# EOQ With Quantity Discounts

## Inventory Model With Single Discount

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

Order Quantity Unit Price (Rs.)
1 ≤ Q1 < b P1
b ≤ Q2 P2

Following are the steps to summarize the approach.

##### Steps

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

 Q2* = (2DCo) -------------- ChP2

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:

On small screens, use horizontal scrollbar to view full calculation

 TC(Q1*) = DP1 + (D/Q1*) X Co + (Q1*/2) X Ch X P1 TC(b) = DP2 + (D/b) X Co + (b/2) X Ch X P2

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:

Order Quantity Unit Price (Rs.)
Upto 699 10.00
700 and above 9.00

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, Ch = 0.12, Co = Rs. 100, P1 = Rs. 10, P2 = Rs. 9.00

##### Step 1

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

Q2* = (2 X 2000 X 100)
----------------
0.12 X 9
= 608.58 pallets/order
Since Q2* < b (i.e., 608 < 700), Q2* is not feasible.
##### Step 2

Q1* =
(2 X 2000 X 100)
----------------
0.12 X 10
= 577.35 pallets/order

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

Order Quantity Unit Price (Rs.)
1 ≤ Q1 < b1 P1
b1 ≤ Q2 < b2 P2
b2 ≤ Q3 P3

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:

Order Quantity Unit Price (Rs.)
Upto 699 10.00
700 to 949 9
950 and above 8

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, Ch = 0.12, Co = 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

 Q3* = (2 X 2000 X 100) ---------------- 0.12 X 8 = 645.49 bins/order

Since Q3* < b2 (i.e., 645.49 < 950), go to step 2 to determine Q2*

##### Step 2

Q2* = (2 X 2000 X 100)
----------------
0.12 X 9
= 608.58 bins/order

Again, since Q2* < b2 and b1 (i.e., 608.58 < 950 & 700) go to step 3 to calculate Q1* and compare total inventory cost corresponding to Q1*, b1 and b2.

##### Step 3

Q1* = (2 X 2000 X 100)
----------------
0.12 X 10
= 577.35 bins/order

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.