Table of Contents
Chapter 2 (Linear Programming: Basic Concepts)
The Wyndor Glass Company Product Mix Problem (Section 2.1)
Formulating the Wyndor Problem on a Spreadsheet (Section 2.2)
The Algebraic Model for Wyndor (Section 2.3)
The Graphical Method Applied to the Wyndor Problem (Section 2.4)
Using the Excel Solver with the Wyndor Problem (Section 2.5)
A Minimization Example—The Profit & Gambit Co. (Section 2.6)
2.2
2.3–2.7
2.8
2.9–2.19
2.20–2.25
2.26–2.31
Introduction to Linear Programming (UW Lecture)
2.32–2.47
These slides are based upon a lecture introducing the basic concepts of linear programming and the
Solver to first-year MBA students at the University of Washington (as taught by one of the
authors). The lecture is largely based upon a production problem using lego building blocks.
The Graphical Method and Properties of LP Solutions (UW Lecture)
2.48–2.56
These slides are based upon a lecture introducing the graphical method and other concepts about
linear programming solutions to first-year MBA students at the University of Washington (as
taught by one of the authors).
McGraw-Hill/Irwin
2.1
© The McGraw-Hill Companies, Inc., 2008
Wyndor Glass Co. Product Mix Problem
•
Wyndor has developed the following new products:
–
–
•
An 8-foot glass door with aluminum framing.
A 4-foot by 6-foot double-hung, wood-framed window.
The company has three plants
–
–
–
Plant 1 produces aluminum frames and hardware.
Plant 2 produces wood frames.
Plant 3 produces glass and assembles the windows and doors.
Questions:
1. Should they go ahead with launching these two new products?
2. If so, what should be the product mix?
McGraw-Hill/Irwin
2.2
© The McGraw-Hill Companies, Inc., 2008
Developing a Spreadsheet Model
•
Step #1: Data Cells
– Enter all of the data for the problem on the spreadsheet.
– Make consistent use of rows and columns.
– It is a good idea to color code these “data cells” (e.g., light blue).
McGraw-Hill/Irwin
2.3
© The McGraw-Hill Companies, Inc., 2008
Developing a Spreadsheet Model
•
Step #2: Changing Cells
– Add a cell in the spreadsheet for every decision that needs to be made.
– If you don’t have any particular initial values, just enter 0 in each.
– It is a good idea to color code these “changing cells” (e.g., yellow with border).
McGraw-Hill/Irwin
2.4
© The McGraw-Hill Companies, Inc., 2008
Developing a Spreadsheet Model
•
Step #3: Target Cell
– Develop an equation that defines the objective of the model.
– Typically this equation involves the data cells and the changing cells in order to
determine a quantity of interest (e.g., total profit or total cost).
– It is a good idea to color code this cell (e.g., orange with heavy border).
G
11
12
McGraw-Hill/Irwin
2.5
Total Profit
=SUMPRODUCT(UnitProfit,UnitsProduced)
© The McGraw-Hill Companies, Inc., 2008
Developing a Spreadsheet Model
•
Step #4: Constraints
– For any resource that is restricted, calculate the amount of that resource used in a
cell on the spreadsheet (an output cell).
– Define the constraint in three consecutive cells. For example, if Quantity A <=
Quantity B, put these three items (Quantity A, <=, Quantity B) in consecutive cells.
A
1
2
3
4
5
6
7
8
9
10
11
12
B
C
D
E
F
G
<=
<=
<=
Hours
Available
4
12
18
Wyndor Glass Co. Product-Mix Problem
Unit Profit
Doors
$300
Windows
$500
Hours Used Per Unit Produced
Plant 1
1
0
Plant 2
0
2
Plant 3
3
2
Units Produced
Doors
0
Hours
Used
0
0
0
Windows
0
Total Profit
$0
E
5
6
7
8
9
Hours
Used
=SUMPRODUCT(C7:D7,UnitsProduced)
=SUMPRODUCT(C8:D8,UnitsProduced)
=SUMPRODUCT(C9:D9,UnitsProduced)
McGraw-Hill/Irwin
2.6
© The McGraw-Hill Companies, Inc., 2008
A Trial Solution
The spreadsheet for the Wyndor problem with a trial solution (4 doors and 3
windows) entered into the changing cells.
McGraw-Hill/Irwin
2.7
© The McGraw-Hill Companies, Inc., 2008
Algebraic Model for Wyndor Glass Co.
Let D = the number of doors to produce
W = the number of windows to produce
Maximize P = $300D + $500W
subject to
D≤4
2W ≤ 12
3D + 2W ≤ 18
and
D ≥ 0, W ≥ 0.
