventory is an asset, but it's also a liability until it has been sold– Inventory is cash in-transit

• Inventory does not improve with time. Inventory becomes less valuable with every passing day – It takes space, heat, light, power, handling, insurance,

interest to carry, …

Effective inventory management is key to running a profitable business

Inventory Management

Apple beats competitors at inventory turn over

Company Days of inventoryApple 5 daysDell 7 daysLenovo 15 daysHP 32 daysIntel 89 daysD-Link 131 days

Source: The Mac Observer, March 5, 2009

Summary

• Lot sizing for a single product– Economic order quantity

• Lot sizing with multiple products (complete aggregation)– Find order frequency (same for all products)

– Calculate order quantity for each product Q = D/n

hC

DSQ

2*

*2* 1

S

ChDn

k

i iii

Safety Inventory

Inventory

Time

AverageInventory

CycleInventory

SafetyInventory

Cycle

Demand (D)Order quantity/lot size (Q)

Example 11-1: Evaluating safety inventory given an inventory policy • Assume that weekly demand for Palms at B&M

Computer World is normally distributed, with a mean of 2,500 and a standard deviation of 500. The manufacturer takes two weeks to fill an order placed by the B&M manager. The store manager currently orders 10,000 Palms when the inventory drops to 6,000. Evaluate the safety inventory carried by B&M and the average inventory carried by B&M. Also evaluate the average time spent by a Palm at B&M

Safety Inventory

Inventory

Time

AverageInventory

CycleInventory

SafetyInventory

CycleLead time (L)

Reorder point (ROP)

Demand (D)

Demand during lead time DL = LD

Order quantity/lot size (Q)

Example 11-1: Evaluating safety inventory given an inventory policy

Inventory

Time0

Reorder point

11,000

6,000

Safety Inventory1,000

Lead time

Order quantity/Lot size

Demand during lead time

Cycle

Example 11-1: Evaluating safety inventory given an inventory policy• Reorder point (ROP)

= demand during lead time + safety inventory= DL + ss

• Average flow time = (avg. inventory) / (avg. demand)

ROP = DL + ss

Role of Safety Inventory

• There is a fundamental tradeoff– Raising the level of safety inventory provides higher

levels of product availability and customer service– Raising the level of safety inventory also raises the

level of average inventory and therefore increases holding costs

Product availability reflects a firm’s ability to fill a customer order out of available inventory

Measuring Product Availability1. Cycle service level (CSL)

• Fraction of replenishment cycles that end with all customer demand met

• Probability of not having a stockout in a replenishment cycle

2. Product fill rate (fr)• Fraction of demand that is satisfied from product in inventory• Probability that product demand is supplied from available

inventory

3. Order fill rate• Fraction of orders that are filled from available inventory

CSL and fr are different!

CSL is 0%, fill rate is almost 100%inventory

time0

CSL is 0%, fill rate is almost 0% inventory

time0

Example 11-2: Evaluating cycle service level given a replenishment policy• Weekly demand for Palms is normally

distributed, with a mean of 2,500 and a standard deviation of 500. The replenishment lead time is two weeks. Assume that the demand is independent from one week to the next. Evaluate the CSL resulting from a policy of ordering 10,000 Palms when there are 6,000 Palms in inventory

Example 11-2: Evaluating cycle service level given a replenishment policy

Inventory

Time0

Reorder point

11,000

6,000

1,000

CSL = Prob(of not stocking out in a cycle)

Lead time

= Prob(demand during lead time ROP)

Measuring Demand Uncertainty

Inventory

Time

AverageInventory

CycleInventory

SafetyInventory

CycleLead time (L)

Reorder point (ROP)

Demand (D)Order quantity/lot size (Q)

Demand during lead time DL = LDStandard deviation of demand over lead time L = (L)D

Example 11-2: Evaluating cycle service level given a replenishment policy

mean

D = 500

D = 2500

stdev

DL = LD = 2 x 2500 = 5000

L = L D = 2 x 500 = 707

mean

stdev

CSL = Prob(demand during lead time ROP)

Example 11-2: Evaluating cycle service level given a replenishment policy

mean

D = 500

D = 2500

stdev

CSL = Prob(demand during lead time ROP)

DL = LD = 2 x 2500 = 5000

L = L D = 2 x 500 = 707

mean

stdev

CSL = F(ROP,DL,L)

Example 11-4: Evaluating safety inventory given a desired service levelAverage demand during lead time

DL =

Standard dev. of demand during lead time

L =

Cycle service level CSL =

LD = 2*2,500 = 5,000

SQRT(L)D = SQRT(2)*500 = 707F(ROP, DL, L) = F(6000, 5000, 707) = 0.92

F(ROP, DL, L) = NORMDIST(ROP, DL, L, 1)

Example 11-4: Evaluating safety inventory given a desired service level• Weekly demand for Lego at a Wal-Mart store is

normally distributed, with a mean of 2,500 boxes and a standard deviation of 500. The replenishment lead time is two weeks. Assuming a continuous-review replenishment policy, evaluate the safety inventory that the store should carry to achieve a CSL of 90 percent.

