CHAPTER THREE Determistic Model

8/11/2019 CHAPTER THREE Determistic Model

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1 Chapter Three

Basically, Deterministic models keeps the condition thatall variables take values which are known exactly.However, Deterministic models do remove some of theassumptions made in the classical analysis or EOQ.

1. Model for discounted unit cost: Many suppliers quotelower prices for larger orders, so a procedure isdescribed for finding the overall lower cost.

2. Finite Replenishment rate: This is typical of a finishedgoods store at the end of a production line. Here unitsarrive at a finite rate and stock accumulates during aproduction run. Remove the assumption ofinstantaneous replenishment used for EOQ andcalculate new optimal batch sizes.

DETERMINISTIC MODELS FOR

INVENTORY CONTROL


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Quantity Discount in Unit Cost

Remove EOQs assumption that all costs arefixed (constant values which never change)

Address problems where costs vary with thequantity ordered

Two main approaches: Valid Total Cost Curve& Rising Delivery Cost


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Valid Total Cost Curve:

In practice, normally a sliding scale of unit costs is

applied, so that every unit bought has a differentprice

Frequently the unit cost decreases in steps with

supplier offering a reduced price on all units if morethan a certain number are bought

There are two main characteristics of valid costcurve:

1. The valid total cost curve always rises to theleft of a valid minimum

2. There are two possible positions for the overallminimum cost, either at valid minimumor cost

breakpoint


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1. Valid Minimum:

The minimum point on the cost curve is

within the range of valid order quantities forthis particular unit cost

2. Invalid Minimum:

The minimum point on the cost curve fallsoutside the valid order range for thisparticular unit cost


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Procedure for finding the lowest Total

Cost with varying unit cost

1. Take the next lowest unit cost curve

2. Calculate the minimum point, Q= 2AS/ic

3. Is this point valid??

4. If not, calculate cost at breakpoint to the leftof valid range

5. If yes, calculate the cost of the valid minimum6. Compare the costs of all the points considered

& select lowest

7. Finish


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INV: Hendrik Lamsali

6 Chapter Three

Exercise:

Annual demand for an item is 2000 units, each

order costs $10 to place and annual holding costis 40% of unit cost. The unit cost depends on thequantity ordered as follows: Unit cost is $1 for order quantities less than 500

$0.80 for quantities between 500 and 999 each $0.60 for quantities of 1000 or more

What is the optimal ordering policy??

Listing the variables:A = 2000 units a year

S = $10 per order

i = 40% of unit cost a year

C = ?? (depend on order quantities)


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Notice that:1) Total cost, TC, for optimal order quantity, Q, is:

TC = c x A + 2 x S x ic x A

1) Total cost for any other order quantity is:

TC = c x A + (S x A)/Q + (ic x Q)/2

Follow the Procedure:Taking the lowest cost curve:

c = $0.60, valid for Q of 1000 or more

Q = (2)(A)(S)/(ic) = (2)(2000)(10)/(0.4)(0.60) = 408.2

This is an invalid minimum as Q is not greater than 1000

So, calculating the cost of the breakpoint:

TC = c x A + (S x A)/Q + (ic x Q)/2

= (0.6)(2000) + (10 x 2000)/1000 + (0.4 x 0.6 x 1000)/2

= $1340 a year (point A)


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Taking the next lowest cost curve:

c = $0.80, valid for Q between 500 and 999

Q = 2AS/ic = (2)(2000)(10)/(0.4)(0.80) = 353.6This is an invalid minimum as Q is not between 500 and 999

So, calculating the cost of the break point at the lower end:

TC = c x A + (S x A)/Q + (ic x Q)/2

= (0.8)(2000) + (10 x 2000)/500 + (0.4x0.8x500)/2

= $1720 a year (point B)


c = $1.00, valid for Q less than 500

Q = 2AS/ic = (2)(2000)(10)/(0.4)(1.00) = 316.2This is valid minimum as Q is less than 500Then, calculating the cost at this valid minimum:


= (1.00)(2000) +2 x 10 x (0.4)(1.00) x 2000

= $2126.49 a year (Point C)


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How to Choose the best ordering policy?

