Managerial Economics 10e - Hirschey by rzv19772

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```									Linear Programming
II
Pioneers of LP
George Dantzig 1947   Narendra Karmarkar
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.
Min C =    50X1 + 40X2 + 0S1 + 0S2 + 0S3
s.t.
.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

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
profit?
Seven Steps to Solve LP problem
using Solver
Step 1: Check whether you have 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
 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.
Seven Steps to Solve LP problem using
Solver
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)
Note: SUMPRODUCT
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.

Magic
Excel will create a
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
Cell       Name        Value       Final Value

\$D\$      Variable
3        levels               0         66000 Max. profit
Original
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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