1. Channel Coordination and Quantity Discounts Z. Kevin Weng Management Science, Volume 41, Issue 9 (September, 1995), 1509-1522. Prepared by: Çağrı LATİFOĞLU

2. Presentation Outline  Introduction  Model  Model Analyses  Allocation of the Profits  Quantity Discounts  Conclusion

3. Introduction  This paper represents a model analyzing the impact of joint decision policies on a channel coordination in a system consisting of a supplier and group of homogenous buyers.

4. Introduction  Joint decision policy is characterized by:  Unit selling prices  The order quantities (coordinated through the quantity discounts and franchies fees)

5. Introduction  Annual Demand Rate  Operating Costs(include purchase, ordering and inventory holding costs) are affected by:  Joint unit selling price  Joint order quantity

6. Introduction  Past studies on this problem is branched into two streams: First Stream: Operating costs are functions of order quantities and demand is treated as a fixed constant. Second Stream: Demand is a decreasing function of buyer’s selling prices and operating costs are assumed to be fixed.

7. Introduction  This research is the generalized version of these two streams, considering channel coordination and operating cost minimization.

8. Model  There is one supplier and one buyer (or a group of homogenous buyers who are all treated same)  It is difficult to extend the model for heterogenous customers since it is difficult to find the avarage inventory in this case.

9. Model  Annual demand rate is a decreasing function of buyer’s selling price  Operating costs of both parties depend on order quantities.

10. Model  Buyers inventory policy is EOQ and quantity discount for buyers are same.  Demand increases with price reduction.

11. Model  Quantity Discounts: to ensure the joint order quantity minimizes the operating costs.  Franchise fees: to enforce joint profit maximization

12. Model p: buyer’s unit purchase price-charged by supplier x: buyer’s unit selling price-charged by buyer h b : buyer’s yearly unit inventory holding cost h s ’: supplier’s yearly unit inventory holding cost S b : buyer’s fixed ordering cost per order S p : supplier’s fixed order processing cost S s ’: supplier’s setup cost for each machine

13. Model  Supplier procures the material by either manufacturing or purchasing where cost of procurement c < p.  Buyer’s lot size Q  Supplier’s lot size mQ where m=1,2,...

14. Model  Holding cost of supplier R=annual production capacity Proc. by purc. : h s Q/2 where h s =Mh s ’ M=m-1 Proc. by mfg. : h s Q/2 where h s =Mh s ’ M=m-1-(m-2)*D(x)/R

15. Model  Supplier‘s order processing and setup ordering cost S s D(x)/Q where S s =S p +S s ’ /m  Supplier’s yearly profit: G s (p)=(p-c)D(x)- S s D(x)/Q- h s Q/2 revenue # of setups inv.holding

16. Model  Buyer’s yearly profit: G b (x,Q)=(x-p)D(x)- S b D(x)/Q- h b Q/2  As we also see in the profits supplier can only control p, while buyer controls Q and x.

17. Model Analyses  In the scenario 1, supplier & buyer will try to maximize their profits by optimizing the decision varibles that are under their control.  In the scenario 2, objective is to maximize the joint profit of both supplier & buyer s.t. both of their profits are greater than the first case.

18. Scenario 1  For supplier’s unit selling price p, x b (p) denotes the buyer’s optimal selling price.  Buyer’s optimal order size is (EOQ): Q b (p)=(2S b D(x b (p))/h b ) ½  where holding & ordering cost is (2S b h b D(x b (p))) ½

19. Scenario 1  G b (x b ) is the corresponding buyer‘s profit: G b (x b |Q b )= (x-p)D(x) - (2S b h b D(x)) ½  The corresponding supplier’s profit: G s (p) = (p-c)D(x b (p))–(S s /S b + h s /h b ) * (S b h b D(x b (p))/2) ½

20. Scenario 1  Lemma 1: With buyer’s EOQ order quantity, Q b (p), supplier’s yearly profit is never higher than the maximum that can be achieved by supplier’s EOQ order quantity.  (S s h b /S b h s + S b h s /S s h b ) >= 2  Buyer’s EOQ will also maximize this profit if S s /S b = h s /h b

21. Scenario 1  p* maximizes G s *(=G s (p*))  x b (p*) maximizes G b *(=G b (x b (p*)))  Total profit maximum profitin case 1 = G s *+ G b *

22. Scenario 2  In this case, the joint policies which enables both supplier & buyer to achieve higher profits, are analyzed, given that they are willing to cooperate.