Compare the results at point A, B & C:

1. Point A: Q=1000, cost= $1340 a year

2. Point B: Q=500, cost= $1720 a year3. Point C: Q=316.2, cost= $2126.49 a year

Therefore, the best policy is to order batchesof 1000 units, placing order every six monthswith total annual costs of $1340


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Exercise 2:

A company works for 50 weeks a year during which

demand for a product is constant at 10 units a week.The cost of placing order, including delivery chargesis estimated to be $150. The company aims for 20%annual return on assets employed. The supplier ofthe item quotes a basic price of $250 a unit withdiscount of 10% on orders of 50 units or more, 15%on orders of 150 units or more and 20% on orders of500 units or more. What is the optimal order quantityfor the item??

Listing the variables:

A = 10 x 50 = 500 units a year

S = $150 per order

i = 20% x c (unit cost)


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Follow the Procedure:

Taking the lowest cost curve:

c = $200, valid for Q of 500 or more

Q = (2)(A)(S)/(ic) = (2)(500)(150)/(0.2)(200) = 61.2This is an invalid minimum as Q is not greater than 500

So, calculating the cost of the breakpoint to the left, Q=500:

TC = c x A + (S x A)/Q + (ic x Q)/2

= (200)(500) + (150 x 500)/500 + (0.2 x 200 x 500)/2

= $110,150 a year (point A)


c = $212.50, valid for Q between 150 and 500

Q = 2AS/ic = (2)(500)(150)/(0.2)(212.50) = 59.4This is an invalid minimum as Q is not between 150 and 500

So, calculating the cost of the break point to the left, Q=150:

TC = c x A + (S x A)/Q + (ic x Q)/2

= (212.50)(500) + (150 x 500)/150 + (0.2x212.50x150)/2

= $109,938 a year (point B)


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c = $225, valid for Q between 50 and 150Q = 2AS/ic = (2)(500)(150)/(0.2)(225) = 57.7

This is a valid minimum as Q is between 50 and 150

So, calculating the cost at this valid minimum:


= (225)(500) +2 x 150 x (0.2)(225) x 500

= $115,098 a year (Point C)

Compare the results at point A, B & C:

1. Point A: Q=500, cost= $110,150 a year

2. Point B: Q=150, cost= $109,938 a year

3. Point C: Q=57.7, cost= $115,098 a year

So, Choose to order in batches of 150 units with annual

cost of $110,000


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Rising Delivery Cost

Extended version of discounted price analysis. Theonly difference is that the valid cost curve isreversed, so we look for optimal order quantities tothe left of a valid minimum and calculate the costof break points at the upper limit of valid ranges.

Address changes in the reorder cost due to theamount of delivery

Example:

Enter Sandman Plc., uses 4 tonnes of fine industrial sand

every day. This sand costs $20 a tonne to buy, and $1.90a tonne to store for a day. Deliveries are made bymodified lorries which carry up to 15 tonnes, and eachdelivery of a load or part load costs $200. Find thecheapest way to ensure continuous supplies of sand??


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Solution:

Notice that unit cost is constantReorder cost is $200 for each lorry load

Orders up to 15 tonnes need one lorry with areaorder cost of $200, orders between 15 and 30

tonnes need two lorries with reorder cost of $400,orders between 30 and 45 tonnes need three lorrieswith reorder costs of $600, and so on.

