sci3133 lec 3 economic order quantity

8/3/2019 SCI3133 Lec 3 Economic Order Quantity

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SCI3133 Inventory Control and MRP Page 1Prof. Mohammad Ishak Desa FSKSM, UTM

Lecture 3:

ECONOMIC ORDER QUANTITY (EOQ)

SCI3133

Inventory Control and MRP


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1. Discuss the reasoning behind EOQ

2. Derive the EOQ model

3. Calculate EOQ for an item

4. Calculate the reorder level

LEARNING OUTCOMES


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Introduction

An inventory problem exist when it is necessary to stock physical items tosatisfy future demands.

Typical decision of every inventory problem (in order to avoid over or under

stocking)

How much to order?

When to Order?

The above two questions are normally answered (determined) through

inventory modeling (eg minimizing the Total Inventory Costs (TC)):

TC = (Purchasing Cost) + (Setup Cost) + (Holding Cost) + (Shortage Cost)


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Purpose of Inventory Modeling

Order Quantity, Q

Annualcost ($)

Holding Cost

Total Cost

Setup Cost

Slope = 0

Minimum

total cost

Optimal orderQopt

Shortage Cost

Purchasing Cost


5/32SCI3133 Inventory Control and MRP Page 5Prof. Mohammad Ishak Desa FSKSM, UTM

Dimensions of Inventory Modeling

DEMAND: Constant vs Variable (Static vs Dynamic) Known vs Random (Deterministic vs Probabilistic)

Stationary vs Nonstationary (in the case of Probabilistic)

Continuous vs Discrete

LEAD TIME: Instantaneous

Constant or Variable

REVIEW TIME: Continuous vs Periodic



Dimensions of Inventory Modeling

DISCOUNTS: None

All Units or Incremental

PLANNING HORIZON:

Single Time Period

Finite Time Period

Infinite Time



BASIC EOQ MODEL

ASSUMPTIONS

1. Demand is known, constant and continuous.

2. Lead time (the time between the placement and receipt of an order) is

zero.

3. Receipt of inventory is instantaneous (inventory from an order arrives in

one batch, at one point in time)

4. Purchase cost is constant; no quantity discounts.

5. The only variable costs: set-up or placing an order (ordering cost) and

holding or storing inventory over time (holding or carrying cost).

6. No shortage/stockout


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Inventory Order Cycle (EOQ Model)

Demandrate (D)

TimeLeadtime

Leadtime

Orderplaced

Orderplaced

Orderreceipt

Orderreceipt

Inve

ntory

Level

Reorder point, R

Order quantity, Q

0


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Variables

Q : Quantity Ordered (Order Quantity)

T : Cycle Time (time between two consecutive

replenishments - Period)

D : Demand (numbers of units to be supplied from

stock in a given time period)

UC : Unit Cost (the price for one unit item)

RC : Reorder Cost (Ordering cost or Setup Cost

the cost of placing a routine order)

HC : Holding Cost (cost of holding one unit item in

stock for one period time)


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DERIVATION EOQ MODEL

Find the total cost of one cycle Divide the total cost by the cycle length Minimize this cost per unit time.


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TotalInventoryCost Per Cycle

=UnitCostPer Cycle

+OrderingCostPer Cycle

HoldingCostPer Cycle

TC per cycle = UC x Q + RC + HC x Q/2 x T

+

BASIC EOQ MODEL (TC FUNCTION)

TC per unit time = (UC x Q)/T + RC/T + HC x Q/2

Given Q = D x T

TC per unit time = UC x D + RC x D + HC x QQ 2


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TC = UC x D + RC x D + HC x QQ 2

Optimal EOQ, Qo =

BASIC EOQ MODEL

d(TC) = - RC x D + HCdQ Q2 2

HC

DRCxx2


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Optimal Cycle Length, To = Qo/D

BASIC EOQ MODELS

Optimal Variable Cost Per Unit Time, VCo = HC x Qo

Optimal Cost Per Unit Time, TCo = UC x D + VCo


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BASIC EOQ MODELSExample 1

Demand for an item is constant at 1,000 units a year. Unitcost is $50, reorder cost is $100, holding cost is 25 per cent ofvalue a year and no shortages are allowed. Determine an

optimal inventory policy for the item.


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Listing the variable given in consistent units:D = 1,000 units a year UC = $50 a unitRC = $100 an order HC = 0.25 50 = 12.50 a unit a year

Optimal order quantity:Qo = (2 RC D/HC) = (2 100 1,000/12.50) = 126.5 units

Cycle length:To = Qo/D = 126.5/1,000 = 0.126 years or 6.6 weeks

Variable cost:VCo = HC Qo = 12.50 126.5 = $1,581 a year

Total cost:TCo = UC D + VCo = 50 1,000 + 1,581 = $51,581 a year



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BASIC EOQ MODELSExercise

X company buys 6,000 units of an item EvEry yEar with a unitcost of RM 30. It costs RM125 to process an order an arrangedelivery, while interest and storage costs amount to Rm6 a

year for each unit held. What is the best ordering policy for theitem?


