Script

INVENTORY MODELING – Quantity Discounts

Slide 1

  • Welcome back.
  • In this module we discuss the situation where, as a retailer, we get cost breaks from our supplier for large order quantities.

Slide 2

  • Discounts for large orders
  • can be given in one of two ways
  • One is that when a large number of items is ordered, all the items are sold at the discounted price –
  • this is an all units discount model and is the model we will concentrate on.
  • Alternatively
  • When large orders are placed, the first few of them (up to some quantity Q1) would be sold at the regular price, with the discount applying to only those items above Q1 purchased. This is called an incremental discount model.

Slide 3

  • To illustrate the all units discount case
  • Let’s refer again to the juicer problem faced by Allen Appliance
  • Recall that this model assumed a 14% holding cost rate, order costs of $12 per order, and a relatively constant yearly demand of 6240.
  • Now suppose Allen is offered the following all-unit discount plan:
  • If Allen orders less than $300, it will pay the full undiscounted cost of $10 per item; if it orders 300 or more but less than 600, Allen will receive a 2.5% price break bringing the unit cost down to $9.75; between 600 and 1000 a 5% price break bringing the unit cost down to $9.50; between 1000 and 5000 a 6% price break bringing the unit cost down to $9.40, and if 5000 or more are ordered at a time it will receive a 10% price break, bring the unit costs down to $9 even.

Slide 4

  • To illustrate the solution approach for this case we will look at each of the 5 possible selling prices as pieces and analyze each piece individually.
  • For each piece, I, with corresponding unit cost C sub I, let us assume that this price is valid everywhere.
  • Then for each piece we calculate its optimal Q-star – they will be slightly different because C sub H in the EOQ formula depends on the value of the item which changes slightly from piece to piece.
  • Now consider the interval over which each cost is actually valid – call it an interval from Q sub L to Q sub U and determine the value of Q that is the lowest cost for that interval. As we show on the next slide, for each interval that value will either be QL, QU or its corresponding Q-star. If the lowest cost occurs at its upper bound QU, we can show that this cannot be the optimal value and this interval is not considered.
  • But for the remaining intervals, we compare their lowest cost by substituting its QL or Q-star into the corresponding cost equation using the appropriate value for C sub I, and we select the one with the minimum overall cost.

Slide 5

  • Now the best value of Q for each interval is
  • Its Q-star if Q* is in the interval, QL if its Q-star is less than QL. If Q* is greater than the upper limit of the interval, QU, it is easy to see that we can get a better discount in the next interval and we need not consider this interval at all.

Slide 6

  • Here we analyze where its Q-star is above the interval. This is the curve as if the price C sub I were valid everywhere.
  • And here is its corresponding Q-star.
  • But suppose the price C sub I was only valid in this interval to the left of Q-star from Q sub L to Q sub U.
  • The lowest point on this curve occurs at Q sub U.
  • But we can see that at Q-star, we will get an even deeper discount giving a lower unit cost of C sub I, so this piece cannot contain the optimal value for Q.

Slide 7

  • Here we show the same picture
  • With the corresponding Q-star
  • But this time, let’s assume the interval over which this curve piece is valid, does include Q-star.
  • We can see that the lowest point on this curve is Q* and thus Q-star. is the optimal value for Q, Q sub opt, for this piece

Slide 8

  • Now let’s see what happens when the Q-star value is to the left of the interval.
  • Again we calculate Q-star as if this piece were valid everywhere.
  • Then we see if the interval over which this value for the cost is actually valid is to the right of Q-star
  • The lowest point for this interval occurs at its lower limit, Q sub L

Slide 9

  • Let’s look at the exact calculations for the juicers ordered by Allen Appliance. The first cost is $10 which is charged for orders between Q sub L = 0 to Q sub U = 300.
  • We first solve for Q-star and find it to be 327.
  • Since this Q-star is to the right of the interval, we have argued that this interval will not contain the overall optimal order quantity.

