1 / 11

Operations Management Linear Programming Module B - New Formulations

Operations Management Linear Programming Module B - New Formulations. New Formulation #1.

summer
Télécharger la présentation

Operations Management Linear Programming Module B - New Formulations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Operations ManagementLinear ProgrammingModule B - New Formulations

  2. New Formulation #1 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.

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

  4. Variables:: xi = % 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 New Formulation #1 Return Risk Investment 0.12 Stocks 5 0.10 8 Real estate 1 T-bills 0.05 0 Cash 0  5 Portfolio

  5. x4  0.15 (Cash) 5x1 + 8x2 + x3  5 (Risk) x1 + x2 + x3 + x4 = 1.0(Total = 100%) x1, x2, x3 , x40 : Maximize: 0.12x1 + 0.10x2 + 0.05x3 New Formulation #1 xi = % 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 Optimal Solution is x1 = 0.85, x2 = 0, x3 = 0 and x4 = 0.15

  6. New Formulation #1 xi = % of portfolio in investment type i. Maximize: 0.12x1 + 0.10x2 + 0.05x3 x4  0.15 (Cash) 5x1 + 8x2 + x3  5 (Risk) x1, x2, x3 , x40 Without the total = 100% constraint, the optimal solution is: x1 = 0, x2 = 0, x3 = 5 and x4 = 0.15 This means invest 500% in T-bills and get a 25% return!!

  7. Midnight - 4 am 4 am - 8 am 9 3 New Formulation #2 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. 8 am - noon Noon - 4 pm 4 pm - 8 pm 8 pm - midnight 15 12 6 13

  8. Midnight - 4 am 4 am - 8 am 9 3 Heuristic Solution 8 am - noon Noon - 4 pm 4 pm - 8 pm 8 pm - midnight 15 12 6 13 Start 9 at midnight, then need 0 to start at 4 am, then need 6 to start at 8 am, then need 7 to start at noon, then need 8 to start at 4 pm, and 4 to start at 8 pm. But those 4 also work from midnight to 4 am! Total = 34

  9. 12-4 am 4-8 am 8-noon 12-4 pm 4-8 pm 8 pm-12 9 3 6 13 15 12 x1 x2 x3 x4 x5 x6 x6 New Formulation #2 xi = 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

  10. New Formulation #2 xi = 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 Minimize: x1 + x2 + x3 + x4 + x5 + x6 x1 + x6  9 x1 + x2  3 x2 + x3  6 x3 + x4  13 x4 + x5  15 x5 + x6  12 x1, x2, x3 , x4 , x5 , x6 0

  11. New Formulation #2 xi = Number of employees who start an 8 hour shift at time i. Minimize: x1 + x2 + x3 + x4 + x5 + x6 x1 + x6  9 Optimal solution = 30 employees x1 = 0, x2 = 3, x3 = 3, x4 = 10, x5 = 5, x6 = 9 x1 = 3, x2 = 0, x3 = 6, x4 = 7, x5 = 8, x6 = 6 x1 + x2  3 x2 + x3  6 x3 + x4  13 x4 + x5  15 x5 + x6  12 x1, x2, x3 , x4 , x5 , x6 0

More Related