# Variable Cost and Following Table

1. The diagram below depicts a system of aqueducts that originate at three rivers (nodes R1, R2 and R3) and terminate at a major city (node T) where the other nodes are junction points in the system. Using units of thousands of acre feet, the tables below show the maximum amount of water that can be pumped through each aqueduct per day and the following diagram shows the network of the system. From/To A R1 75 R2 40 R3 B 65 50 80 C 60 70 From/To D A 60 B 70 C E 45 55 70 F 45 90 From/To T D 120 E 190 F 130

The city water manager wants to determine a flow plan that will maximize the flow of water of the city. Formulate this problem as a max flow problem by identifying a

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The relevant data are given in the table below. Once the production is under way, the marginal net revenue (purchase price minus the marginal production cost) from each airplane produced is shown. Maximize the company’s total profit. Formulate a model with both integer and binary variables for this problem. Customer 1 $3 million $2 million 20% 3 planes 2 $2 million $3 million 40% 2 planes 3 0 $0.8 million 20% 5 planes

Start-up cost Marginal net revenue Capacity used per plane Maximum order

8. Consider the situation in which orders from m different destinations are delivered from a central warehouse. Each destination receives its order in one delivery. Feasible routes are assigned to different carriers, and each carrier may combine at most r orders. Suppose that there are n feasible routes with each route specifying the destinations to which orders are delivered. Assume further that the cost of the jth route is Cj. Overlapping is expected so that the same destination can be reached by more than one carrier. Formulate this problem as an integer model. 9. Consider the job-shop scheduling problem involving eight operations on a single machine with a total of two end products. The sequencing of operations is shown in Figure 1. Let bj be the processing time for the jth operation (j=1,2,..,8). Delivery dates for products 1 and 2 are restricted by d1 and d2 time units measured from the zero datum. Since each operation