1. Linear Programming II

2. George Dantzig 1947 Narendra Karmarkar Pioneers of LP

3. Computer Solution of the LP Problem Simplex Method A step-by-step mathematical procedure for finding the optimal solution to a LP problem. This procedure moves from corner point to corner point of the feasible solution space. Most practical applications of LP use computer programs to perform the calculations and to obtain the optimal solution. The output of these programs usually includes the optimal solution to the primal problem as well as the optimal values of the dual variables.

4. Cost minimization: Silverado Mining Company Min C =50X 1 + 40X 2 + 0S 1 + 0S 2 + 0S 3 s.t..75X 1 +.25X 2 – S 1 = 36(High-Grade Ore).25X 1 +.25X 2 – S 2 = 24(Medium-Grade Ore).50X 1 + 1.50X 2 – S 3 = 72(Low-Grade Ore) X 1, X 2, S 1, S 2, S 3 ≥ 0 Primal Solution Dual Solution VariableValueVariableValue X124W120 X272W2 140Interpretation ? S348W3 0 Binding constraints on high-grade ore & low-grade ore Change in total cost (marginal cost) due to a one-unit (ton) change in the required output.

5. Computing Analysis of LP: Example A firm produces two types of hot tubs – Aqua Spa (A) and Hydro Lux (H). Their unit profit are P A =$350 and P H =$300, and inputs are pump, tubing, and labor. The resource requirements for the two tubs are: What are the optimal input and output mix and the optimal profit? InputsUnitsInput required to product one unit of output Resource Limit AH PumpNumbers11200 TubingFeet12162880 LaborHours961566

6. Seven Steps to Solve LP problem using Solver Step 1: Check whether you have Solver on the Tools menu. If not, you have to add-in Solver on the Tools menu following the instruction below: The Solver Add-in is an Excel add-in program that is available when you install Microsoft Office or Excel. To use it in Excel, however, you need to load it first.  Click Add-Ins On the Tools menu.  Select the check box next to Solver Add-in in Add-Ins available box, click OK. If Solver Add-in is not listed, click Browse to locate it.  If you see a message that tells you the Solver Add-in is not currently installed on your computer, click Yes to install it.  Click Tools on the menu bar. When you load the Solver Add-in, the Solver command is added to the Tools menu.

7. Seven Steps to Solve LP problem using Solver Step 2: Set up the profit maximization problem in Excel Formulas in Excel Cells. B3:C3Optimal value of output (A, H) that Excel will find for you D3Target Cell that computes profit=SUMPRODUCT(B3:C3,B2:C2) D4# of pumps used=SUMPRODUCT(B3:C3,B4:C4) D5Feet of tubing used=SUMPRODUCT(B3:C3,B5:C5) D6Hours of labor used=SUMPRODUCT(B3:C3,B6:C6) Note: SUMPRODUCT

8. Seven Steps to Solve LP problem using Solver Step 3: Click Solver on the Tool Menu. A window will pop up for you to define the following: (a) Enter a cell reference or name for the target cell. (b) Choose Max for max. problems or Min for min. problem. (c) Enter a name or reference for each adjustable cell in By Changing Cells. Separating nonadjacent references with commas. You can specify up to 200 adjustable cells. (d) Enter constraints in the Subject to the Constraints box.

9. Seven Steps to Solve LP problem using Solver Step 4: Check Assuming Linear Model and Nonnegative in Option Box; clink OK. You will get back the Solver Parameters :

10. Seven Steps to Solve LP problem using Solver Step 5: Click Solve

11. Seven Steps to Solve LP problem using Solver Step 6: To keep the solution values on the worksheet, click Keep Solver Solution in the Solver Results dialog box. To restore the original data, click Restore Original Values. Magic Excel will create a spreadsheet of Answer Report.

12. Output of H Output of A Max. profit Seven Steps to Solve LP problem using Solver Adjustable Cells CellName Original ValueFinal Value $B$ 3 Variable levels A0 $C$ 3 Variable levels H0 Step 7: Interpretation of Answer Report Target Cell (Max) CellName Original ValueFinal Value $D$ 3 Variable levels0 See handout Solving a LP problem using Solver in Excel for more information. 66000 80 120

13. Summary of LP Linear Programming Graphical analysis Computational analysis