1. Inventory Models Quantity Discount Models

2. TYPES OF QUANTITY DISCOUNT MODELS Quantity Discount Models are of two types –All Units Discounts Above a certain order level, Q 1, all units in the order are discounted (Usual Case) –Incremental Discounts (Not Discussed Here) Above a certain order level, Q 1, all items above Q 1 are discounted – the first Q 1 items are sold at a non-discounted price

3. ALL UNIT QUANTITY DISCOUNTS Reconsider the juicer example where: H =.14, C O = 12, D = 6240/yr Suppose the following all-unit discount pricing plan applies: Quantity Unit Ordered Cost <300$10.00 300- 600$ 9.75 600-1000$ 9.50 1000-5000$ 9.40 ≥5000$ 9.00

4. PIECEWISE APPROACH Consider each quantity discount for C i as if it were valid everywhere from 0 -  STEP 1: For each value of C i, calculate the corresponding value of Q i * -- it will change slightly since C h = HC i and C i changes slightly. STEP 2: For each quantity discount for C i, consider the interval (from a lower limit Q L to an upper limit Q U ) over which the price C i is valid and determine the value of Q that gives the lowest cost for the interval. (See next slide.) STEP 3: Compare the lowest costs for each interval, and choose the lowest of these interval lowest costs.

5. DETERMINIG THE BEST VALUE OF Q FOR EACH INTERVAL AND OVERALL BEST VALUE OF Q FOR AN INTERVAL BETWEEN Q L AND Q U –If Q* >Q U, ignore this interval –if Q i * is in interval, Q opt = Q i * –If Q* < Q L, Q opt = Q L

6. Q OPT for the interval in which Q i * > Q U TC Q Qi*Qi* QLQL QUQU Lowest point occurs at Q U. Q opt = Q U But we can see that Q* (which will have an even deeper discount) gives a lower total cost, TC.

7. Q OPT When Q i * Is In the Interval TC Q Q* QLQL QUQU Q* is the lowest point in the interval. Q opt = Q*

8. Q OPT for the interval in which Q i * < Q L TC Q Q* QLQL QUQU Lowest point occurs at Q L. Q opt. = Q L

9. Calculations for C = $10 Interval (0,300) Since Q* > 300, the optimal solution for the model cannot come from this interval.

10. Calculations for C = $9.75 Interval (300,600) Since Q* = 331 is in the interval (300,600), for this interval: Q opt = Q* = 331

11. Calculations for C = $9.50 Interval (600,1000) Since Q* = 336 < 600, for this interval: Q opt = 600

12. Calculations for C = $9.40 Interval (1000,5000) Since Q* = 337 < 1000, for this interval: Q opt = 1000

13. Calculations for C = $9.00 Interval (5000,  ) Since Q* = 345 < 5000, for this interval: Q opt = 5000

14. QUANTITY DISCOUNT APPROACH FOR ALLEN QuantityUnit Cost C h Q* Q opt TC < 300 $10.00 $1.40 327 ---- ---- 300-600 $ 9.75 $1.365 331 331 $61,292 600-1000 $ 9.50 $1.33 336 600 $59,804 1000-5000 $ 9.40 $1.316 337 1000 $59,389  5000 $ 9.00 $1.26 345 5000 $59,325 ORDER 5000ORDER 5000 Note: This is over a 9 month supply -- is this OK?

15. REORDER POINT ANALYSIS Reorder point and safety stock determination are not affected by quantity discounts. They are found in the same way as before: –r* = LD + SS if demand is constant over lead time –r* is found using service levels if demand varies during lead time

16. Using the Template All-Units Worksheet Enter Values Enter Discount Breaks and Discount Prices Optimal Values

17. Review Types of Discount Models: –All-unit and Incremental All-Units Quantity Discount Model –Q i * is found for each interval from Q L to Q U –Best point for interval is Q i * if Q i * is in interval –If Q* < Q L, Q L is best point for interval Best value of Q is found by finding the lowest cost for each interval and taking the Q with the lowest total cost. Reorder point analysis is not affected. Use of template