Other variables:

A = 4 tonnes a dayc = $20 a tonne

I = $1.90 a tonne a day


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Taking the Lowest Cost Curve:

S = $200, valid for Q less than 15 tonnes

Q = (2)(A)(S)/I = (2)(200)(4)/1.90 = 29.0 tonnes

This is invalid minimum as Q is not less than 15 tonnesCalculating the cost at the break point:

TC = c x A + (S x A)/Q + (i x Q)/2

= (20)(4) + (200x4)/15 + (1.90x15)/2

= $147.58 (Point A)


S = $400, valid for Q between 15 and 30 tonnes

Q = (2)(A)(S)/I = (2)(400)(4)/1.90 = 41.0 tonnes

This is invalid minimum as Q is not between 15 and 30 tonnes

Calculating the cost at the break point (to the right of the range):

TC = c x A + (S x A)/Q + (i x Q)/2

= (20)(4) + (400x4)/30 + (1.90x30)/2

= $161.83 (Point B)


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Q = (2)(A)(S)/I = (2)(600)(4)/1.90 = 50.3 tonnes

This is invalid minimum as Q is not between 30 and 45 tonnesCalculating the cost at the break point (to the right of the range):

TC = c x A + (S x A)/Q + (i x Q)/2

= (20)(4) + (600x4)/45 + (1.90x45)/2

= $176.08 (Point C)



Q = (2)(A)(S)/I = (2)(800)(4)/1.90 = 58.0 tonnes

This is a valid minimum as Q is between 45 and 60 tonnes

Calculating the cost at the valid minimum:

TC = c x A + (2)(S)(I)(A)

= (20)(4) + (2)(800)(1.90)(4)

= $190.27 a day (Point D)


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Compare the results at point A, B,C & D:

1. Point A: Q=15 tonnes cost= $147.58

2. Point B: Q=30 tonnes cost= $161.83

3. Point C: Q=45 tonnes cost= $176.084. Point D: Q=58 tonnes cost= $190.27

So, the best choice is to order 15 tonnes at a time,

with deliveries needed every 15/4 = 3.75 days


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Finite Replenishment Rates

Used when the rate of production is greater than

demand, thus goods will accumulate at a finiterate while the line is operating ( PD)

Concern with the stock of finished goods at the

end of a production line

Main Assumptions: A single item is considered

Demand is known, constant and continuous All costs are known exactly and do not vary

No shortages are allowed

Replenishment of stock often occurs at a finiterate rather than instantaneously


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The purpose of this analysis is to find the optimalbatch size (optimal order quantity) when the

production rate is greater than demand

Other significant objectives are finding totalcost, production time, cycle time and the perfect

time to place an order (ROL)

Modified Formula EOQ:

Q = 2DCo/iCc x P/P - D


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Example:

Demand for an item is constant at 1800 units a year. Theitem can be made at a constant rate of 3500 units a year.

Unit cost is $50, batch set-up cost is $650 and holding cost is30% of value a year. What is the optimal batch size for theitem? If production set-up time is 2 weeks, when should thisbe started?

SOLUTION:

Listing the variables we know:

D = 1800 units a year

P = 3500 units a year

c = $50 a unit

Co = $650 a batch

i = 0.3 x $50 = $15 unit a year


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Find the optimal batch order(optimal order quantity):

Q = 2DCo/iCc x P/P - D

= (2)(650)(1800)/15 x 3500/3500-1800

= 566.7

Production time, PT, is: PT = Q/P

= 566.7/3500 = 0.16 years

= 8.4 weeks

Cycle Time, CL, is:

CL = 2Co/iCcx A x P/PA

= (2)(650)/(15)(1800) x 3500/3500-1800

= 0.31 years/16.4 weeks


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Variable Cost, VC:

VC = 2 x Co x iCc x D x PD/P

= (2)(650)(15)(1800) x (35001800)/3500

= $4129 a year

Total Cost: TC = C x D + VC

= 50 x 1800 x 4129

= $94,129 a year

If it takes 2 weeks to set up production (lead time), then the

time to start production (replenishment) can be found from thecalculation of the Reorder level.

ROL = LT x d = 2 x (1800/52) = 70 (round up)

Then, the best policy is to start making a batch of 567 unitswhenever stocks fall to 70 units.

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CHAPTER THREE Determistic Model