4. linear programming using excel solver

ADVANCED OPERATIONS

RESEARCH

By: -Hakeem–Ur–Rehman

IQTM–PU 1

RA OLINEAR PROGRAMMING USING EXCEL SOLVER

LINEAR PROGRAMMING USING EXCEL SOLVER

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Excel LingoOn the toolbar at the bottom of the screen, click on: Start All Programs Microsoft Office Microsoft Office Excel 2007

Spreadsheet: A two-dimensional array of rectangles. Cell: Each rectangle in excel (Four types of information can be typed into a cell:

Number, Fraction, Function, and Text.) It is identified by its column and rowlocation on the spreadsheet, which are designated by letter and numbers,respectively (i.e. cell A1).

SUMPRODUCT: A function that first multiplies the numbers in n consecutive cells(i.e. A1 through E1) by the numbers in another set of n consecutive cells (i.e. A5through E5), respectively, then takes the sum of the n number of products (i.e.A1*A5 + B1*B5 + C1*C5 + D1*D5 + E1*E5), and finally deposits that sum in thecell you have selected (i.e. F1).

Cell reference: Lets you repeat patterns of information between cells, whichoccurs a selected cell refers to information typed in another cell. Absolute reference: A cell that always refers to the originally referred cell; if

the location of the selected cell changes, the referred cell will not change. Itincludes a “$” sign before the cell’s column (i.e. $A1), row (i.e. A$1), or both(i.e. $A$1).

Relative reference: A cell that initially refers to the originally selected cell; ifthe location of the selected cell changes, the referred cell will change and thelocation of the new referred cell will reflect the location change of the selectedcell. It omits the “$” sign (i.e. A1).


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How to activate Solver:


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Solver can find a solution to:

Systems of equations

Inequalities

Optimization problems Linear programs*** Integer programs Nonlinear programs

EXAMPLE

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XYZ manufacturing company has a division that produces two models ofgrates, model–A and model–B. To produce each model–A grate requires ‘3’ g.of cast iron and ‘6’ minutes of labor. To produce each model–B grate requires‘4’ g. of cast iron and ‘3’ minutes of labor. The profit for each model–A grateis Rs.2 and the profit for each model–B grate is Rs.1.50. One thousand g. ofcast iron and 20 hours of labor are available for grate production each day.Because of an excess inventory of model–A grates, Company’s manager hasdecided to limit the production of model–A grates to no more than 180 gratesper day.Solve the given LP problem and perform sensitivity analysis.

LP MODEL: Let X1 and X2 be the number of model–A and model–B gratesrespectively.The complete LP model is as follow:

Maximum: Z = 2X1 + 1.5X2 2X1 + (3/2)X2

Subject to:3X1 + 4X2 ≤ 1000 (Cast Iron Constraint)6X1 + 3X2 ≤ 1200 (Labor Hour Constraint)X1 ≤ 180 (Production limit of Model-A Constraint)

X1, X2 ≥ 0


STEP – I: ENTER THE DATA & FUNCTION

Cell I8: Enter:=SUMPRODUCT($G$6:$H$6,G8:H8)

Drag to cells G11:H11


STEP – II: RECORD THE SOLVER PARAMETERS

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The “Solver Parameters” dialog box:WINDOWS

“Set Target Cell” window:Identifies the cell that Solver will use to record the optimal z-value for the problem.

“By Changing Cells” window: Identifies the cells that Solver will use to record the optimal solution for the decision variables.

“Subject to the Constraints” window:Identifies the non-negativity constraints and the constraints given by the problem.

Buttons

“Options” button: Identifies the type of optimization problem; remember to check off the “Assume Linear Model” option.

“Add” button: Used to insert the constraints; identified constraints are displayed in the “Subject to the Constraints” window.

“Solve” button: Used to determine the optimal value for the objective z and the decision variables.


STEP – II: RECORD THE SOLVER PARAMETERS (Cont…)



With the CURSOR in the“Set Target Cell Box”: Clickon Cell “I8”

SET TARGET CELL:



LEAVE THE BUTTON FOR Max

HIGHLIGHTED

EQUAL TO:



WITH THE CURSOR IN THE“BY CHANGING CELLSBOX”: HIGHLIGHT CELLS“G6” & “H6”

BY CHANGINGCELLS:



SUBJECT TO THE CONSTRAINTS:

In the “SolverParameters” dialogbox, click on the “Add”button.

