Chapter 4

Inventory Control Subject to Known

Demand

McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.

4-2

Breakdown of the Total Investment in Inventories in the U.S. Economy (1999)

4-3

Reasons for Holding Inventories

Economies of Scale Uncertainty in delivery leadtimes Speculation. Changing Costs Over Time Smoothing. Demand Uncertainty Costs of Maintaining Control System

4-4

Characteristics of Inventory Systems Demand

May Be Known or Uncertain May be Changing or Unchanging in Time

Lead Times - time that elapses from placement of order until it’s arrival. Can assume known or unknown.

Review Time - Is system reviewed periodically or is system state known at all times?

Treatment of Excess Demand. Backorder all Excess Demand Lose all excess demand Backorder some and lose some

Inventory that changes over time perishability obsolescence

4-5

Relevant Costs

Holding Costs - Costs proportional to the quantity of inventory held. Includes:

a) Physical Cost of Space (3%)

b) Taxes and Insurance (2 %) c) Breakage Spoilage and Deterioration (1%) *d) Opportunity Cost of alternative investment. (18%)

(Total: 24%)

Note: Since inventory may be changing on a continuous basis, holding cost is proportional to the area under the inventory curve. (See examples.)

4-6

Inventory as a Function of Time

4-7Relevant Costs (continued)

Ordering Cost (or Production Cost).Includes both fixed and variable components.

slope = c

K

C(x) = K + cx for x > 0 and =0 for x = 0.

4-8

Relevant Costs (continued)

Penalty or Shortage Costs. All costs that accrue when insufficient stock is available to meet demand. These include: Loss of revenue for lost demand Costs of bookeeping for backordered demands Loss of goodwill for being unable to satisfy

demands when they occur. Generally assume cost is proportional to

number of units of excess demand.

4-9

Simple EOQ Model Assumptions:

1. Demand is fixed at units per unit time.

2. Shortages are not allowed.

3. Orders are received instantaneously. (this will be relaxed

later).

4. Order quantity is fixed at Q per cycle. (can be proven

optimal.)

5. Cost structure:

a) Fixed and marginal order costs (K + cx)

b) Holding cost at h per unit held per unit time.

4-10

Inventory Levels for the EOQ Model

Saw structure is typical. First order when inventory is 0. Reordering Q everytime when inventory is 0 must be optimal

4-11

D = λ is the demand rate (in units per year)

c = unit production cost, not counting setup or inventory costs (in dollars per unit)

K = setup costs (per placed order) in dollars

h = holding cost (in dollars per unit per year), if the holding cost consists entirely of interest on money tied up in inventory,

h = ic, where i – is an annual interest rate

Q = lot size (order size) in units

T = time between orders (cycle length)

G(Q) = average annual cost

The EOQ Model: Notation

Q

T

T

cQ

T

KQhQG

2

4-12

Ordering Costs: (Order amount Q)

C(Q) = K + cQ

Holding Cost:

h = Ic =(Interest Rate)(Cost of Inv.)

Average Inventory Size?

Under constant demand: Q/2

Time Between Orders:

Q/T

T = Q/

Relationships

T

Q

Rate of consumption

Time (t)

Inve

nto

ry (

I(t)

) Assume Constant Demand

T

Q

Time between orders

slope = -

InstantaneousReplenishment

4-13

Total Costs

What is the average annual cost?

G(Q) = average order cost + average holding cost

G(Q) K cQ

T

hQ

2

Average ordering cost per time T

Average inventory level at any time

4-14

Total Costs

What is the average annual cost?

cQ

KhQ

hQQ

cQK

hQ

T

cQKQG

2

2

2)(

4-15

The Average Annual Cost Function G(Q)

c

Q

KhQ

T

cQ

T

KQhQG

22

4-16

The Average Annual Cost Function G(Q)

4-17

Take the derivative of G(Q)

Is this a minimum?

EOQ:

Minimize Annual Costs

2)(

2)(

2

h

Q

KQG

hQc

Q

KQG

G (Q)2KQ3 0,Q 0 YES!