McGraw-Hill/Irwin
2.8
© The McGraw-Hill Companies, Inc., 2008
Graphing the Product Mix
W
P rod uct ion rat e (un it s p e r we ek) fo r w in do w s
8
A product mix of
D = 4 and W
7
=6
(4, 6)
6
5
A product mix of
4
D = 2 and
3
W
=3
(2, 3)
2
1
Origin
-2
-1
0
-1
1
2
3
4
5
6
7
8
D
Production rate (units per week) for doors
-2
McGraw-Hill/Irwin
2.9
© The McGraw-Hill Companies, Inc., 2008
Graph Showing Constraints: D ≥ 0 and W ≥ 0
W
8
P ro du cti on rate fo r wi n do ws
6
4
2
0
2
4
6
8
D
Production rate for doors
McGraw-Hill/Irwin
2.10
© The McGraw-Hill Companies, Inc., 2008
Nonnegative Solutions Permitted by D ≤ 4
W
8
D= 4
Pro du c tio n ra te fo r w in d o w s
6
4
2
0
McGraw-Hill/Irwin
2
4
Production rate for doors
2.11
6
8
D
© The McGraw-Hill Companies, Inc., 2008
Nonnegative Solutions Permitted by 2W ≤ 12
Production rate for windows
W
8
2 W = 12
6
4
2
0
2
4
6
8
D
Production rate for doors
McGraw-Hill/Irwin
2.12
© The McGraw-Hill Companies, Inc., 2008
Boundary Line for Constraint 3D + 2W ≤ 18
Production rate for windows
W
10
(0, 9)
8
1
(1, 7 )_
2
(2, 6)
6
3 D + 2 W = 18
(3, 4
1
_
)
2
4
(4, 3)
2
(5, 1
1
_
)
2
(6, 0)
0
2
4
6
8
D
Production rate for doors
McGraw-Hill/Irwin
2.13
© The McGraw-Hill Companies, Inc., 2008
Changing Right-Hand Side Creates Parallel Constraint
Boundary Lines
Production rate for windows
W
12
10
3D + 2W = 24
8
6
3D + 2W = 18
4
2
0
3D + 2W = 12
2
4
8
6
10
D
Production rate for doors
McGraw-Hill/Irwin
2.14
© The McGraw-Hill Companies, Inc., 2008
Nonnegative Solutions Permitted by
3D + 2W ≤ 18
Production rate for windows
W
10
8
6
3D + 2W = 18
4
2
0
2
4
6
8
D
Production rate for doors
McGraw-Hill/Irwin
2.15
© The McGraw-Hill Companies, Inc., 2008
Graph of Feasible Region
Production rate for windows
W
10
3 D + 2 W = 18
8
D= 4
6
2 W =12
4
Feasible
region
2
0
McGraw-Hill/Irwin
2
4
Production rate for doors
2.16
6
8
D
© The McGraw-Hill Companies, Inc., 2008
Objective Function (P = 1,500)
Production rate
W
for windows
8
6
4
P = 1500 = 300D
Feasible
region
+ 500W
2
0
McGraw-Hill/Irwin
2
4
Production rate for doors
2.17
6
8
D
© The McGraw-Hill Companies, Inc., 2008
Finding the Optimal Solution
Production rate
W
for windows
8
P = 3600 = 300D + 500W
P = 3000 = 300D + 500W
Optimal solution
(2, 6)
6
Feasible
4
region
P = 1500 = 300D + 500W
2
0
2
4
Production rate for doors
McGraw-Hill/Irwin
2.18
6
8
10
D
© The McGraw-Hill Companies, Inc., 2008
Summary of the Graphical Method
•
Draw the constraint boundary line for each constraint. Use the origin (or any
point not on the line) to determine which side of the line is permitted by the
constraint.
•
Find the feasible region by determining where all constraints are satisfied
simultaneously.
•
Determine the slope of one objective function line. All other objective function
lines will have the same slope.
•
Move a straight edge with this slope through the feasible region in the
direction of improving values of the objective function. Stop at the last instant
that the straight edge still passes through a point in the feasible region. This
line given by the straight edge is the optimal objective function line.
•
A feasible point on the optimal objective function line is an optimal solution.
McGraw-Hill/Irwin
2.19
© The McGraw-Hill Companies, Inc., 2008
Identifying the Target Cell and Changing Cells
•
•
•
•
Choose the “Solver” from the Tools menu.
Select the cell you wish to optimize in the “Set Target Cell” window.
Choose “Max” or “Min” depending on whether you want to maximize or minimize the
target cell.
Enter all the changing cells in the “By Changing Cells” window.
