# Week 7: Nonlinear Optimization

The purpose of this assignment is to introduce how to find solutions to business problems using nonlinear optimization techniques.

Using specified data files, chapter example files, and templates from the “Topic 7 Student Data, Template, and Example Files” topic material, complete Chapter 14, Problems 69, 70 (ignore the SolverTable portion), 72 (ignore the SolverTable portion), and 94. Use Microsoft Excel’s Solver Add-In to complete these problems. For Problems 70, 72, and 94, run the Solver’s Answer and Sensitivity Reports. Interpret and summarize the key results.

Problem #69:

In the electricity pricing model, the demand functions have positive and negative coefficients of prices. The negative coefficients indicate that as the price of a product increases, demand for *that *product decreases. The positive coefficients indicate that as the price of a product increases, demand for the *other *product increases.

**a. **Increase the magnitudes of the negative coefficients from –0.013 and –0.015 to –0.018 and –0.023, and rerun Solver. Are the changes in the optimal solution intuitive? Explain.

**b. **Increase the magnitudes of the positive coefficients from 0.005 and 0.003 to 0.007 and 0.005, and rerun Solver. Are the changes in the optimal solution intuitive? Explain.

**c. **Make the changes in parts **a **and **b **simultaneously and rerun Solver. What happens now?

Problem #70:

In the electricity pricing model, we assumed that the capacity level is a decision variable. Assume now that capacity has already been set at 0.65 million of mWh . (Note that the cost of capacity is now a sunk cost, so it is irrelevant to the decision problem.) Change the model appropriately and run Solver. Then use SolverTable to see how sensitive the optimal solution is to the capacity level, letting it vary over some relevant range. Does it appear that the optimal prices will be set so that demand is always equal to capacity for at least one of the two periods of the day?

Problem #72:

Add a new stock, stock 4, to the portfolio optimization model. Assume that the estimated mean and standard deviation of return for stock 4 are 0.125 and 0.175, respectively. Also, assume the correlations between stock 4 and the original three stocks are 0.3, 0.5, and 0.8. Run Solver on the modified model, where the required expected portfolio return is again 0.12. Is stock 4 in the optimal portfolio? Then run SolverTable as in the example. Is stock 4 in any of the optimal portfolios on the efficient frontier?

Problem #94:

You have $50,000 to invest in three stocks. Let *R**i *be the random variable representing the annual return on $1 invested in stock *i*. For example, if *R**i *= 0.12, then $1 invested in stock *i *at the beginning of a year is worth $1.12 at the end of the year. The means are *E*(*R*1) = 0.14, *E*(*R*2) = 0.11, and *E*(*R*3) = 0.10. The variances are *V**ar R*1 = 0.20, *V**ar R*2 = 0.08, and *V**ar R*3 = 0.18. The correlations are *r*12 = 0.8, *r*13 = 0.7, and *r*23 = 0.9. Determine the minimum variance portfolio that attains an expected annual return of at least 0.12.

To receive full credit on the assignment, complete the following.

- Ensure that all Solver settings are defined through the use of the Solver dialog box.
- Ensure that Excel files include the associated cell functions and/or formulas if functions and/or formulas are used.
- Include a written response to all narrative questions presented in the problem by placing it in the associated Excel file.
- Include Answer and Sensitivity Reports interpretation and summary of key results.
- Place each problem in its own Excel file. Ensure that your first and last name are in your Excel file names.

APA style is not required, but solid academic writing is expected.

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