## Optimization

Problem 1 (50 points)

For a certain civil engineering system, we have two types of resources: Labor, and equipment We refer to these resources by the index j. Likewise each resource can be assigned to two types of activities: Maintenance and operations. We refer to these activities by the index i. Furthermore, the variables xij represent the amount of hours used by the resource j (labor/equipment) performing activity i (maintenance/ operations).

Maintenance. Maintaining the system involves the use of equipment and labor.

The part of the labor that goes into maintenance has a cost of $2/hour; and the part of equipment use that goes into maintenance has a cost of $4/hour.

The total labor and equipment costs for maintenance should not exceed the maximum maintenance budget of $21 million

Operations. Operating the system involves the use of equipment and labor.

The part of labor that goes into operations has a cost of $5/hour; and the part of equipment use that goes into operations has a cost of $3/hour.

The total labor and equipment costs for operations should not exceed and maximum operations budget of $18 million

From past records of similar systems, it has been found that the performance of the system, Z, depends on the amount of effort put in maintenance and operations. Specifically, each hour of labor represents $3 million of profit for the company and each hour of equipment represents $2 million of profit for the company (it does not matter whether these hours where used for maintenance or operation). As a manager in the agency, you seek to maximize Z under the given budgetary constraints.

Write the complete optimizations problem mathematically (the objective function and all constraints).

Remember that xij should be non-negative.

Using software for solution:

Solve the problem using Excel SOLVER and attach a copy of your output, clearly highlighting the optimal solution using any highlighter.

Problem 2 (50 points)

For construction of a civil engineering facility, a contractor has found natural reserves of sand and gravel at Bloomingdale and Valley Springs where he may purchase such material. The unit cost including delivery from Bloomingdale and Valley Springs is $5 and $7, respectively. After the material is brought to the site it is mixed thoroughly and uniformly, and contract specification state that the mix should contain a minimum of 30% sand.

A total volume of 100,000 m3 of mixed material is needed for the project. The Bloomingdale Pit contains 25% sand and the Valley Springs Pit contains 50% sand.

As the new young construction engineer on the project, you are asked to determine how much material should be taken from each pit in order to minimize the cost of material.

(a) Define the decision variables

(b) Write the constraints in mathematical form

(c) Write the objective function in mathematical form

(d) Find the optimum solution using the graphical method (how much material should the contractor take from each pit in order to minimize the overall cost of the material?)

(e) If you had not been hired, the contractor would have used 60,000 m3 from Bloomingdale and 40,000 m3 from Valley Springs. How much did you save the company by giving them the experts advice as your answer to Question (d)?

(f) Solve part (d) using Excel Solver and attach a printout of the program output to your homework submission. Mark clearly the optimal solution in the output using a highlighter.

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