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Industrial Engineering

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Dec 6, 2023

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Introduction In this assignment, we were asked to solve a problem in class regarding binary optimization and present it to our classmates at the end of the class. The objective of the problem was to find out of the 8 projects (Appendix A - Slide 2) given by the professor in the question, which ones should be selected to maximize the division’s revenue. The question of the assignment was: “A software support division of Blain Information Services has eight projects that can be performed. Each project requires different amounts of development time and testing time. In the coming planning period 1,150 hrs of development and 900 hrs of testing time are available, based on the skill mix of the staff. The internal transfer price (revenue to the IT division) and the times required for each project are shown in the table below. Which projects should be selected to maximize the division’s revenue?” The goal of our group in this assignment was to find which project we should use to maximize the revenue of the division. We received a table of data given in the problem (Appendix A -Slide 2). Also, we had to find the total transfer price, the slack of the development, and the slack in testing time. In Appendix A - Slide 2, we can see a table of data that was used to find out the answer to the problem. Explaining that data, the first column was the Project number. The second column, represented the development time. The third column, however, represented the test time. Lastly, the fourth column was used to give the transfer price. All these data combined was used to get to our answer, which will be explained in further sections of this report. Case Problem As stated before, our job is to maximize the division's revenue. There are 8 projects to choose from, from which the ones that will allow the company to maximize revenues will be selected. The first step is to establish a Mathematical model, using a linear optimization model. Linear optimization model, it is a method of applying the best solution in the decision-making process (e.g. profit maximization, cost minimization). Then find the decision variables, which in this case are projects 1 through 8. The next step is to identify the objective variables, which we can find by multiplying transfer price and decision variables, by doing that we get $23,520.00*x1 + $72,912.00*x2 + $62,054.00*x3 + $32,340.00*x4 + $70,560.00*x5 + $57,232.00*x6 +

$19,184.00*x7 + $32,340.00*x8. Next, we need to set the constraints, i.e. the sum of development time must be equal to or less than 1,150h, the formula is 80*x1 + 248*x2 + 41*x3 +10*x4 + 240*x5 + 195*x6 + 269*x7 + 110*x8 <= 1,150h, the sum of testing time must be equal to or less than 900h, the formula is 67*x1 + 208*x2 + 180*x3 + 92*x4 + 202*x5 +1 64*x6 + 226*x7 + 92*x8 <=900h and our variables must be binary. When we have everything set, we can go to excel, which, thanks to the solver function, will help us which projects should be selected to maximize revenues. Mathematical Model We used a binary optimization model to develop our mathematical model. Our objective function consists of the sum of the proposed projects times their respective transfer prices and its mathematical description is 23520*1 + 72912*2 + 62054*3 + 32340*4 + 70560*5 + 57232*6 + 19184*7 + 32340*8 (where 23520, 72912, 62054, 32340, 70560, 57232, 19184, 32340 are the transfer prices of each project and 1, 2, 3, 4, 5, 6, 7, 8 are the proposed projects). Our model has three constraints. The binary constraint makes the changing variable cells yield only 0 or 1 as result. The constraint on hours of development is described by the equation 80*1 + 248*2 + 41*3 + 10*4 + 240*5 + 195*6 + 269*7 + 110*8 <= 1150 (where 80, 248, 41, 10, 240, 195, 269, 110 are the hours of development required by each project, 1, 2, 3, 4, 5, 6, 7, 8 are the proposed projects, and 1150 the constraint on hours of development). The constrain on hours of testing is described by the equation 67*1 + 208*2 + 180*3 + 92*4 + 202*5 + 164*6 + 229*7 + 92*8 <= 900 (where 67, 208, 180, 92, 202, 164, 226, 92 are the hours of testing required by each project, 1, 2, 3, 4, 5, 6, 7, 8 are the proposed projects, and 900 the constraint on hours of testing). Statistical & Business Insights After setting our model, we used the application Solve to get our results and the answer report. We set the objective cell maximized, selected the changing variable cells, plugged in the binary constraint on the changing variable cells, the constraint on hours of development, the constraint on hours of testing, and then clicked on Solve. Our result was a Total Transfer Price (Total Revenue) of $295,098 by selecting Projects 2, 3, 4, 5, and 6 out of the eight projects proposed. Project 2 yielded $72,912 in revenue, Project 3 yielded $62,054 in revenue, Project 4 yielded $32,340, Project 5 yielded $70,560 in revenue, and Project 6 yielded $57,232 in revenue. The sum of the revenues yielded by the five selected projects equals the Total Transfer Price of $2,905,098. The answer report revealed that to reach the optimum level of Total Transfer Price our model did not have to employ either all the hours of development or all the hours of testing established by the constraints. The answer report shows a slack of 416 hours in time of development and a slack of 54 hours in time of testing. Our advice to the software support division of Blain Information Services is to set the number of hours spent on development to 734 (1150 - 416) and set the

number of hours spent on testing to 846 (900 - 54). This way the firm can eliminate the costs of idle resources associated with the slacks on development time and testing time. Additional information Linear optimization is applying the best solution in the decision-making process (e.g., profit maximization, cost minimization). It can be used to solve problems in a variety of businesses and industries. In our example, we used linear optimization to find the project or combination of projects that would maximize the division's revenue. Other businesses might use linear optimization to minimize costs, maximize profits, or solve other problems. A linear optimization is a valuable tool that can be used to solve many different types of business problems. The model can help a company optimize resources in two ways: finding the project or combination of projects that will yield the greatest return on investment and finding the optimal mix of development and testing hours to complete the selected project or combination of projects. To find the project or combination of projects that will yield the greatest return on investment, the company would use the linear optimization model to identify the decision variables (the projects) and the objective variable (the total transfer price). The company would then set the constraints, including the binary constraint that the decision variables must be either 0 or 1, and solve the model ( Baki et al., pg. 64) . The result would be the project or combination of projects that would yield the greatest return on investment. The company would use the answer report from the linear optimization model to find the optimal mix of development and testing hours to complete the selected project or combination of projects. The answer report would reveal the slack in development and testing hours, which the company could then use to set the hours for each activity. This would ensure that the company can avoid incurring the costs of idle resources associated with the slacks. The objective of our model was to find out which of the 8 projects the professor gave in the question, which ones should be selected to maximize the division's revenue. In other words, our goal was to find which projects would be the most profitable for the company. To do this, we used a binary optimization model. This model allowed us to set constraints on each project's development and testing time. We also had to identify the decision variables and the projects themselves. Once we had everything set, we used the application Solve to get our results. Our results showed that the most profitable projects for the company were Projects 2, 3, 4, 5, and 6. These projects yielded a Total Transfer Price (Total Revenue) of $295,098. The answer report also revealed that to reach the optimum level of Total Transfer Price, our model could employ some of the hours of development or all the hours of testing established by the constraints. This means the company could eliminate the costs of idle resources associated with the slacks on development and testing time. The answer report from the optimization model can help a company optimize its capital resources. The report shows the total transfer price that can be achieved from selecting certain

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