Example 11-4: Evaluating safety inventory given a desired service level

mean

D = 500

D = 2500

stdev

CSL = Prob(demand during lead time ROP)

DL = LD = 2 x 2500 = 5000

L = L D = 2 x 500 = 707

mean

stdev

CSL = F(ROP,DL,L)

ROP = F-1(CSL,DL,L)

Example 11-4: Evaluating safety inventory given a desired service levelAverage demand during lead time

DL =

Standard dev. of demand during lead time

L =

Desired cycle service level

CSL =

Safety inventory ss =

LD = 2*2,500 = 5,000

SQRT(L)D = SQRT(2)*500 = 7070.90

Option 1:ROP = F-1(CSL, DL, L) =F-1(0.90, 5000, 707) = 5,906

ss = ROP – DL = 5906 – 5000 = 906

F-1(CSL, DL, L) = NORMINV(CSL, DL, L)

Ethical Dilemma

Wayne Hills Hospital in Wayne, Nebraska, faces a problem common to large urban hospitals, as well as to small remote ones as itself.

That problem is deciding how much of each type of whole blood to keep in stock. Because blood is expensive and has a limited shelf

life, Wayne Hills naturally wants to keep its stocks as low as possible. Unfortunately, past disasters such as a major tornado and a train wreck demonstrated that lives would be lost when not enough

blood was available to handle massive needs.

Source: Operations Management, J. Heizer and B. Render

Summary

L: Lead time for replenishment

D: Average demand per unit time

D:Standard deviation of demand per period

DL: Average demand during lead time

L: Standard deviation of demand during lead time

CSL: Cycle service level

ss: Safety inventory

ROP: Reorder point),,(

),,(1

LL

LL

L

DL

L

DD

D

D

CSLFROP

ROPFCSL

ssROP

L

LD

Average Inventory = Q/2 + ss

Safety Inventory

What actions can be taken to improve product availability without hurting safety inventory?

ROP = F-1(CSL,DL,L)

ROP – DL = F-1(CSL,0,L)

(ROP – DL)/L = F-1(CSL,0,1)

ss/L = Fs-1(CSL)

ss = Fs-1(CSL)L

Safety Inventory

Why is it that successful retailers and manufacturers (i.e. Wal-Mart, Seven-Eleven Japan, Dell) carry only little inventory but

still have high levels of product availability?

Measuring Product Availability1. Cycle service level (CSL)

• Fraction of replenishment cycles that end with all customer demand met

• Probability of not having a stockout in a replenishment cycle

2. Product fill rate (fr)• Fraction of demand that is satisfied from product in inventory• Probability that product demand is supplied from available

inventory

3. Order fill rate• Fraction of orders that are filled from available inventory

Product Fill Rate

ESC = 10

inventory

inventory

time

time

0

0

Q = 1000

Q = 1000

fr = 1 – 10/1000 = 1 – 0.01 = 0.99

fr = 1 – 970/1000 = 1 – 0.97 = 0.03

ESC = 970

Expected Shortage per Replenishment Cycle• Expected shortage during the lead time

• If demand is normally distributed

L

ROPx

Df(x)dxxfROPxESC of pdf is where)()(

LsL

Ls

ssf

ssFssESC

1

Does ESC decrease or increase with ss?

Does ESC decrease or increase with L?

Product Fill Rate

• fr: is the proportion of customer demand satisfied from stock.

Probability that product demand is supplied from inventory.

• ESC: is the expected shortage per replenishment cycle (is the demand not satisfied from inventory in stock per replenishment cycle)

• ss: is the safety inventory

• Q: is the order quantity

LSL

LS

ssf

ssFssESC

Q

ESCfr

}1{

1

Example 11-3: Evaluating fill rate given a replenishment policy• Recall that weekly demand for Palms at B&M is

normally distributed, with a mean of 2,500 and a standard deviation of 500. The replenishment lea time is two weeks. Assume that the demand is independent from one week to the next. Evaluate the fill rate resulting from the policy of ordering 10,000 Palms when there are 6,000 Palms in inventory.

Example 11-3: Evaluating fill rate given a replenishment policyLot size Q =

Average demand during lead time

DL =

Standard dev. of demand during lead time

L =

Expected shortage per replenishment cycle

ESC =

Product fill rate fr =

10,000

LD = 2*2,500 = 5,000

SQRT(L)D = SQRT(2)*500 = 707-ss(1-Fs(ss/L))+Lfs(ss/L) =-1000*(1-Fs(1,000/707) +707fs(1,000/707) =

25.13

1 – ESC/Q = 1 – 25.13/10,000 = 0.9975

Cycle Service Level versus Fill Rate

What happens to CSL and fr when the safety inventory (ss) increases?

What happens to CSL and fr when the lot size (Q) increases?

Managing Inventory in Practice• India’s retail market

– Retail market (not inventory) projected to reach almost $308 billion by 2010

– Due to its infrastructure (many mom-and-pop stores and often poor distribution networks) lead times are long

ss = Fs-1(CSL)L

Managing Inventory in Practice• Department of Defense

– DOD reported (1995) that it had a secondary inventory (spare and repair parts, clothing, medical supplies, and other items) to support its operating forces valued at $69.6 billion

– About half of the inventory includes items that are not needed to be on hand to support DOD war reserve or current operating requirements