Problem 1 (50 points)For a certain civil engineering system, we have two types ofresources: Labor, and equipment We refer to these resources by the indexj. Likewise each resource can be assigned to two types ofactivities: Maintenance and operations. We refer to these activities bythe 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 acost of $4/hour. The total labor and equipment costs for maintenance should not exceed the maximum maintenance budget of $21 millionOperations. 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 millionFrom 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 thecompany and each hour of equipment represents $2 million of profit forthe company (it does not matter whether these hours where used formaintenance or operation). As a manager in the agency, you seek tomaximize 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 ofyour output, clearly highlighting the optimal solution using anyhighlighter.Problem 2 (50 points)For construction of a civil engineering facility, a contractor hasfound natural reserves of sand and gravel at Bloomingdale and ValleySprings 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 mixedthoroughly and uniformly, and contract specification state that the mixshould contain a minimum of 30% sand.A total volume of 100,000 m3 of mixed material is neededfor the project. The Bloomingdale Pit contains 25% sand and the ValleySprings Pit contains 50% sand.As the new young construction engineer on the project, you are askedto determine how much material should be taken from each pit in order tominimize 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 muchmaterial should the contractor take from each pit in order to minimizethe 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 expert’s advice as your answer to Question (d)?(f) Solve part (d) using Excel Solver and attach a printout of theprogram output to your homework submission. Mark clearly the optimalsolution in the output using a highlighter.
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