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Klein Chemicals, Inc., produces a special oil-based material that is currently in short supply. Four of Klein's customers have already placed orders that together exceed the combined capacity of Klein's two plants. Klein's management faces the problem of deciding how many units it should supply to each customer. Because the four customers are in different industries, different prices can be charged because of the various industry pricing structures. However, slightly different production costs at the two plants and varying transportation costs between the plants and customers make a "sell to the highest bidder" strategy unacceptable. After considering price, production costs, and transportation costs, Klein established the following profit per unit for each plant-customer alternative. Customer Plant 2 D2 D2 4 DA 11 Clifton Springs $32 $34 $32 $40 2 Danville $34 $30 $28 $38 The plant capacities and customer orders are as follows. Plant Capacity (units) Distributor Orders (units) Clifton Springs 5,000 D1 2,000 Danville 3,000 D 5,000 3,000 D4 2,000 (a) How many units should each plant produce for each customer to maximize profits? Optimal Solution Units Cost Clifton Springs-D1 $ Clifton Springs-D2 $ Clifton Springs-D3 $ Clifton Springs-D4 $ Danville-D1 $ Danville-D2 $ Danville-D3 $ Danville-D4 Total Cost $ (b) Which customer demands will not be met? Distributor 1 will have a shortfall of units. Distributor 2 will have a shortfall of units. Distributor 3 will have a shortfall of units. Distributor 4 will have a shortfall of units. (c) Show your network model and linear programming formulation. (i) network model (Submit a file with a maximum size of 1 MB.) Choose File No file chosen This answer has not been graded yet. (ii) linear programming formulation Let Xij = number of units i shipped to client ], using the indices from the given table. (It may be necessary to combine plants or distributors in a single node in order to solve this problem. Use index number 5 for this type of node. Enter "DNE" in any unused answer blanks.) Max s.t. Orders from Clifton Springs Orders from Danville Orders from/for Dummy Node Orders for D1 Orders for D Orders for DA Orders for D4 Xij 20 for all i, j