Quantity discounts all unit

P. L. Abad. Determining optimal selling price and lot size when the supplier offers all-unit quantity discounts. Decision Sciences, 19(3) 622-634, 1988. 

All-unit" quantity discounts Under this scenario, the entire purchase will be charged at a lower price based on the order quantity. Figure 5.1 illustrates an example of "all-unit" quantity discount price structure. 

All-unit quantity discount models are commonly used by trucking companies for the freight rate structure in road transportation. 

Figure 5.2 also represents a nonlinear cost function. It can be linearized using binary variables, similar to the all-unit quantity discount model. Let 8i, 82, and 83 be the binary variables for each price range and Xj, X2 and X3 are the quantity purchased under price ranges 1,2, and 3. Then, the linear IP formulation becomes as follows ... 

In all unit quantity discounts, the pricing schednle contains specified break points qo,qi,..., Qr, where qo = 0. If an order placed is at least as large as qi bnt smaller than q j, each unit is obtained at a cost of C,. In general, the unit cost decreases as the quantity ordered increases that... 

In Example 11-7, we evaluate the optimal lot size given an all unit quantity discount (see worksheets Example 11-7 and Example 11-7 check in spreadsheet Chapterll-examples7-8). 

If the manufacturer in Example 11-7 sold all bottles for 3, it would be optimal for DO to order in lots of 6,325 bottles. The quantity discount is an incentive for DO to order in larger lots of 10,000 bottles, raising both the cycle inventory and the flow time. The impact of the discount is further magnified if DO works hard to reduce its fixed ordering cost to 5 = 4 (from the current 100). Then, the optimal lot size in (he absence of a discount is 1,265 bottles. In the presence of the all unit quantity discount, the optimal lot size is still 10,000 bottles. In this case, the presence of quantity discounts leads to an eightfold increase in average inventory and flow time at DO. 

Prefab, a furniture manufacturer, uses 20,000 square feet of plywood per month. Its trucking company charges Prefab 400 per shipment, independent of the quantity purchased. The manufacturer offers an all unit quantity discount with a price of 1 per square foot for orders under 20,000 square feet, 0.98 per square foot for orders between 20,000 square feet and 40,000 square feet, and 0.96 per square foot for orders larger than 40,000 square feet Prefab incurs a holding cost of 20 percent. What is the optimal lot size for Prefab What is the annual cost of such a policy What is the cycle inventory of plywood at Prefab How does it compare with the cycle inventory if the manufacturer does not offer a quantity discount but sells aU plywood at 0.96 per square foot ... 

Demand for phones at Amazon is 5,000 per month. The holding cost at Amazon is 25 percent and the company incurs a fixed cost of 500 for each order placed. The supplier offers an all unit quantity discount with a price of 200 per phone for all orders under 10,000, a price of 195 for all orders of 10,000 or more but under 20,000 and a price of 190 for all orders of 20,000 or more. How many phones should Amazon order per replenishment ... 

If the steel mill and the service center could work in a coordinated manner, what is the optimal order size that minimizes their joint fixed and holding costs What annual savings could the supply chain expect as a result of coordination Design an all unit quantity discount that the integrated steel mill could use to get the service center to order the coordinated amount without increasing annual costs at the service center. 

Looking ahead, this type of quantity discount structure also appears in Chapter 5, where we discuss techniques for representing this "all-unit-discount" cost structure in mixed-integer programming models. 

As discussed in Chapter 16, setting a fixed price for all units does not maximize profits for the manufacturer. In principle, the manufacturer can obtain the entire area under the demand curve above its marginal cost by pricing each unit differently based on customers marginal willingness to pay at each quantity. Quantity discounts are one mechanism for price discrimination because customers pay different prices based on the quantity purchased. 

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