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Derivation


For the incremental discount policy the discount applies only for quantities exceeding the price break quantities. If the price break quantity would be 1,000 units and the order quantity would be 1,500 units then the first 999 units wouldn't qualify for a discount. Only for 501 units, exceeding the price break quantity, a discount could be used. In comparison, for an all-units discount policy the total order quantity of 1,500 units would get the lower discount price.

Let's clarify the necessary calculation steps for the incremental discount policy with the following sample data.

Base Data:

Demand per year A=600 units, the fixed order cost K=$8, the unit cost c0=$0.30 for quantities less than x1=500 units, c1=$0.29 for quantities less than x2=1,000 units but greater than or equal to x1=500 units, and c2=$0.28 for quantities greater than or equal to x2=1,000 units. The inventory carrying cost rate is 20% annually.

For this incremental discount example - fixed and variable purchase costs - can be expressed mathematically as

For our sample data, see above, and assumed order quantities of 300, 700 and 1,200 units, we would get the following - fixed and variable - purchasing costs:

a) Q = 300 units

b) Q = 700 units

c) Q = 1,200 units

It is important to note that for the incremental discount policy the fixed order cost is different for each price break region. This could be thought of as a kind of surcharge which must be paid compared to the all-unit discount policy. The figure below illustrates this important fact.

Because the intercept with the vertical axis represents fixed order costs, the incremental discount policy has three different fixed order costs, one for each region, compared to the all-unit discount policy which has only one. This fact is important to understand for the later order quantity calculations. For our sample data we get

 

The solution steps for the incremental discount policy can be done as described by Nahmias (Production and Operation Analysis, 1997, pp 238 - 240):

  1. Determine an algebraic expression for the variable purchasing cost for each price interval. Use this expression to determine the average variable purchasing cost.
  2. Substitute the expressions derived for the average variable purchasing cost into the average annual cost function. Compute the minimum value of Q corresponding to each price interval separately.
  3. Determine which minima computed is realizable (that is, falls into the correct interval). Compare the values of the average annual costs at the realizable EOQ values and pick the lowest.

Hadley and Whitin (Analysis of Inventory Systems, 1963) found out, that the minimum total cost will never be at a price break! Remember, for the all-unit discount an optimal solution could be at a break point.

For each price interval the variable unit cost VPC(Q) will be calculated to get the average variable unit cost VPC(Q)/Q. Taking the same sample data as above results in

Dividing the variable purchase cost by the order quantity gives the average unit cost for each price break region

 

The average annual cost function for the incremental discount policy can then be derived as

For each interval the minima of the curves must be computed by differentiating TC(Q) with respect to Q, setting the results equal to zero and solving for Q. For Qs falling in their correct intervals annual costs must be calculated. The lowest realizable value represents then the solution.

Taking again our sample data we get three different order quantities.

The major difference in the three formulas, besides the different variable unit cost, is only the fixed portion of the order cost, which we already derived previously without differentiation. From the equations it becomes clear as well that there are two ways to find the order quantities for the incremental discount policy

a) By differentiation or

b) By calculating the fixed order cost K in a separate step as shown above 

For our sample data, only Q0 and Q1 are valid order quantities, not Q2. Comparing the resulting annual costs for Q0 and Q1 leads to the solution of Q0 with 400 units and total annual costs of $204. The offered discounts would be rejected!


Finally, for the interested reader I included the detailed differentiation steps for finding the optimal Qs.

  


To review the above formulas an Excel simulation sheet is included to clarify the calculation steps. Move your mouse cursor over the below picture and click to open the spreadsheet. Press the button 'Enable Macros' to gain advantage of the full functionality. Review the sample data or click the button 'Clear Input Fields' in order to enter your own data. Don't forget to click the button 'Calculate' in order to get your results.

The system calculates for you

a) the optimal annual total costs, b) the optimal order quantity and c) the cycle time in weeks.

 


 

 

 

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Last modified: January 20, 2001