1. B-1 Operations Management Linear Programming Module B - Harder Formulations

2. B-2 You are creating an investment portfolio from 4 investment options: stocks, real estate, T-bills (Treasury-bills), and cash. Stocks have an annual rate of return of 12% and a risk measure of 5. Real estate has an annual rate of return of 10% and a risk measure of 8. T-bills have an annual rate of return of 5% and a risk measure of 1. Cash has an annual rate of return of 0% and a risk measure of 0. The average risk of the portfolio can not exceed 5. At least 15% of the portfolio must be in cash. Formulate an LP to maximize the annual rate of return of the portfolio. Harder Formulation Example

3. B-3 Investment Formulation Constraints:: Average risk  5 At least 15% in cash Investment ReturnRisk Stocks 0.12 5 8 Portfolio Real estate T-bills 0.10 0.05 1  5 Cash 0 0

4. B-4 Investment Formulation Investment ReturnRisk Stocks 0.12 5 8 Portfolio Real estate T-bills 0.10 0.05 1  5 Cash 0 0 Variables:: x i = % of portfolio in investment type i. i = 1 is Stocks; i = 2 is Real estate; i = 3 is T-bills; i=4 is is Cash

5. B-5 Investment Formulation : Maximize: 0.12x 1 + 0.10x 2 + 0.05x 3 x i = % of portfolio in investment type i. i = 1 is Stocks; i = 2 is Real estate; i = 3 is T-bills; i=4 is is Cash x 4  0.15 (Cash) 5x 1 + 8x 2 + x 3  5 (Risk) x 1 + x 2 + x 3 + x 4 = 1.0 (total = 100%) x 1, x 2, x 3, x 4  0 Optimal Solution is x 1 = 0.85, x 2 = 0, x 3 = 0 and x 4 = 0.15

6. B-6 Investment Formulation x i = % of portfolio in investment type i. Maximize: 0.12x 1 + 0.10x 2 + 0.05x 3 x 4  0.15 (Cash) 5x 1 + 8x 2 + x 3  5 (Risk) x 1, x 2, x 3, x 4  0 Without the total = 100% constraint, the optimal solution is: x 1 = 0, x 2 = 0, x 3 = 5 and x 4 = 0.15 This means invest 500% in T-bills and get a 25% return!!

7. B-7 A business operates 24 hours a day and employees work 8 hour shifts. Shifts may begin at midnight, 4 am, 8 am, noon, 4 pm or 8 pm. The number of employees needed in each 4 hour period of the day to serve demand is in the table below. Formulate an LP to minimize the number of employees to satisfy the demand. Another Formulation Example Midnight - 4 am 4 am - 8 am 3 6 8 am - noon 13 Noon - 4 pm 15 4 pm - 8 pm 12 8 pm - midnight 9

8. B-8 Another Formulation Example 12-4 am8-noon12-4 pm4-8 pm8 pm-124-8 am 313151296 x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x6x6 x i = number of employees who start an 8 hour shift at time i i = 1 is midnight, i = 2 is 4 am, i = 3 is 8 am; i=4 is noon, i = 5 is 4 pm, i = 6 is 8 pm

9. B-9 Another Formulation Example Minimize: x 1 + x 2 + x 3 + x 4 + x 5 + x 6 x 1 + x 6  3 x 1, x 2, x 3, x 4, x 5, x 6  0 x i = Number of employees who start an 8 hour shift at time i. i = 1 is midnight, i = 2 is 4 am, i = 3 is 8 am; i=4 is noon, i = 5 is 4 pm, i = 6 is 8 pm x 1 + x 2  6 x 2 + x 3  13 x 3 + x 4  15 x 4 + x 5  12 x 5 + x 6  9

10. B-10 Another Formulation Example Minimize: x 1 + x 2 + x 3 + x 4 + x 5 + x 6 x 1 + x 6  3 x 1, x 2, x 3, x 4, x 5, x 6  0 x i = Number of employees who start an 8 hour shift at time i. x 1 + x 2  6 x 2 + x 3  13 x 3 + x 4  15 x 4 + x 5  12 x 5 + x 6  9 Optimal solution is 30 employees: x 1 = 3, x 2 = 3, x 3 = 10, x 4 = 5, x 5 = 9, x 6 = 0