KQ2

h

20 Q*

2Kh

QQG offunction linear non

4-18

4-19

Properties of the EOQ Solution

2KQ

h

Q is increasing with both K and and decreasing with h

Q changes as the square root of these quantities Q is independent of the proportional order cost,

c. (except as it relates to the value of h = Ic)

4-20Properties of the EOQ Solution

This formula is well-known economic order quantity, is also known as economic lot size

This is a tradeoff between lot size and inventory

“Garbage in, garbage out” - usefulness of the EOQ formula for computational purposes depends on the realism of input data

Estimating setup cost is not easily reduced to a single invariant cost K

c

Q

KhQQG

2 h

KQ

2

4-21

Example Uvic requires 3600 gallons of paint annually for

scheduled maintenance of buildings. Cost of placing an order is $16 and the interest rate (annual) is 25%. Price of paint is $8 per gallon.

How much paint should be ordered, and how often?

240600,57)8(25.

)3600)(16(22*

h

KQ

days 18 = days working17.5 =

days/year) working(250 * years 07.3600

240 Q

T

4-22

Order Point for the EOQ Model

Assumption: Delivery is immediateThere is no time lag between production and availability to

satisfy demand Relax this assumption! Let the order lead time to be equal to τ

τ ττ

Does it matter if τ < T or τ >

T ?

Keep track oftime left to zero inventory or set automatic reorder at a particularinventory level, R.

R = λ*τ, if τ < T R = λ*MOD(τ/T),

if τ > T

τ

4-23

Sensitivity Analysis

Let G(Q) be the average annual holding and set-up cost function given by

c

Q

KhQQG

2

( ) 1 *

* 2 *

G Q Q Q

G Q Q

h

KQ

2

independent of Q

Holding & Setup costs

Q

KhQQG

2

hKG 2

0.000

1.000

2.000

3.000

4.000

5.000

6.000

0 2 4 6 8 10

and let G* be the optimal average annual holding and setup cost. Then it can be shown that:

Cost penalties are quite small

4-24

Finite Replenishment Rate: Economic Production Quantity (EPQ)

Assumptions for EOQ:

Production is instantaneousThere is no capacity constraint, and entire lot is produced simultaneously

Delivery is immediateThere is no time lag between production and availability to satisfy demand

Example: Parts produced at the same factory –

production rate is P (P > λ), arriving continuously.

4-25

Inventory Levels for Finite Production Rate Model

4-26

The EPQ Model: NotationD = λ is the demand rate (in units per year)c = unit production cost, not counting setup or inventory costs

(in dollars per unit)K = setup costs (per placed order) in dollarsh = holding cost (in dollars per unit per year), if the holding cost

consists entirely of interest on money tied up in inventory, h=ic, where i is an annual interest rate

Q = size of each production run (order) in unitsT = time between initiation of orders arrival (cycle length)

T1 = production (replenishment) timeT2 = downtimeH = maximum on-hand inventoryG(Q) = average annual setup & holding cost

21 TTT

4-27

The EPQ Model: Formula

21 TTT

T

KHhQG

2

P

QT 1

P

QQTTT

12

PslopeT

H

1

QP

PTPH

1

Q

K

P

hQ

T

KHhQG

1

22

Phhwhere

h

KQ

1,

2

Q

KhQQG

2For EOQ:

h

KQ

2

4-28

Quantity Discount Models

One of the most severe assumptions: the unit variable cost c did not depend on the replenishment quantity

In practice: quantity discounts exist based on the purchase price or transportation costs – take advantage of these can result in substantial savings

All Units Discounts: the discount is applied to ALL of the units in the order. Gives rise to an order cost function such as that pictured in Figure 4-9 in Ch. 4.7

Incremental Discounts: the discount is applied only to the number of units above the breakpoint. Gives rise to an order cost function such as that pictured in Figure 4-10

4-29All-Units Discount Order Cost Function

QforQ

QforQ

QforQ

QC

000,128.0

000,150029.0

500030.0

64.149$)516(

00.145$)500(

70.149$499

C

C

C

4-30

G(Q)

Q500 1,000

All-Units Discount Average Annual Cost

Function

QforQ

QforQ

QforQ

QC

000,128.0

000,150029.0

500030.0

G0(Q)

G1(Q)

G2(Q)

Gmin(Q)

4-31Incremental Discount Order Cost Function

QforQQ

QforQQ

QforQ

QC

000,128.015000,128.0295

000,150029.0550029.0150

500030.0

4-32

Average Annual Cost Function for Incremental Discount Schedule

4-33

Properties of the Optimal Solutions For all units discounts, the optimal will occur

at the minimum point of one of the cost curves or at a discontinuity point One compares the cost at the largest realizable

EOQ and all of the breakpoints succeeding it

For incremental discounts, the optimal will always occur at a realizable EOQ value. Compare costs at all realizable EOQ’s.

