EOQ With Quantity Discounts

Inventory Model With Single Discount

The purchase inventory model with single discount may be expressed as follows:

Following are the steps to summarize the approach.

Steps

1. Compute the optimal order quantity for the lowest price (highest discount), i.e.,

and compare the value of Q₂* with the quantity b which is required to avail the discount.
If Q₂* ≥ b, then place orders for quantities of size Q₂* and obtain discount; otherwise move to step 2.

2. Compute Q₁* for price P₁ and compare TC(Q₁*) with TC(b). The values of TC(Q₁*) and TC(b) may be determined as follows:

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If TC(Q₁*) > TC(b), then place orders for quantities of size b to get the discount.

Example

A big cold drinks company, the Piyo - Pilao Company, buys a large number of pallets every year, which it uses in the warehousing of its bottled products. A local vender has offered the following discount schedule for pallets:

The average yearly replacement is 2000 pallets. The carrying costs are 12% of the average inventory and ordering cost per order is Rs. 100.

Solution.

Given
D = 2000 pallets/year, C_h = 0.12, C_o = Rs. 100, P₁ = Rs. 10, P₂ = Rs. 9.00

Step 1

The lowest price (highest discount) is RS. 9.00.

TC(Q₁*) = TC(577.35) = 2000 X10 + (2000/577.35) X 100 + (577.35/2 ) X 0.12 X 10
= Rs. 20692.82

TC(b) = TC (700) = 2000 X 9 + (2000/700) X 100 + (700/2) X 0.12 X 9
= Rs. 18663.71

Since TC(b) < TC(Q₁*) and hence the optimal order quantity is the price discount quantity, i.e., 700 units.

b) Inventory model with double discount

Where b₁ and b₂ are the quantities, which determine the price discount.

Following are the steps to summarize the approach.

Steps

1. Compute the optimal order quantity for the lowest price (highest discount), i.e., Q₃* and compare it with b₂

1. If Q₃* ≥ b₂, then place order equal to this optimal quantity Q₃*
2. If Q₃* < b₂, then go to step 2

2. Compute Q₂* and since Q₃* < b₂, this implies Q₂* is also less than b₂. Thus, either Q₂* < b₁ or b₁ ≤ Q₂* < b₂

1. If Q₂* < b₂, but ≥ b₁, then proceed as in the case of single discount, i.e., compare TC(Q₂*) and TC(b₂) to determine the optimal purchase quantity.
2. If Q₂* < b₂ and b₁, then move to step 3

3. Compute Q₁* and compare TC(b₁), TC(b₂) and TC(Q₁*) to determine the purchase quantity.

Example

A large dairy firm, the Cow and Buffalo Company, buys bins every year, which it uses in the warehousing of its bottled products. A local vender has offered the following discount schedule for bins:

The average yearly replacement is 2000 bins. The carrying costs are 12% of the average inventory and ordering cost per order is Rs. 100.

Solution.

Given
D = 2000 bins/year, C_h = 0.12, C_o = Rs. 100, P₁ = Rs. 10, P₂ = Rs. 9, P₃ = Rs. 8

Step 1

The lowest price (highest discount) is Rs. 8. Thus calculating Q₃* = corresponding to this range as follows:

Use horizontal scrollbar to view full calculation

TC(Q₁*) = TC(577.35) = 2000 X10 + (2000/577.35) X 100 + (577.35/2 ) X 0.12 X 10
= Rs. 20692.82

TC(b₁) = TC(700) = 2000 X 9 + (2000/700) X 100 + (700/2) X 0.12 X 9
= Rs. 18663.71

TC(b₂) = TC(950) = 2000 X 8 + (2000/950) X 100 + (950/2) X 0.12 X 8
= Rs. 16666.52

The lowest total inventory cost is TC(b₂) = Rs. 16666.52 and hence the optimal order quantity is the price discount quantity of 950 units, i.e., Q* = b₂ = 950 units.