Fill in the “CellReference” and“Constraint” windowsby clicking on thechanging cells and thefunction cells.

Click on the “OK”button after addingeach constraint.



With the cursor in the cell reference box: highlight cells “I9 throughI11”. Leave the direction as “≤”. With the cursor in the constraintbox: : highlight cells “K9 through K11”.

If more constraints were to be added, click “Add” and follow thesame procedure.

SUBJECT TO THE CONSTRAINTS (Cont…):



OPTIONS:



SOLVE:



REPORT:


Analyzing the Excel Spreadsheet


THE ANSWER REPROT


THE SENSITIVITY REPROT

Range of Optimality Changing the profit coefficient of the objective function

Will the original optimal solution still be optimal? Range of Optimality?

Profit coefficient for X1

2, range of optimality (2 + 1, 2 – 0.875) = (3, 1.125) Profit coefficient for X2

1.5, range of optimality (1.5 + 1.167, 1.5 – 0.5) = (2.667, 1)



Changing the RHS – CAST IRONS Binding Constraints

3X1 + 4X2 ≤ 1000 (Cast Irons Constraint) 3(120) + 4 (160) = 1000

Suppose we increase one gram Cast Iron, what’s the impact onthe optimal profit?

The unit change in the objective function is the shadow priceof the resource.

Shadow price of Cast Iron Gram = 0.2 Range of Feasibility: (1000 + 600, 1000 – 300) = (1600, 700)



Changing the RHS – LABOUR HOUR Binding Constraints

6X1 + 3X2 ≤ 1200 (Labor Hours Const.) 6(120) + 3(160) = 1200

Suppose we increase one Labour hour, what’s the impact on theoptimal profit?

The unit change in the objective function is the shadow priceof the resource.

Shadow price of Labour Hour = 0.23333 Range of Feasibility: (1200 + 225, 1000 – 450) = (1425, 550)



Changing the RHS – LABOUR HOUR NON–Binding Constraints

X1 ≤ 180 (Model-A Production Cont.) Optimum: 120 + 0 = 120 (Model–A Grates)

We have 60 excessive Model–A Grates (slack) Increasing the Grates? Decreasing the Grates?

Shadow price of Model–A = 0 Range of Feasibility: (180 + ∞, 180 – 60) = (∞, 120)

EXAMPLE: PRODUCTION SCHEDULING

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Cool-bike Industries manufactures boys and girls bicycles in both 20-inch and 26-inch models. Each week itmust produce at least 200 girl models and 200 boy models. The following table gives the unit profit and thenumber of minutes required for production and assembly for each model.

X1 = Number of 20-inch girls bicycles produced this week; X2 = Number of 20-inch boys bicyclesproduced this week; X3 = Number of 26-inch girls bicycles produced this week; X4 = Number of 26-inchboys bicycles produced this week

MAX 27X1 + 32X2 + 38X3 + 51X4

S.T.X1 + X3 200 (Min girls models)X2 + X4 200 (Min boys models)

12X1 + 12X2 + 9X3 + 9X4 4800 (Production minutes)6X1 + 9X2 + 12X3 + 18X4 4800 (Assembly minutes)2X1 + 2X2 500 (20-inch tires)

2X3 + 2X4 800 (26-inch tires)All X's 0

Bicycle Unit Profit Production Minutes Assembly Minutes

20-inches girls $27 12 6

20-inches boys $32 12 9

26-inches girls $38 9 12

26-inches boys $51 9 18

The Production and assembly areas run two (eight-hour) shifts per day, five days per week. This week thereare 500 tires available for 20-inch models and 800 tires available for 26-inch models. Determine Cool-bike’soptimal schedule for the week. What profit will it realize for the week?

EXAMPLE: PRODUCTION SCHEDULING (Cont…)

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X1 = Number of 20-inch girls bicycles produced this week; X2 = Number of 20-inch boys bicyclesproduced this week; X3 = Number of 26-inch girls bicycles produced this week; X4 = Number of 26-inchboys bicycles produced this week

MAX 27X1 + 32X2 + 38X3 + 51X4





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MAX 27X1 + 32X2 + 38X3 + 51X4





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MAX 27X1 + 32X2 + 38X3 + 51X4




QUESTIONS

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Engineering

4. linear programming using excel solver