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(a) The Economic Order Quantity is determined by 2 2(2000)(200)
16000 126.491
h 50 EOQ DCo
C
Thus, the economic order quantity is about 126.5 units
(b) Yearly Demand 2000
Number of orders per year = 15.81
EOQ 126.491 Thus, the number of orders per year is approximately 16 orders.
(c) Number of Days per Year 365
Length of cycle (days) = 23.1
Number of Orders per Year 15.81 The cycle length is about 23 days
Example
A Ltd buys 400 units of an item at a purchase cost ₦5000 per unit and ordering cost of
₦2,000 per order placed. The carrying cost is estimated at 24% of cost of an item p.a.
The Co. has received a 2% discount offer for purchases of 100 or more units.
Required:
a) Determine the best inventory policy for this item.
b) Determine the discount level at which the firm will be indifferent between taking and not taking the discount offer.
Steps
1. Calculate the EOQ with no discount and find the resulting TC
2. Find TC when discount is taken. The EOQ will be lowest quantity to just qualify for discount
3. Compare TC in 1 and 2 and hence make the decision
Solution
a)1) No discount 2
h
EOQ DCo
C =
24 . 0 5000
2000 400
2
= 37 units
TC = DCp + 2 h o
Q C D C
Q
= 400 x 5000 + 2000
37 24 400 . 0 2 5000
37 x = ₦2,043,822
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TC = 400 x 4900 + 100/2 x 4900 x 0.24 + 2000
100
400x = ₦ 2,026,800 3) Decision:
Take the discount offer since it is less costly than NOT taking it.
EOQ = 100 units Timing of orders
Annual No. of orders, N =
100
400 Q
D = 4 orders
Therefore, make an order every 12/4 = 3 months or quarterly
b) Let x represent discount level for cost indifference point. At this point the following conditions will hold:
TC (with discount) = TC (no discount)
2043822 100 2000
24 400 . 0 ) 1 ( 2 5000 ) 100
1 ( 5000
400
x
2043822 8000
24 . 0 ) 1 ( 250000 )
1 (
2000000
2043822 8000
60000 60000
2000000
2000000
2043822 2060000
2068000
2060000 2043822
2068000
= 0.0117
=1.17% = 1.2% (1 dp) Decision Rule
If discount is greater than 1.2% take it but less than 1.2% do not take it.
Multiple discount offers (price breaks)
This is an extension of the single discount offer in the sense that a price / quantity schedule is available or provided instead of a single offer.
The solution approach can be broken down into the following steps:
1. Calculate EOQ for each price/quantity range.
2. The EOQ calculated in 1 will fall in one of 3 categories which will be differently treated as follows:
a) Below range – Ignore the calculated EOQ but calculate TC for the least quantity in the range.
b) Within range – Evaluate total cost for the EOQ calculated.
c) Above range – Ignore the range since there would be another range which will yield lower total cost.
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3.4.2 The Adapted Model
The adapted model assumes gradual replenishment unlike the basic model. This assumption is on the premises that when stocks are received from the production line, it is very likely that finished items are received continuously over a period of time.
Stock is therefore, subject to gradual replenishment. Like the basic model, this model also assumes constant demand rate, D, and the two cost factors: the order cost per cycle, CS, and the holding cost per item, Ch.
The adapted model is often referred to as the production run model whereby:
(i) a production run is started every time the inventory level decreases to zero, and stops whenqitems have been produced or supplied. The run lasts for time t and is known as the run time
(ii) the quantity ordered per cycle is referred to as the run size, and items are supplied at the rate of Pper annum
(iii) the effective replenishment rate is defined as PDitems The adapted model uses the following optimal statistics:
i. Yearly Demand
Number of run per year =
EBQ
ii. Number of Days per Year
Length of cycle (days) =
Number of Runs per Year iii. EBQ - Numbers of days per year
Run time (days) =
Annual production rate
iv. Peak inventory level = Effective Replenishment rate – run time v. Average inventory level = ½ X peak inventory level
The value of the run size that minimises inventory cost for the adapted model is defined by Economic Batch Quantity (EBQ),
Economic Batch Quantity (EBQ), also known as the optimum production quantity (EPQ), is the order size of a production batch that minimizes the total cost. Economic Batch Quantity (EBQ) is a formula for calculating the quantity of inventory that a company should order in cases where the resupply is gradual e.g. when the company produces its own inventory and takes a while to complete production. Think of a company assembling vehicles.
