ISE 216 – Production Systems Analysis

Chapter 4 – Inventory Control Subject to Known Demand

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

This Chapter

• Considers inventory control policies for individual item when product demand is assumed to follow a known pattern

• Assumes zero forecast error. – Is this realistic? Hardly, but it is easier.– Do not worry we will get to the more realistic

cases

Why do we study inventory?

Investment in Inventories in the U.S. Economy (1999) Inventory is money.

Reasons for Holding Inventories

• Economies of scale• Uncertainty in delivery lead times• Speculation (Changing costs over time)• Smoothing• Demand uncertainty• Costs of maintaining control system

Characteristics of Inventory Systems • Demand

– May be known or uncertain– May be changing or constant in time

• Lead Times (time elapses from placement of order until its arrival)– known – unknown

• Review policy: Is the system reviewed – periodically– continuously

Characteristics of Inventory Systems • Treatment of Excess Demand

– Backorder all excess demand– Lose all excess demand– Backorder some and lose some

• Inventory that changes over time– perishability– obsolescence

Relevant Costs• Holding Costs - Costs proportional to the quantity

of inventory held. Includes:– Physical space cost (3%)– Taxes and insurance (2 %)– Breakage, spoilage and deterioration (1%)– Opportunity Cost of alternative investment (18%)

(in total: 24% of all costs)

Note: Since inventory level is changing on a continuous basis, holding cost is proportional to the area under the inventory curve

Inventory as a Function of Time

Relevant Costs (continued)

• Penalty or Shortage Costs (opposite of holding ): All costs that happen when the stock is insufficient to meet the demand– Loss of revenue for lost demand– Costs of bookkeeping for backordered demands– Loss of goodwill for being unable to satisfy the

demand– Generally assumed cost is proportional to

number of units of excess demand

Relevant Costs (continued)• Ordering Cost (or Production Cost) - has both

fixed and variable components

slope = c

K

x : order or production quantity C(x): ordering or production cost C(x) = K + cx for x > 0 C(x) = 0 for x = 0.

Simple Economic Order Quantity (EOQ) Model – Assumptions

1. Demand is constant and uniform ( l units/year, l/4 units per quarter)

2. Shortages are not allowed

3. Orders are received instantaneously

4. Order quantity is fixed (Q per cycle) - can be proven to be optimal

5. Costs– Fixed and marginal ordering costs (K + cQ)

– Holding cost per unit held per unit time (h)

Inventory Levels for the EOQ Model

First order when inventory is 0. Reorder Q units every time when inventory is 0. It must be optimal

The EOQ Model: NotationParameters λ : the demand rate (in units per time)time

c : unit ordering/roduction cost (in dollars per unit), setup or inventory costs are not included

K : setup cost (per placed order) in dollars

h : holding cost (in dollars per unit per year) if the holding cost consists of interest of the

money tied up in inventory,

h = ic, where i is the annual interest rate

The EOQ Model: NotationDecision variable is Q

Q: lot size (order size) in units

T : time between two consecutive orders (cycle length)

G(Q) = average cost per unit time

Q

T

Relationships

• Ordering Cost per cycle:C(Q) = K + cQ

• Holding Cost per cycle = unit holding cost * area of the triangle

h * QT/2 or unit holding cost * average inventory size * cycle length h*Q/2*T

Relationships• Average Inventory Size = Q/2 why ?

• Time Between Orders (Cycle length) = l Q/T

T = Q/lT

Q

Rate of consumption l

Time (t)

Inve

nto

ry (

I(t)

) Assume Constant Demand

T

Q

Time between orders

slope = -l

InstantaneousReplenishment

Total Costs

• What is the average annual cost?

G(Q) = average total (order + holding) cost = total cost per cycle / cycle length

2( )2

QK cQ hT K cQ Q

G Q hT T

Ordering cost per cycle

Average inventory level at any time

Average total cost

• What is the average annual cost?

cQ

KhQ

hQQ

cQK

hQ

T

cQKQG

2

2

2)(

The Average Annual Cost Function G(Q)

c

Q

KhQ

T

cQ

T

KQhQG

22

Q that minimizes the annual cost

Let Q* be the optimal Q value. Is G’(Q*)=0 ?

2

3

( )2

( )2

2( ) 0, when 0

K hQG Q c

Q

K hG Q

Q

KG Q Q

Q

YES!2

20 *

2

K h KQ

Q h

is a nonlinear function of G Q Q

Properties of the EOQ Solution

• Q* is increasing function of K and and decreasing function of h in square roots

• Q* is independent of c (except the case we calculate h = Ic), Why?

2*

KQ

h

Properties of the EOQ Solution

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

• This is a tradeoff between lot size (Q) and inventory• “Garbage in, garbage out” - usefulness of the EOQ

formula for computational purposes depends on the realism of the input data

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

c

Q

KhQQG

2 h

KQ

2

Example• UVIC annually requires 3600 liters of paint 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 liter. How many liters of paint should be ordered and how often?

2 2(16)(3600)* 57600 240

0.25(8)

KQ

h

17.5=18 working days =

0.07 years * 250 working days a/year 3600

240 ===lQ

T

Order Point for the EOQ Model

Assumption: Delivery is immediate or order lead time τ = 0

τ ττ τ

Order Point for the EOQ Model• Does it matter if τ < T or τ > T ?

• Keep track of time left to zero inventory or set to automatic reorder at a particular inventory level, R.

