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Channel Coordination and Quantity Discounts

Channel Coordination and Quantity Discounts. Z. Kevin Weng Management Science, Volume 41, Issue 9 (September, 1995), 1509-1522. Prepared by: Çağrı LATİFOĞLU. Presentation Outline. Introduction Model Model Analyses Allocation of the Profits Quantity Discounts Conclusion. Introduction.

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Channel Coordination and Quantity Discounts

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  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 hb: buyer’s yearly unit inventory holding cost hs’: supplier’s yearly unit inventory holding cost Sb: buyer’s fixed ordering cost per order Sp: supplier’s fixed order processing cost Ss’: 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 mfg. : hsQ/2 where hs=Mhs’ M=m-1-(m-2)*D(x)/R Proc. by purc. : hsQ/2 where hs=Mhs’ M=m-1

  15. Model • Supplier‘s order processing and setup ordering cost SsD(x)/Q where Ss=Sp+Ss’/m • Supplier’s yearly profit: Gs(p)=(p-c)D(x)- SsD(x)/Q- hsQ/2 revenue # of setups inv.holding

  16. Model • Buyer’s yearly profit: Gb(x,Q)=(x-p)D(x)- SbD(x)/Q- hbQ/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, xb(p) denotes the buyer’s optimal selling price. • Buyer’s optimal order size is (EOQ): Qb(p)=(2SbD(xb(p))/hb)½ • where holding & ordering cost is (2SbhbD(xb(p)))½

  19. Scenario 1 • Gb(xb) is the corresponding buyer‘s profit: Gb(xb|Qb)= (x-p)D(x) - (2SbhbD(x))½ • The corresponding supplier’s profit: Gs(p) = (p-c)D(xb(p))–(Ss/Sb+ hs/hb) * (SbhbD(xb(p))/2)½

  20. Scenario 1 • Lemma 1: With buyer’s EOQ order quantity, Qb(p), supplier’s yearly profit is never higher than the maximum that can be achieved by supplier’s EOQ order quantity. • (Sshb/Sbhs+ Sbhs/Sshb) >= 2 • Buyer’s EOQ will also maximize this profit if Ss/Sb= hs/hb

  21. Scenario 1 • p* maximizes Gs*(=Gs(p*)) • xb(p*) maximizes Gb*(=Gb(xb(p*))) • Total profit maximum profitin case 1 = Gs*+ Gb*

  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: Gj(x,Q) = Gs(p) + Gb(x,q) Qj(x) = (2SjD(x)/hj)½ where Sj=Ss+Sb and hj=hs+hb

  24. Scenario 2 • Joint profit function: Gj(x|Qj(x)) =(x-c)D(x) - (2SjD(x)hj)½ For buyer’s unit selling price xb(p*) and Qj(xb(p*)) = (2SjD(xb(p*))/hj)½ • Lemma 2: Gj(xb(p*)|Qj(xb(p*))) >= Gs*+ Gb*

  25. Scenario 2 • For a given policy (x, Qj(x)) Gs(p|Qj(x))= (p-c)D(x)-SsD(x)/Qj(x)- hsQj(x)/2 Let pmin(x) is the smallest price that satisfies Gs(p|Qj(x))>= Gs* pmin(x) = c +{Gs*/D(x) + (Ss/Sj+ hs/hj) * (Sjhj/2D(x))½

  26. Scenario 2 In that case buyer’s profit will be Gb(x, Qj(x))= (x-p)D(x)-SbD(x)/Qj(x)- hbQj(x)/2 Let pmax(x) is the largest buyers purchasing price that satisfies Gb(x, Qj(x)) >= Gb* pmax(x) = x -{Gb*/D(x) + (Sb/Sj+ hb/hj) * (Sjhj/2D(x))½

  27. Scenario 2 Gj(x|Qj(x)) - (Gs*+ Gb*) = D(x)*[pmax(x) - pmin(x)] Increased Unit Profit Yearly increase in Profit For achiving this buyer should select x rather than xb(p*) where x<= xb(p*)

  28. Allocation of the Profits • For the joint optimal policy (x*, Qj(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: • pj=d pmin(x)+(1-d) pmax(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 pj D(x*) and in return supplier charges (1-ß) pj avarage unit selling price. In this case the buyer’s optimal selling price x*((1-ß) pj) is equal to optimal joint selling price x*.

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

  31. Quantity Discounts – All Unit Qai is a price breakpoint where the corresponding all-unit discount price is rai p* If Ss/Sb= hs/hb then Qb(rai p*) = Qai Else Qb(rai p*) ≠ Qai

  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 Gb(x*(rap*))>= Gb*) • Supplier’s yearly profit increase ß % (>=0) In that case; rap*=pj = pmax(x*) - λGb*/ D(x*) Qa = Qj(x*) = [2SjD(x*)/hj]½ λ Gb* + ß Gs* =[pmax(x*) - pmin(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 pj 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. • r1’=r1(1-Q/Q1) + Q/Q1 • Gb(xb(r1’p*)|Q)=(xb(r1’p*)- r1’p*) D(xb(r1’p*)) - Sb D(xb(r1’p*))/Q1- hbQ1/2

  36. Incremental Quantity Discount • Q1= [2(Sb+p*(1-r1’Q) D(xb(r1’p*))/hb] • Gs1( r1’p*|Q)= (r1’p*-c) D(xb(r1’p*)) - Ss D(xb(r1’p*))/Q1- hsQ1/2

  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 ra= r1’p*=pj and Qa= Q1= Qj(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

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