4-34

Example Supplier of paint to the maintenance department has

announced new pricing:

$8 per gallon if order is < 300 gallons

$6 per gallon if order is ≥ 300 gallons

Data remains as before: K = 16, I = 25%, = 3600

Is this a case of all units or incremental discount?

4-35

Solution Step 1: For Price 1:

Step 2: As Q(1) < 300, EOQ is realizable.

Step 3: Price 2:

Step 4: As Q(2) < 300, EOQ is not realizable.

Q(1) 2KIc1

2(16)(3600)

(.25)(8)240 gallons

Q(2) 2KIc2

2(16)(3600)

(.25)(6)277 gallons

4-36

Cost Function

Q

C(Q

)

G(Q|p1)

G(Q|p2)

240 277 300

Not Realizable

Realizable

4-37

Q

C(Q

)

G(Q|p1)

G(Q|p2)

240 277 300

Only possible solutions

Cost Function

4-38

Solution Step 5: Compare costs of possible solutions.

For $8 price, Q=240:

For $6 price, Q=300:

Q=300 is the optimal quantity.

G(Q) c j K

Q

Ic jQ

2

G(240) (3600)(8)(3600)(16)

240

(.25)(8)(240)

2$29,280 per year

G(300) (3600)(6)(3600)(16)

300

(.25)(6)(300)

2$22,017 per year

300 1* 300 and year

3600 12

QQ T

4-39Resource Constrained Multi-Product

SystemsClassic EOQ model is for a single item. Setup plan for n items.

Option A: Treat one system with multiple items as multiple systems with one item

Works if: There are no interactions among items, such as sharing common resources – budget, storage capacity, or both

Option B: Modify classic EOQ to insure no violation of the resource constraints

Works if: Have not made any mistakes and know how to use Lagrange multipliers

4-40Resource Constrained Multi-Product

SystemsConsider an inventory system of n items in which the total amount available to spend is C and items cost respectively c1, c2, . . ., cn. Then this imposes the following budget constraint on the system

, where Qi is the order size for product i

, where wi is the volume occupied by product i

Minimize

s.t. and

CQcn

iii

1

Q

KQhQG

2For EOQ:

n

i i

iiiin Q

KQhQQG

11 2,...,

n

i i

iiiin Q

KQhQQG

11 2,...,

CQcn

iii

1

WQwn

iii

1

WQwn

iii

1

4-41Resource Constrained Multi-Product

SystemsMinimize

s.t.CQc

n

iii

1

n

i i

iiiin Q

KQhQQG

11 2,...,

WQwn

iii

1

Budget constraint

Space constraint

Lagrange multipliers method: relax one or more constraints

Minimize

by solving necessary conditions:

n

iii

n

iii

n

i i

iiiin QwWQcC

Q

KQhQQG

12

11

1211 2

,,,...,

2,1;,...,10,0

jniforG

Q

G

ji

4-42

Resource Constrained Multi-Product Systems: Steps to Find Optimal Solution

Single constraint:

1. Solve the unconstrained problem. If constraint is satisfied, this solution is the optimal one.

2. If the constraint is violated, rewrite objective function using Lagrange multipliers

3. Obtain optimal Qi* by solving (n+1) equations

;,...,10,0 niforG

Q

G

i

4-43

Resource Constrained Multi-Product Systems: Steps to Find Optimal SolutionDouble constraints:

1. Solve the unconstrained problem. If both constraints are satisfied, this solution is the optimal one.

2. Otherwise rewrite objective function using Lagrange multipliers by including one of the constraints, say budget, and solve one-constraint problem to find optimal solution. If the space constraint is satisfied, this solution is the optimal one.

3. Otherwise repeat the process for the only space constraint.

4. If both single-constraint solutions do not yield the optimal solution, then both constraints are active, and the Lagrange equation with both constraints must be solved.

