ILOG CPLEX 11.0 User's Manual

Variable Shipping CostsNow consider the costs of shipping from a given factory to a given showroom. Assume thatfor every pair (factory, showroom), there are different rates, varying according to thequantity shipped. To illustrate the difference between convex and concave piecewise linearfunctions, in fact, this example assumes that there are two different tables of rates forshipping cars from factories to showrooms. The first table of rates looks like this:◆ a rate of 120 per car for quantities between 0 and 200;◆ a rate of 80 per car for quantities between 200 and 400;◆ a rate of 50 per car for quantities higher than 400.These costs that vary according to quantity define the piecewise linear function representedin Figure 18.3. As you see, the slopes of the segments of that function are decreasing, so thatfunction is concave.Figure 18.3400003000020000100000Figure 18.3 A concave piecewise linear cost function200 400Also assume that there is a second table of rates for shipping cars from factories toshowrooms. The second table of rates looks like this:◆ a rate of 30 per car for quantities between 0 and 200;◆ a rate of 80 per car for quantities between 200 and 400;◆ a rate of 130 per car for quantities higher than 400.The costs in this second table of rates that vary according to the quantity of cars shippeddefine a piecewise linear function, too. It appears in Figure 18.4. The slopes of the segmentsin this second piecewise linear function are increasing, so this function is convex.ILOG CPLEX 11.0 — USER’ S MANUAL 343