Question: Problem 6-06 (Algorithmic) Klein Chemicals, Inc., produces a special oil-based material that is currently in short supply. Four of Klein's customers have already placed orders
Problem 6-06 (Algorithmic)
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. Kleins 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 plantcustomer alternative:
| Customer | ||||||||
| Plant | D1 | D2 | D3 | D4 | ||||
| Clifton Springs | $128 | $136 | $128 | $160 | ||||
| Danville | $136 | $120 | $112 | $152 | ||||
The plant capacities and customer orders are as follows:
| Plant | Capacity (units) | Distributor Orders (units) | |||
| Clifton Springs | 15000 | D1 6000 | |||
| D2 15000 | |||||
| Danville | 9000 | D39000 | |||
| D46000 | |||||
How many units should each plant produce for each customer in order to maximize profits? Which customer demands will not be met? Choose correct network model and Show linear programming formulation. If the constant is "1" it must be entered in the box. If your answer is zero, enter "0".
Note: Dummy origin has supply of 12000.
Let xij = amount of units produced by plant node i for customer node j.
| Max | x11 | + | x12 | + | x13 | + | x14 | + | x21 | + | x22 | + | x23 | |||||
| + | x24 | + | x31 | + | x32 | + | x33 | + | x34 | |||||||||
| s.t. | x11 | + | x12 | + | x13 | + | x14 | |||||||||||
| x21 | + | x22 | + | x23 | + | x24 | ||||||||||||
| x31 | + | x32 | + | x33 | + | x34 | Dummy | |||||||||||
| x11 | + | x21 | + | x31 | = | |||||||||||||
| x12 | + | x22 | + | x32 | = | |||||||||||||
| x13 | + | x23 | + | x33 | = | |||||||||||||
| x14 | + | x24 | + | x34 | = | |||||||||||||
| xij 0 for all i, j | ||||||||||||||||||
| Units | Cost, $ | |
| Clifton Springs - D1 | ||
| Clifton Springs - D2 | ||
| Clifton Springs - D3 | ||
| Clifton Springs - D4 | ||
| Danville - D1 | ||
| Danville - D2 | ||
| Danville - D3 | ||
| Danville - D4 | ||
| Dummy - D1 | ||
| Dummy - D2 | ||
| Dummy - D3 | ||
| Dummy - D4 | ||
| Total Cost |
___________ demand has a shortfall of ________.
__________ demand of __________ is not satisfied.
(0) (ii) 128 6000 D1 136 DI 6000 0 128 136 CS 15000 15000 C.S. 128 136 160 120 15000 D2 D2 15000 0 0 0 9000 9000 D 12% 112 9000 D3 0 D3 9000 136 Dummy 12000 160 12000 Dummy 132 132 6000 D4 D4 6000 (IV) 136 6000 D1 6000 D1 128 128 136 15000 C.S C.S 15000 128 120 160 136 15000 D2 D2 15000 136 0 120 9000 D 112 9000 112 152 128 D3 9000 D3 9000 0 0 12000 Dummy Dummy 12000 152/ 160 0 D4 6000 6000 D4Step by Step Solution
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