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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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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)
Taking the next lowest cost curve:
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:
TC = c x A + 2 x S x ic x A
= (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)
Taking the next lowest cost curve:
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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Taking the next lowest cost curve:
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:
TC = c x A + 2 x S x ic x A
= (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)
Taking the next lowest cost curve:
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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Taking the next lowest cost curve:
S = $600, valid for Q between 30 and 45 tonnes
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)
Taking the next lowest cost curve:
S = $800, valid for Q between 45 and 60 tonnes
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.