5. Obtain optimal Qi* by solving (n+2) equations

2,1;,...,10,0

jniforG

Q

G

ji

n

iii

n

iii

n

i i

iiiin QwWQcC

Q

KQhQQG

12

11

1211 2

,,,...,

4-44

Problem: determine optimal procedure for producing n products on a single machine

Consider n items with known demand rates , production rates , holding costs , and set-up costs . The objective is to minimize the cost of holding and setups, and to have no stock-outs. For the problem to be feasible we must have that

Assumption: rotation cycle policy – exactly one setup for each product in each cycle; production sequence stays the same in each next cycle

EOQ Models for Production Planning

1

1.n

j

j jP

jjP

jh jK

4-45

The method of solution is to express the average annual cost function in terms of the cycle time, T to assure no stock-outs. The optimal cycle time has the following mathematical form, where sj is a setup time

And the optimal production quantities are given by:

where T = max {T*, Tmin},

see pp.216-217

1

1

2

*'

n

jj

n

j jj

K

Th

,TQ jj

n

j j

j

n

jj

P

s

T

1

1min

1

4-46Homework:

Read Ch. 4

Problems 4.5, 4.12, 4.15, 4.16 4.17, 4.18, 4.22, 4.24, 4.25 4.26, 4.27, 4.28, 4.30

Work on appendix 4-A,

4-47

References

Presentations by McGraw-Hill/Irwin and by Wilson,G.R.

“Production & Operations Analysis” by S.Nahmias

“Factory Physics” by W.J.Hopp, M.L.Spearman

“Inventory Management and Production Planning and Scheduling” by E.A. Silver, D.F. Pyke, R. Peterson

“Production Planning, Control, and Integration” by D. Sipper and R.L. Bulfin Jr.

4-48

Reorder Point Calculation for Example 4.1

4-49

Reorder Point Calculation for Lead Times Exceeding One Cycle

4-50

Sensitivity Analysis

Let G(Q) be the average annual holding and set-up cost function given by

and let G* be the optimal average annual cost. Then it can be shown that:

( ) / / 2G Q K Q hQ

( ) 1 *

* 2 *

G Q Q Q

G Q Q

4-51

EOQ With Finite Production Rate

Suppose that items are produced internally at a rate P > λ. Then the optimal production quantity to minimize average annual holding and set up costs has the same form as the EOQ, namely:

Except that h’ is defined as h’= h(1- λ/P)

2

'

kQ

h

4-52

Inventory Levels for Finite Production Rate Model

4-53

Quantity Discount Models

All Units Discounts: the discount is applied to ALL of the units in the order. Gives rise to an order cost function such as that pictured in Figure 4-9

Incremental Discounts: the discount is applied only to the number of units above the breakpoint. Gives rise to an order cost function such as that pictured in Figure 4-10.

4-54

All-Units Discount Order Cost Function

4-55

Incremental Discount Order Cost Function

4-56

Properties of the Optimal Solutions

For all units discounts, the optimal will occur at the bottom of one of the cost curves or at a breakpoint. (It is generally at a breakpoint.). One compares the cost at the largest realizable EOQ and all of the breakpoints succeeding it. (See Figure 4-11).

For incremental discounts, the optimal will always occur at a realizable EOQ value. Compare costs at all realizable EOQ’s. (See Figure 4-12).

4-57

All-Units Discount Average Annual Cost Function

4-58

Average Annual Cost Function for Incremental Discount Schedule

4-59

Resource Constrained Multi-Product Systems Consider an inventory system of n items in which the total amount

available to spend is C and items cost respectively c1, c2, . . ., cn. Then this imposes the following constraint on the system:

When the condition that

is met, the solution procedure is straightforward. If the condition is not met, one must use an iterative procedure involving Lagrange Multipliers.

1 1 2 2 ... n nc Q c Q c Q C

1 1 2 2/ / ... /n nc h c h c h

4-60

EOQ Models for Production Planning Consider n items with known demand rates,

production rates, holding costs, and set-up costs. The objective is to produce each item once in a production cycle. For the problem to be feasible we must have that

1

1.n

j

j jP

4-61

The method of solution is to express the average annual cost function in terms of the cycle time, T. The optimal cycle time has the following mathematical form.

And the optimal production quantities are given by:

1

1

2

*'

n

jj

n

j jj

K

Th

* *j jQ T