The EOQ Model

Tuesday, April 11, 2023 Industrial Management - THE EOQ MODEL 1

THE EOQ MODEL

Syed Mohammed SajlSemester 6

B Tech – PS & E


To a pessimist, the glass is half empty.To an optimist, it is half full.

- Anonymous


SOME BASIC DEFINITIONS

• An INVENTORY is an accumulation of a commodity that will be used to satisfy some future demand.

• Inventories may be of the following form:- Raw material- Components (subassemblies)- Work-in-process- Finished goods- Spare parts- Pipeline


EOQ History

• Introduced in 1913 by Ford W. Harris, “How Many Parts to Make at Once”

• Interest on capital tied up in wages, material and overhead sets a maximum limit to the quantity of parts which can be profitably manufactured at one time; “set-up” costs on the job fix the minimum. Experience has shown one manager a way to determine the economical size of lots.

• Early application of mathematical modeling to Scientific Management


EOQ MODELING ASSUMPTIONS

1. Production is instantaneous – there is no capacity constraint and the entire lot is produced simultaneously.

2. Delivery is immediate – there is no time lag between production and availability to satisfy demand.

3. Demand is deterministic – there is no uncertainty about the quantity or timing of demand.

4. Demand is constant over time – in fact, it can be represented as a straight line, so that if annual demand is 365 units this translates into a daily demand of one unit.

5. A production run incurs a fixed setup cost – regardless of the size of the lot or the status of the factory, the setup cost is constant.

6. Products can be analyzed singly – either there is only a single product or conditions exist that ensure separability of products.


TOTAL INVESTMENT IN INVENTORIES IN US


CHARACTERISTICS OF INVENTORY SYSTEMS

1. Demand• constant versus variable• known versus uncertain

2. Replenishment Lead Time3. Review

• periodic or continuous

4. Replenishment• periodic or continuous

5. Excess demand • backordered or lost

5. Changing Inventory• perishability or obsolescence


COST STRUCTURE

• Order costs – fixed (Set-up cost, K)– variable, per-unit (c)

• Holding costs (h)• Penalty cost (p)

– if backordering allowed: loss of goodwill – if demand lost: loss of goodwill + loss of profit


THE ECONOMIC ORDER QUANTITY (EOQ)

MODEL


ASSUMPTIONS LEADING TO EOQ

• Demand rate, λ, is constant and deterministic over time (units/day, units/year, etc.)

• Shortages not permitted

• No replenishment lead time

• The cost structure includes– fixed ordering cost, K, per order placed

– unit variable cost, c, independent of order size (no discounts)

– holding cost, h

• Infinite planning horizon


WHY EOQ?

• Easy to compute

• Does not require data that is hard to obtain

• Policies are surprisingly robust

• Assumptions can be relaxed

• Gives a good overall idea

• Can be starting point for more complicated models


EOQ - NOTATION

Q = order quantity (we want to find the optimal Q)

K = fixed order costc = unit variable cost, $ / unith = holding cost, $ / (unit * (unit time))h = I cλ = demand rate, units / (unit time)


EOQ – THE OBJECTIVE

• Our aim is to determine the best replenishment strategy (when and how much to order) under the criterion that the relevant costs (order and holding costs) will be minimized over time.


EOQ – THE INTUITION

• “When” should we place a new order ?– assume zero starting inventory– demand is deterministic and at a constant rate– lead time is negligible– no backorders are allowed



• “How much” we have to order each time ?– parameters do not change over time

– there is no reason for ordering different quantities

– so, order the same quantity Q in each order



Decision Variable: Q, the order size

TTime

Q

Inventory

Slope= -λ


DERIVING THE EOQ: THE “CYCLE”

• The same picture occurs over and over again– Why?

– any one of the triangles will be called a “cycle”

• Why not minimize the total cost in a cycle?

• Minimize the average cost per unit time (annual cost)

• Convert the order and holding costs to “average annual cost”, using the total cost in a cycle

length Cycle

cycle ain cost Total

T

timeof units Tper Costslim

T


DERIVING THE EOQ: THE ORDER COST

• Cycle length T =Q / λ (slope, - λ, = -Q/T)

• Since Q / λ is the time between two orders (the cycle time), λ /Q is the number of orders per unit time

• In each order, we pay K+Qc

• Order cost per unit time


DERIVING THE EOQ: THE HOLDING COST

• Holding cost per unit time =

22

length cycle

cycle oneover cost holding

2

0 Qh

Q

Qh

QT

dttQh

QT

2

levelinventory AverageQhh

• This can also be calculated as


DERIVING THE EOQ

2)('

2

h

QKQG

02

)(''3

Q

KQG

h

KQ

2* = I c


THE AVERAGE ANNUAL COST CURVE

unit time

cost

Q

2

hQG(Q)

Q

K

Q*

Annual fixed ordering and holding cost

The minimum


AN EXAMPLE

• Suppose that you are working for a retail store and have to determine how many boxes of detergent to order.

• Suppose every time you make an order you pay $20 for transportation and $10 to prepare the ordering request.

• Suppose every dollar tied in the inventory would earn 10 cents annually, if you had invested it.

• Suppose there is a regular weekly demand of 20 boxes and each box costs you $10.

• What is the order quantity?• What is the trade-off here?


THE SOLUTION

• λ=20*52 = 1040 boxes annually• K (fixed ordering cost) = $30 each time• h (holding cost)


INVENTORY TURNOVER RATIO WITH EOQ MODEL

• Turnover Ratio: A measure of effective inventory control – “How many times I sold my inventory?”

demand rate 2

avg. inventory / 2opt

h

Q K


EOQ IN A PRODUCTION ENVIRONMENT

• So far we have discussed a case in which we “order” from an outside supplier

• If we produce in-house, the relevant problem is called the “Economic Production Quantity”

• In this case, K denotes the “setup cost”• Why do we have a setup cost? Examples include:

– cost of production setup independent of the number produced: heating an oven, cost of dyes, etc.

– time required for the setup: lost production!– first few units might need to be scrapped


THANK YOU!

Education

The EOQ Model