Batch production is a technique which is commonly used today for distributing the total production in a series of small batches rather than mass producing in one go.
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Sometimes the production of goods in batches is necessary because, for example, certain equipment used in manufacturing (e.g. dyes) may wear out and need replacement before the production can run again.
Batch production may be desirable in other cases as well. For example, where the objects being produced are perishable, the entire production requirement for say a year can‘t be manufactured in a week as it might cause the goods to expire after some time. Batch production also reduces the risk of obsolescence as any minor changes required in the specification of goods (e.g. size, colour, etc.) can be made in future batches according to the feedback received from customers or retailers instead of producing everything in one go and hoping for the best.
Whereas EOQ is suitable for determining the order size when the parts, materials or finished goods are ready to be delivered by external suppliers when the order is placed, EBQ is used to determine the size of a production run (i.e. batch size) when the manufacturing takes place internally and any raw materials or parts required for production are either acquired internally or are supplied incrementally by other companies according to the production requirement.
Formula for EBQ is given as;
Economic Batch Quantity 2
( )
(1 / )
s h
EBQ DC
C D P
Where:
Cs is the setup cost of a batch
D= Annual Demand
PP is the annual production capacity
Chis the annual cost of holding one unit of finished inventory
The formula for calculating EBQ is very similar to EOQ with one notable difference in the denominator. The cost of holding in EBQ formula is decreased by the amount of inventory that will be produced and sold on the same day therefore not contributing to the annual cost of holding the inventory.
Example
Abu owns and operates a small factory that manufactures plastic bottles which he sells to bottling companies.
Additional information:
Annual demand is 1 million bottles spread evenly over the year
Setup cost is $5000 per batch
Holding cost is $3 per annum for each bottle
Maximum production capacity is 2 million bottles per annum
Currently, bottles are manufactured in 10 batches
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A. Find the optimum production quantity that Sarah should produce to minimize her costs
B. Calculate the current annual holding cost and setup cost
C. Calculate the savings to Sarah if she adopts the EBQ Solution A: Optimum Production Quantity
Economic Batch Quantity
2 2 5000 1,000,000
( )
(1 / ) 3 (1 (1,000,000 / 2,000,000)
s h
EBQ DC
C D P
10,000,000,000
6,666,666,666 81,650
1.5
Abu should manufacture bottles in batches of 81,650 units.
Solution B: Current Costs
Batch Quantity = Annual Demand ÷ Number of batches = 1,000,000 ÷ 10
= 100,000 units
Annual Holding Cost = (Batch Quantity/2) × Ch × (1- D/P) = (100,000/2) × 3 × (1-(1,000,000/2,000,000)) = $75,000
Setup Cost = Number of setups × setup cost = 10 × 5000
= $50,000
Total Current Cost = ($75,000 + $50,000) = $125,000 Solution C: Savings from EBQ
Annual Holding Cost:
= (Batch Quantity/2) × Ch × (1- D/P)
= (81,650/2) × 3 × (1-(1,000,000/2,000,000)) = $61,238 (A)
Setup Cost
Number of batches = 1,000,000 ÷ 81,650 = 12.2475 Setup Cost = Number of batches × Cost of setup = 12.2475 × $5000 = $61,23 (B)
Total Cost (EBQ) = (A) + (B) = $122,476 (C) Total Current Cost = 125,000 (D)
Savings = (D) - (C) = 2,524
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SELF ASSESSMENT EXERISE
A manufacturing activity requires a continuous supply of 3000 items per year from store, replenished by production runs, each of which operates at the constant rate of 5000 items per year. Each production run has a set-up cost of N300, and the holding cost per item per annum is N25. Compute the Economic Batch Quantity (EBQ), and find the number of runs per year, the length of cycle, the run time, the peak inventory level, and the average inventory level.