LINEAR PROGRAMMING INTRODUCTION

Chapter 7 of Quantitative Methods for Business by Anderson, Sweeney and Williams

Read sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7 and appendix 7.1

One of the most widely used M.S. tools Used to solve a wide variety of problems

• Human Resource Scheduling• Routing of Delivery Vehicles• Selection of Advertising Media• Planning Product Levels in Manufacturing• Blending in Oil Refineries• Selection of Investment Opportunities • and many, many more

LINEAR PROGRAMMING

Most widely used solution technique: The Simplex Method• Developed by George Dantzig in the 1940’s• First used to solve military operations problems

during WWII• Coincided with the development of the first widely

used computers

LINEAR PROGRAMMING

Objective is to minimize or maximize some linear function of the decision variables

Constraints are linear equations or inequalities

LINEAR PROGRAMMING MODELS

Before the next class, you should complete the following homework problems in chapter 7

#1

Homework status

Extremely large models easily solvable using existing computer programs

Consider using L.P. Models whenever you are faced with allocating scarce resources among competing activities

LINEAR PROGRAMMING MODELS

3 Steps to Formulate A Linear Programming Problem:

1. DEFINE DECISION VARIABLES2. FORMULATE OBJECTIVE3. IDENTIFY CONSTRAINTS

LINEAR PROGRAMMING--Introduction to Formulation

A glass company is considering using its excess capacity to manufacture two new products: • wood frame windows, which earn $3 profit each,

and• aluminum frame doors which earn $5 profit,

each.

Glass Manufacturer Example

There are • 4 man-hours / day available in plant 1 (Plant 1 is

used for wood frames)• 12 man-hours / day available in plant 2 (Plant 2

for aluminum frames)• 18 man-hours / day available in plant 3 (Plant 3

for glass and assembly)

Glass Manufacturer Example

Each unit of the new products would require the following plant resources (in man-hours):

WINDOWS DOORSPLANT 1 1 --PLANT 2 -- 2PLANT 3 3 2

Glass Manufacturer Example

Formulate the problem:

1. DEFINE DECISION VARIABLES

Glass Manufacturer Example

2. OBJECTIVE FUNCTION

Glass Manufacturer Example

3. IDENTIFY CONSTRAINTS

Glass Manufacturer Example

1. Associate the horizontal axis with one decision variable, and associate the vertical axis with the other.

2. Draw the constraints.3. Identify the feasible region as the area

where all the constraints intersect (are “satisfied”).

GRAPHICAL SOLUTION TECHNIQUE

4. Find the optimal solution (the feasible point which gives the best value of the objective).

a. Graph the objective function line for any 2 arbitrary values of Z

b. Identify the improving direction for Zc. Move a pen parallel to the Z lines in the

improving direction as far as possibled. Last feasible point pen touches is the optimal

solution

GRAPHICAL SOLUTION TECHNIQUE

How to find the coordinates of the optimal point?

1. Identify the (2) constraints which go through the point

2. Solve those (2) constraint equations simultaneously

GRAPHICAL SOLUTION TECHNIQUE

Before the next class, you should complete the following homework problems in chapter 7

#3, #7, #8, #11 (by hand), #24 (by hand), and #31 (by hand)

Homework status

◦ A unique (extreme point) optimal solution◦ Alternate optimal solutions◦ An unbounded solution◦ An infeasible problem

POSSIBLE ANSWERS

LINEARITY OF OBJECTIVE AND CONSTRAINTS• i.e. they can be written in the form:

C1X1+C2X2+…+CnXn RHS

where the Xis are the decision variables, the Cis are (constant) coefficients, and RHS is the (constant) RHS

DIVISIBILITY OF DECISION VARIABLES• i.e. they may take fractional values

IMPORTANT ASSUMPTIONS OF LINEAR PROGRAMMING PROBLEMS

1. Label one row for each of these: Objective function coefficients Values of the decision variables Each constraint.

2. Label one column for each of these: Each decision variable Total value of the left hand side Right hand side value.

Using Excel to solve an LP

3. Key in coefficients from constraints (blue border)

4. Key in right-hand-side values for constraints (blue border)

5. Key in objective function coefficients (blue border)

Using Excel to solve an LP

6. Designate cells for decision values (red border)7. Designate cell for objective function value

(double black border)

Using Excel to solve an LP

8. Specify formula for left-hand-sides of constraints, using the SUMPRODUCT function

9. Specify formula of Objective Function Value cell, using the SUMPRODUCT function

Using Excel to solve an LP

• (Key in any values for decision variables• Try different values, to see what happens to

left-hand-side values and objective value)10. Click Data Analysis Solver11. Specify “Target Cell” as the objective function

value (cell with double black border)

Using Excel to solve an LP

13. Click MAX (or MIN)14. Specify “changing cells” as decision

variable value cells (cells with red border)15. Click “Add” to add constraints, one (or

several) at a time by selecting each:• Total Left-hand-side cell(s)• Appropriate inequality/equality sign• Right-hand-side cell(s)

Using Excel to solve an LP

16. Be sure the box next to “Make Unconstrained Variables Non-negative” is checked.

17. By “Select a Solving Method”, choose “Simplex LP”

18. Click SOLVE19. Study the solution to be sure it is

reasonable (including feasible) and modify/correct the model and resolve if it is not.

Using Excel to solve an LP

See Solver Hints file (on Blackboard, under Module2 Linear Programming) if you need help.

Homework help

Before the next class, you should complete the following homework problems in chapter 7

#11 (on Excel), #24 (on Excel), #31 (on Excel)

Homework status