Managerial Economics 10e - Hirschey by rzv19772

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									Linear Programming
 Pioneers of LP
George Dantzig 1947   Narendra Karmarkar
Computer Solution of the LP
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
The output of these programs usually includes the
optimal solution to the primal problem as well as the
optimal values of the dual variables.
Cost minimization: Silverado Mining Company
  Min C =    50X1 + 40X2 + 0S1 + 0S2 + 0S3
             .75X1 + .25X2 – S1 = 36             (High-Grade Ore)
             .25X1 + .25X2 – S2 = 24             (Medium-Grade Ore)
             .50X1 + 1.50X2 – S3 = 72      (Low-Grade Ore)
                          X1, X2, S1, S2, S3 ≥ 0
  Primal Solution           Dual Solution
  Variable Value          Variable       Value
      X1       24             W1           20
      X2       72             W2          140 Interpretation ?
      S3       48             W3             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.
 Computing Analysis of LP: Example
• A firm produces two types of hot tubs – Aqua Spa (A) and
  Hydro Lux (H). Their unit profit are PA=$350 and PH=$300,
  and inputs are pump, tubing, and labor.
• The resource requirements for the two tubs are:

      Inputs    Units    Input required to product   Resource
                             one unit of output        Limit
                             A             H
      Pump     Numbers       1             1           200
      Tubing    Feet         12            16         2880
      Labor     Hours        9             6          1566

• What are the optimal input and output mix and the optimal
  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.
Seven Steps to Solve LP problem using
  Step 2: Set up the profit maximization problem in Excel
          Formulas in Excel Cells.

  B3:C3   Optimal value of output (A, H)
            that Excel will find for you
  D3      Target Cell that computes profit =SUMPRODUCT(B3:C3,B2:C2)
  D4      # of pumps used                  =SUMPRODUCT(B3:C3,B4:C4)
  D5      Feet of tubing used              =SUMPRODUCT(B3:C3,B5:C5)
  D6      Hours of labor used              =SUMPRODUCT(B3:C3,B6:C6)
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.
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 :
Seven Steps to Solve LP problem
         using Solver
          Step 5: Click Solve
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.

                        Excel will create a
                         spreadsheet of
                        Answer Report.
Seven Steps to Solve LP problem using Solver
    Step 7: Interpretation of Answer Report
                                                                See handout Solving
                                                                a LP problem using
  Target Cell (Max)
                                                                Solver in Excel for
                                 Original                       more information.
            Cell       Name        Value       Final Value

           $D$      Variable
             3        levels               0         66000 Max. profit
 Adjustable Cells
          Cell        Name        Value        Final Value
          $B$      Variable
            3        levels A              0           80    Output of A
          $C$      Variable
            3        levels H              0
                                                      120    Output of H
                Summary of LP
• Linear Programming
  •Graphical analysis
  •Computational analysis

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