• Two cases: 1. if τ < T R = λ*τ, 2. if τ > T R = λ* (τ mod T)

Sensitivity Analysis

G(Q) : the average annual holding and set-up cost function

c

Q

KhQQG

2

h

KQ

2

independent of Q and omitted

Q

KhQQG

2

hKG 2

G*: the optimal average annual holding and setup cost

Sensitivity Analysis

( ) 1 *

* 2 *

G Q Q Q

G Q Q

Sensitivity of EOQ to Order Quantity

0,000

1,000

2,000

3,000

4,000

5,000

6,000

0 2 4 6 8 10

Ratio of Optimal to Suboptimal Quant.

Rat

io o

f o

pt.

to

Su

bo

pti

mal

C

ost

Cost penalties are quite small

Finite Replenishment Rate: Economic Production Quantity (EPQ)

Assumptions of EOQ: 1. Production is instantaneous: There is no capacity

constraint, and entire lot is produced simultaneously

2. Delivery is immediate: There is no time lag between production and availability to satisfy demand

Inventory Levels for Finite Production Rate Model

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

The EPQ Model: NotationParameters λ : the demand rate (in units per year)P : the production rate (in units per year)c : unit production cost (in dollars per unit)K : setup costs (per placed order) in dollarsh : holding cost (in dollars per unit per year) if the interest rate is

given calculated as h=ic,

Decision variables Q : size of each production run (order) in unitsT : time between two consecutive production orders (cycle

length)

T1 : production (replenishment) timeT2 : no production (down time) H : maximum on-hand inventory level

21 TTT

The EPQ Model: Formula

21 TTT

P

QT 1

P

QQTTT

12

1 ( )Q

H P T PP

The EPQ Model: Formula

2 12

HTh K cQ Q K

G Q h cT P Q

2, where 1

KQ h h

h P

For EOQ:

h

KQ

2

For EPQ:

Quantity Discount Models

• The assumption is “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

• Two types of discounts: – All Units Discounts: the discount is applied to ALL of the units

in the order. • 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. • Order cost function such as that pictured in Figure 4-10

All-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

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)

Incremental Discount Order Cost Function

.

. . ,

. , . ,

Q for Q

C Q Q Q for Q

Q Q for Q

0 30 500

150 0 29 500 5 0 29 500 1 000

295 0 28 1 000 15 0 28 1 000

Average Annual Cost Function for Incremental Discount Schedule

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.

Example• Supplier of paint to the maintenance department

has announced new pricing:$8 per liter if order is < 300 liters$6 per liter if order is ≥ 300 liters

• Other data is same as before:K = 16, i= 25%, l = 3600

• Is this a case of all units or incremental discount?

Solution• Step 1: For price 1:

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

• Step 3: For price 2:

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

*1

1

2 2(16)(3600)240 liters

(0.25)(8)

KQ

ic

*2

2

2 2(16)(3600)277 liters

(0.25)(6)

KQ

Ic

Cost Function

Q

C(Q

)

G(Q|c1)

G(Q|c2)

240 277 300

Not Realizable

Realizable

Q

C(Q

)

G(Q|c1)

G(Q|c2)

240 277 300

Only possible solutions

Cost Function

Solution• Step 5: Compare costs of possible solutions.

– For $8 price, Q=240:

– For $6 price, Q=300:

– Q=300 is the optimal quantity.

( )2j j j

Q KG Q ic c

Q

1

(3600)(16) (0.25)(8)(240)(240) (3600)(8) $29280 per year

240 2G

2

(3600)(16) (0.25)(6)(300)(300) (3600)(6) $22017 per year

300 2G

* 300 1* 300 and * year

3600 12

QQ T

Resource Constrained Multi-Product Systems• Classic EOQ model is for a single item. • If we have multiple (n) items

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

• Works if there is no interaction among the 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 you know how to use Lagrange multipliers!

Resource Constrained Multi-Product Systems

• Consider an inventory system of n items in which the total amount available to spend is C

• Unit costs of items are c1, c2, . . ., cn, respectively • This imposes the following budget constraint on the

system (Qi is the order size for product i)

• Let wi be the volume occupied by product i

CQcn

iii

1

WQwn

iii

1

Resource Constrained Multi-Product Systems

11

minimize ,...,2

subject to

ni i i

n ii i

Q KG Q Q h

Q

CQcn

iii

1

WQwn

iii

1

Budget constraint

Space constraint

Resource Constrained Multi-Product Systems

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

,,,...,

0, 0 for 1,..., ; 1, 2i j

G Gi n j

Q

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

Single constraint case :1. Solve the unconstrained problem (Find the EOQ values).

If the 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

0, 0 1,...,i

G Gfor i n

Q

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

Double constraints case:1. Solve the unconstrained problem (Find individual EOQ

values). 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. Solve one-constraint problem. If the other constraint is satisfied, this solution is the optimal one.

3. Otherwise repeat the step 2 for the other constraint.

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

Double constraints case :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

0, 0 for 1,..., ; 1, 2i j

G Gi n j

Q

n

iii

n

iii

n

i i

iiiin QwWQcC

Q

KQhQQG

12

11

1211 2

,,,...,

EOQ Models for Production PlanningProblem: determine optimal procedure for producing n products on a single

machineλj : demand rate (in units per year) of product jPj : producton rate of product j cj : unit production cost (in dollars per unit) of product jKj : setup cost (per placed order) in dollars of product jhj : holding cost (in dollars per unit per year) of product j

The objective is to minimize the cost of holding and setups, and to have no stock-outs. To have a feasible solution:

Assumption: We apply a rotation cycle policy Exactly one setup for each product in each cycle - production sequence

stays the same in each cycle.

1

1.n

j

j jP

•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 T = max{T*, Tmin}, where sj is setup time of product j

The optimal production quantities are given by

1

1

2

*'

n

jj

n

j jj

K

Th

j jQ T

n

j j

j

n

jj

P

s

T

1

1min

1