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Optimization Problems in Electricity Supply and Manufacturing

1. BC hydro supplies electricity to households in Vancouver, Coquitlam and Burnaby. The company buys electricity from two dams, the Capilano dam and the Coquitlam dam. Their demand forecasts for the winter 2021 are as follows:

Surrey 650 units, Burnaby 550 units and Vancouver 800 units. BC Hydro has contracts with the two dams to provide the necessary units as well. Capilano is to provide 950 units and Coquitlam is to provide 1050 units. Distribution costs (in thousand dollars) are as follows:

To |
|||

From |
Surrey |
Burnaby |
Vancouver |

Capilano |
16 |
12 |
8 |

Coquitlam |
7 |
10 |
13 |

a. Develop a network representation of the problem (diagram). (2 marks)

b. Determine the objective function and the constraints. (4 marks)

c. Find the optimal solution to this problem. (6 marks)

d. Recent increases in the population of Surrey might cause an increase in demand of their electricity in the winter by 150 units (800 units needed instead of 650). Considering the extra demand, how many units should be sent to each city from each of the dams? (6 marks)

2. Harkirat Manufacturing Company produces connecting rods for 3- and 5-cylinder automobile engines using the same production line. The cost required to set up the production line to produce the 3-cylinder connecting rods is $1750, and the cost required to set up the production line for the 5-cylinder connecting rods is $2250. Manufacturing costs are $11 for each 3-cylinder connecting rod and $14 for each 5-cylinder connecting rod. Harkirat makes the decision at the end of each week as to which product will be manufactured the following week. If there is a production changeover from one week to the next, the weekend is used to reconfigure the production line. Once the line has been set up, the weekly production capacities are 3500 5-cylinder connecting rods and 6800 3-cylinder connecting rods. Let

x_{3} represent the number of 3-cylinder connecting rods produced next week

x_{5} represent the number of 5-cylinder connecting rods produced next week

s_{3} = 1 if the production line is set up for the 3-cylinder connecting rods; 0 if otherwise

s_{5} = 1 if the production line is set up for the 5-cylinder connecting rods; 0 if otherwise

a. Using the decision variables x_{3} and s_{3}, write a constraint that limits next week’s production of the 3-cylinder connecting rods to either 0 or 6800 units. (2 mark)

b. Using the decision variables x_{5} and s_{5}, write a constraint that limits next week’s production of the 5-cylinder connecting rods to either 0 or 3500 units. (2 mark1)

c. Write the constraints that, taken together, limit the production of connecting rods for next week. (2 marks)

d. Write the objective function for minimizing the cost of production for next week. (2 marks)

3. Martha Manufacturing a product through three separate production lines “A”, “B” and “C”. Costs for producing a product through each line as well as maximum outputs of each line and fixed cost associated with setting up the line is shown below:

Production line |
Cost per unit |
Maximum production |
Fixed cost |

A |
1.5 |
28 |
150 |

B |
1.0 |
32 |
170 |

C |
1.25 |
25 |
160 |

The company has received an order of 60 units to be produced. Model the problem for optimum number of products to be made from each production line.

a. Formulate a linear programming model for maximizing total profit contribution. (7 marks)

b. Solve the linear program formulated in part (a). How much of each product should be produced, and what is the projected total profit contribution? (7 marks)