Slide 10

  • A cost of $9.75 per item will be charged to Allen for orders of between Q sub L = 300 to Q sub U = 600.
  • Plugging into the EOQ formula (and noting C sub H is now .14 times 9.75 not .14 times 10), we find that Q-star for this interval is 331.
  • This is in the interval from 300 to 600 and hence is the lowest point for this piece.
  • Substituting 331 for Q sub OPT in the total annual cost equation gives a total annual cost of
  • $61,292

Slide 11

  • A cost of $9.50 per item will be charged to Allen for orders of between Q sub L = 600 to Q sub U = 1000.
  • Plugging into the EOQ formula (and noting C sub H is now .14 times 9.50), we find that Q-star for this interval is 336.
  • Since 336 is to the left of the lower limit for the interval, the lowest cost over the interval from 600 to 1000 occurs at its lower limit of 600.
  • Substituting 600 for Q sub OPT in the total annual cost equation gives a total annual cost of
  • $59,804

Slide 12

  • A cost of $9.40 per item will be charged to Allen for orders of between Q sub L = 1000 to Q sub U = 5000.
  • Plugging into the EOQ formula (and noting C sub H is now .14 times 9.40), we find that Q-star for this interval is 337.
  • Since 337 is to the left of the lower limit for the interval, the lowest cost over the interval from 1000 to 5000 occurs at its lower limit of 1000.
  • Substituting 1000 for Q sub OPT in the total annual cost equation gives a total annual cost of
  • $59,389

Slide 13

  • Finally a cost of $9 per item will be charged to Allen for orders of between Q sub L = 5000 to Q sub U of infinity.
  • Plugging into the EOQ formula (and noting C sub H is now .14 times 9), we find that Q-star for this interval is 345.
  • Since 345 is to the left of the lower limit for the interval, the lowest cost over the interval from 5000 to infinity occurs at its lower limit of 5000.
  • Substituting 5000 for Q sub OPT in the total annual cost equation gives a total annual cost of
  • $59,325

Slide 14

  • Summarizing
  • For Allen Appliance, the lowest total costs over each of the pieces were: irrelevant for the first piece since its Q-star was to the left of its upper limit, $61,292 for the second piece, $59,804 for the third piece, $59,389 for the fourth piece, and $59,325 for the last piece
  • Thus using the minimum cost criterion only, the optimal solution is to order 5000 units at a time to minimize its total annual cost.
  • But note that 5000 units is over a 9-month supply. Management must determine if this is really okay – do we have a large enough inventory area, could new models be introduced in that time interval, could costs change in this time period – these are all questions that should be considered before making the final recommendation.

Slide 15

  • The reorder point analysis
  • is independent of the order quantity determination.
  • The reorder point is found in the same ways as before:
  • It is L times D plus safety stock if demand is assumed to be constant over the lead time
  • And it is found using the cycle service level analysis if demand is assumed to have a probability distribution during lead time.

Slide 16

  • The optimal order quantity and the quantities of interest can be calculated using
  • the All-Units Discounts worksheet of the inventory template.
  • We enter the required values for the input parameters in the usual way, except that here, the per unit cost C in cell B6, refers to the undiscounted cost.
  • Then in the yellow areas of columns B and C from rows 17 on down we enter the lower limits of each discount and the corresponding discounted price
  • The optimal order quantity, the total annual cost using this quantity and other quantities of interest are then generated in the Optimal Outputs section.

Slide 17

  • Let’s review what we’ve done in this module.
  • We stated there are two types of discount models
  • All-unit and incremental models
  • For the all units discounted model
  • We found Q-star for each piece
  • And stated that Q-star is the best point for this piece if it is interval
  • Q sub L is the best point if Q-star lies to the left of Q sub L, and if Q-star lies to the right of Q sub U, the interval would not be the one with the optimal solution.
  • The minimum costs for each interval are then compared and the best value of Q for the entire problem is the one corresponding to the minimum of these total costs.
  • We stated that the calculation of the reorder point is unaffected by any approach to calculate the optimal order quantity.
  • And we showed how to use the All Units Discount worksheet of the inventory template.

That’s it for this module. Do any assigned homework and I’ll be back to talk to you again next time.