23. Scenario 2  Joint profit function: G j (x,Q) = G s (p) + G b (x,q) Q j (x) = (2S j D(x)/h j ) ½ where S j =S s +S b and h j =h s +h b

24. Scenario 2  Joint profit function: G j (x|Q j (x)) =(x-c)D(x) - (2S j D(x)h j ) ½ For buyer’s unit selling price x b (p*) and Q j (x b (p*)) = (2S j D(x b (p*))/h j ) ½  Lemma 2: G j (x b (p*)|Q j (x b (p*))) >= G s *+ G b *

25. Scenario 2  For a given policy (x, Q j (x)) G s (p|Q j (x))= (p-c)D(x)-S s D(x)/Q j (x)- h s Q j (x)/2 Let p min (x) is the smallest price that satisfies G s (p|Q j (x))>= G s * p min (x) = c +{G s */D(x) + (S s /S j + h s /h j ) * (S j h j /2D(x)) ½

26. Scenario 2 In that case buyer’s profit will be G b (x, Q j (x))= (x-p)D(x)-S b D(x)/Q j (x)- h b Q j (x)/2 Let p max (x) is the largest buyers purchasing price that satisfies G b (x, Q j (x)) >= G b * p max (x) = x -{G b */D(x) + (S b /S j + h b /h j ) * (S j h j /2D(x)) ½

27. Scenario 2 G j (x|Q j (x)) - (G s *+ G b *) = D(x)*[p max (x) - p min (x)] Increased Unit Profit Yearly increase in Profit For achiving this buyer should select x rather than x b (p*) where x<= x b (p*)

28. Allocation of the Profits  For the joint optimal policy (x*, Q j (x*))  If the d percentage of the increased profit goes to buyer, (1-d) percentage will go to supplier and so the price that will be charged by the supplier will be:  p j =d p min (x)+(1-d) p max (x)

29. Allocation of the Profits  To make buyer choose the joint optimum order quantity(rather than the amount that maximizes its profit alone) quantity discounts are offered.  For making him choose the joint optimum unit selling price, franchise fees are used.  Once a year buyer pays the supplier ß p j D(x*) and in return supplier charges (1-ß) p j avarage unit selling price. In this case the buyer’s optimal selling price x*((1-ß) p j ) is equal to optimal joint selling price x*.

30. Quantity Discounts  All unit: If buyer orders an amount Q x (>Q i ), the discount is applied to whole order(Q x ).  Incremental: If buyer orders an mount Q x (>Q i ), the discount is applied to additional units (Q x -Q i ).

31. Quantity Discounts – All Unit Q ai is a price breakpoint where the corresponding all-unit discount price is r ai p* If S s /S b = h s /h b then Q b (r ai p*) = Q ai Else Q b (r ai p*) ≠ Q ai

32. Quantity Discounts – All Unit  It is also proposed that there should be only one price breakpoint and it should be at joint optimal order quantity(since it is unique).

33. Quantity Discounts – All Unit  Buyer’s yearly profit increase λ % (>=0) (which satisfies G b (x*(r a p*))>= G b *)  Supplier’s yearly profit increase ß % (>=0) In that case; r a p* =p j = p max (x*) - λG b */ D(x*) Q a = Q j (x*) = [2S j D(x*)/h j ] ½ λ G b * + ß G s * =[p max (x*) - p min (x*)]D(x*)

34. Quantity Discounts – All Unit  From the formulations we can see that all unit discount percentage and buyer’s profit increase percentage have a linear relationship due to the fact that p j linearly affects purchase cost but it has no impact on the other costs.  Another observation is the negative linear relation between supplier percentage profit increase and all-unit quantity discount

35. Incremental Quantity Discount  In this policy, the discount is applied to the units that are over the price breakpoint Q.  r 1 ’ =r 1 (1-Q/Q 1 ) + Q/Q 1  G b (x b (r 1 ’ p*)|Q)=(x b (r 1 ’ p*)- r 1 ’ p*) D(x b (r 1 ’ p*)) - S b D(x b (r 1 ’ p*))/Q 1 - h b Q 1 /2

36. Incremental Quantity Discount  Q 1 = [2(S b +p*(1-r 1 ’ Q) D(x b (r 1 ’ p*))/h b ]  G s1 ( r 1 ’ p*|Q)= (r 1 ’ p*-c) D(x b (r 1 ’ p*)) - S s D(x b (r 1 ’ p*))/Q 1 - h s Q 1 /2

37. Equivalence of AQD and IQD  Given that both AQD and IQD increase buyer’s profit by an equal amount (since they have the same unit selling price, x*) the increase in supplier’s profits should be same. (details are in the paper)  It is found that r a = r 1 ’ p*=p j and Q a = Q 1 = Q j (x*)

38. Conclusion  Quantity discounts alone are not sufficient to guarantee joint profit maximization, franchise fees should be implemented as a control mechanism  Whether the demand is constant or not, AQD and IQD perform identically,  Dependency of demand on unit selling price and operating cost dependency on order quantities is more critical.

39. Q & A