Answer Happy

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. Plant Clifton Springs Danville E 5 Danville Plant Clifton Springs D₁ $32 Customer D₂ D3 D₁ $34 $32 $40 The plant capacities and customer orders are as follows. $34 $30 $28 Capacity (units) 5,000 3,000 $38 Distributor Orders (units) D₁ D₂ D3 DA 2,000 5,000 3,000 2,000
(a) How many units should each plant produce for each customer to maximize profits? Optimal Solution Clifton Springs-D₁ Clifton Springs-D₂ 4000 Clifton Springs-D3 Clifton Springs-D4 Danville-D₁ Danville-D₂ Danville-D3 Danville-D4 Total Cost 0 Units 1000. 2000 0 0 1000 (b) Which customer demands will not be met? Distributor 1 will have a shortfall of Distributor 2 will have a shortfall of Distributor 3 will have a shortfall of Distributor 4 will have a shortfall of $0 $ 136000 $0 Cost 40000 $68000 $0 $0 $ 38000 units. units. units. units. (c) Show your network model and linear programming formulation.
(ii) linear programming formulation Let x = number of units i shipped to client j, using the indices from the given table. (It may be necessary to combine plant 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 D₂ Orders for D3 Orders for Da x 2.0 for all i, f.