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Sensitivity Analysis

Change of variable cost moving away from the EOQ:

VC 1 | Qo Q |

---- = --- x | ---- + ---- |

VCo 2 | Q Qo|


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Variable Cost Curve Around the EOQ

VCo

1.05 VCo

1.1 VCo


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To keep variable costs within 10 per cent of the minimum, theorder size mustbe between 0,64 and 1.56 times the EOQ, or between:

0.64 126.5 = 81 units and 1.56 125.5 = 197 units

For deliveries of 200 units the variable cost isVC = RC DQ +HC Q2 =100 1,000200 + 12.50 200 2 = $1,750 a year

TC = UC D + VC = 50 1,000 + 1,750 = $51,750 a year.



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Orders for Discrete Items

If the EOQ suggest an order for 5.5 computers, we must ordereither 5 or 6.

For expensive item, which one is better, to round up or round

down, ie Q which the first integer above the Qo, or Q-1 which

is the integer below the Qo.

Selection Procedure:

1. Calculate the EOQ, Qo

2. Find the integers Q and Q-13. If Q x (Q-1) is less than or equal to Qo2,order Q. Else order Q-1.


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Uncertainty in Demand and Costs

In Basic EOQ model, demand and costs were knownwith certainty. Not likely true in practice

Suppose there is an error of E in forecasting the demandD. Then the actual D = D x (1+E).

The resulting error in the variable cost

VC 1 | 1 (1+E)1/2 |

---- = --- x | ---- + ---- |VCo 2 | (1+E)1/2 1 |


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Uncertainty in Demand and Costs

Suppose there is an error of E1 in estimating the RC,and an error of E2 in HC . Then

Qo = SQRT ((2 x RC (1 +E1) x D)/HC x (1 + E2)

The resulting error in the variable cost

VC 1 | (1+E1)1/2 (1+E2)1/2 |

---- = --- x | ---- + ---- |VCo 2 | (1+E2)1/2 (1+E1)1/2 |


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Lead Time - Causes

Def: Total time between placing order and receiving the order.

The Lead Time occurs because of:

Time for order preparation

Time to get the order to right place in suppliers

Time at the supplier (process and prepare for deliveries)

Time to get materials delivered from suppliers

Time to process the delivery.

Ways to reduce Lead Time

E-business

Mode of transportation

Get near suppliers


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Reorder Level for Basic EOQ Model

Because Demand and Lead Time are Constant, the Reorder Level

(ROL) which indicate the time to place an order is given by,

ROL = Lead Time x Demand per Unit Time

= LT x D

The rule is to order a batch of size Qo whenever the stock level

falls to LT x D.


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Longer Lead Times

Lead Time LT > Cycle Time T

Eg: D = 100 units per week

Qo = 250 units

T= Qo/D = 2.5 weeksSuppose LT = 2 weeks, then ROL = 200 units

When LT =3, ROL = 300 units which is greater than the

maximum stock level!. What to do?


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Longer Lead Times (cont.)

If Lead Time LT > Cycle Time T, there will be an outstanding

order (or orders) when it is time to place an order.

In this case, ROL = LT x D = Stock on Hand + Stock on Order.

i.e the Actual ROL = LT x DQo

In general when LT is between nT and (n+1)T, then

ROL = LT x Dn x Qo


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Example 2

Demand for an item is constant at 40 units aweek, and the optimal economic order quantity is

calculated to be 100 units.

What is the reorder level if lead time is constant

at four weeks?

What happens if the lead time (a) falls to two

weeks; (b) rises to six weeks?


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Example 2

Qo = 100 units T= 100/40 = 2.5 weeks.

LT = 4 weeks, (there will be one delivery

outstanding when it is time to place anotherorder). T < LT < 2T (i.e n=1)

Then the reorder level is:

LT D Qo = 4 40 100

= 160 100 units = 60 units


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Example 2

If the LT falls to 2 weeks, there are no deliveriesoutstanding (n=0) when it is time to place

another order, and the reorder level is:

LT D = 2 40 = 80 units.

If the LT rises to 6 weeks, there are two

deliveries outstanding (n=2) when it is time to

place another order, and the reorder level is:

LT D 2 Qo = 6 40 2 100 = 40 units.


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Summary

By making a series of assumptions we can describe a simple, or

idealized, inventory system. Then we can build a model to relate

the overall cost of this system to the four cost components and the

order quantity. In particular, we can find the order quantity that

minimizes the total cost per unit time.

This order quantity, which is conventionally called the economic

order quantity (EOQ).

(EOQ), sets the overall features of the stocks, including length ofthe stock cycle, average stock level and costs.


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Summary

In practice, these calculations are always done by computer and

there is a wealth of software available. The use a spreadsheet is

helpful.

The analysis is based on a series of assumptions, but one of its

strengths is that costs rise slowly around the economic orderquantity. As a result, the EOQ gives good guidelines for the best

order size in a variety of circumstances.

It is easy to calculate the effects of moving away from the EOQperhaps for convenience or discrete demandand to find the

effects of errors and approximations in the calculations.


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Summary

The lead time is the total time between ordering materials and

having them delivered and available for use.

We can use the lead time demand to define a reorder level which

shows when to place an order.

sci3133 lec 3 economic order quantity

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