Linear Programming II

George Dantzig 1947 Narendra Karmarkar

Pioneers of LP

Computer Solution of the LP ProblemSimplex MethodA 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.

Cost minimization: Silverado Mining CompanyMin 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 SolutionVariable Value Variable Value

X1 24 W1 20X2 72 W2 140 Interpretation ?S3 48 W3 0

Binding constraints on high-grade ore & low-grade oreChange in total cost (marginal cost) due to a one-unit (ton) change in the required output.

Computing Analysis of LP: Computing Analysis of LP: ExampleExample

• 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:

• What are the optimal input and output mix and the optimal profit?

Inputs Units Input required to product one unit of output

Resource Limit

A H

Pump Numbers 1 1 200Tubing Feet 12 16 2880Labor Hours 9 6 1566

Seven Steps to Solve LP Seven Steps to Solve LP problem using Solverproblem 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 Seven Steps to Solve LP problem using Solverproblem using Solver

Step 2: Set up the profit maximization problem in ExcelFormulas 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 Seven Steps to Solve LP problem using Solverproblem 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 Seven Steps to Solve LP problem using Solverproblem 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 Seven Steps to Solve LP problem using Solverproblem using Solver

Step 5: Click Solve

Seven Steps to Solve LP Seven Steps to Solve LP problem using Solverproblem 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.

Output of H

Output of A

Max. profit

Seven Steps to Solve LP Seven Steps to Solve LP problem using Solverproblem using Solver

Adjustable Cells

Cell NameOriginal

Value Final Value$B$

3Variable

levels A 0

$C$3

Variable levels H 0

Step 7: Interpretation of Answer Report

Target Cell (Max)

Cell NameOriginal

ValueFinal Value

$D$3

Variable levels 0

See handout Solving a LP problem using Solver in Excel for more information.

66000

80

120

Summary of LPSummary of LP• Linear Programming

•Graphical analysis•Computational analysis