B
C
3
4
Unit Profit
D
Doors
Windows
$300
$500
5
6
7
8
9
Plant 1
Plant 2
Plant 3
Hours Used Per Unit Produced
1
0
0
2
3
2
E
Hours
Used
1
2
5
F
<=
<=
<=
G
Hours
Available
1
12
18
10
Doors
11
12
McGraw-Hill/Irwin
Units Produced
1
2.20
Windows
1
Total Profit
$800
© The McGraw-Hill Companies, Inc., 2008
Adding Constraints
•
•
To begin entering constraints, click the “Add” button to the right of the
constraints window.
Fill in the entries in the resulting Add Constraint dialogue box.
B
C
3
Unit Profit
4
D
Doors
Windows
$300
$500
5
6
Plant 1
Plant 2
Plant 3
7
8
9
Hours Used Per Unit Produced
1
0
0
2
3
2
E
Hours
Used
1
2
5
F
<=
<=
<=
G
Hours
Available
1
12
18
10
Doors
11
12
Units Produced
McGraw-Hill/Irwin
Windows
1
1
2.21
Total Profit
$800
© The McGraw-Hill Companies, Inc., 2008
The Complete Solver Dialogue Box
McGraw-Hill/Irwin
2.22
© The McGraw-Hill Companies, Inc., 2008
Some Important Options
•
Click on the “Options” button, and click in both the “Assume Linear Model”
and the “Assume Non-Negative” box.
– “Assume Linear Model” tells the Solver that this is a linear programming model.
– “Assume Non-Negative” adds nonnegativity constraints to all the changing cells.
McGraw-Hill/Irwin
2.23
© The McGraw-Hill Companies, Inc., 2008
The Solver Results Dialogue Box
McGraw-Hill/Irwin
2.24
© The McGraw-Hill Companies, Inc., 2008
The Optimal Solution
McGraw-Hill/Irwin
2.25
© The McGraw-Hill Companies, Inc., 2008
The Profit & Gambit Co.
•
Management has decided to undertake a major advertising campaign that will
focus on the following three key products:
– A spray prewash stain remover.
– A liquid laundry detergent.
– A powder laundry detergent.
•
The campaign will use both television and print media
•
The general goal is to increase sales of these products.
•
Management has set the following goals for the campaign:
– Sales of the stain remover should increase by at least 3%.
– Sales of the liquid detergent should increase by at least 18%.
– Sales of the powder detergent should increase by at least 4%.
Question: how much should they advertise in each medium to meet the sales
goals at a minimum total cost?
McGraw-Hill/Irwin
2.26
© The McGraw-Hill Companies, Inc., 2008
Profit & Gambit Co. Spreadsheet Model
B
4
C
D
Television
3
Unit Cost ($millions)
E
F
G
Print Media
1
2
5
Increased
Sales
3%
18%
6
7
8
9
10
Stain Remover
Liquid Detergent
Powder Detergent
Increase in Sales per Unit of Advertising
0%
1%
3%
2%
-1%
4%
8%
>=
>=
Minimum
Increase
3%
18%
>=
4%
11
12
Television
13
14
Advertising Units
McGraw-Hill/Irwin
Print Media
4
3
2.27
Total Cost
($millions)
10
© The McGraw-Hill Companies, Inc., 2008
Algebraic Model for Profit & Gambit
Let TV = the number of units of advertising on television
PM = the number of units of advertising in the print media
Minimize Cost = TV + 2PM (in millions of dollars)
subject to
Stain remover increased sales:
PM ≥ 3
Liquid detergent increased sales: 3TV + 2PM ≥ 18
Powder detergent increased sales: –TV + 4PM ≥ 4
and
TV ≥ 0, PM ≥ 0.
McGraw-Hill/Irwin
2.28
© The McGraw-Hill Companies, Inc., 2008
Applying the Graphical Method
Amount of print media advertising
PM
Feasible
10
region
8
6
4
PM = 3
2
-TV + 4 PM = 4
-4
-2
0
2
3 TV + 2 PM = 18
4
6
8
10
TV
Amount of TV advertising
McGraw-Hill/Irwin
2.29
© The McGraw-Hill Companies, Inc., 2008
The Optimal Solution
PM
10
Cost = 15 = TV
Cost = 10 = TV
Fe as ib le
re gi o n
+ 2 PM
+ 2 PM
4
(4,3)
optimal
solution
0
5
10
15
TV
A m ou nt o f TV adv erti s ing
McGraw-Hill/Irwin
2.30
© The McGraw-Hill Companies, Inc., 2008
Summary of the Graphical Method
•
Draw the constraint boundary line for each constraint. Use the origin (or any
point not on the line) to determine which side of the line is permitted by the
constraint.
•
Find the feasible region by determining where all constraints are satisfied
simultaneously.
•
Determine the slope of one objective function line. All other objective function
lines will have the same slope.
•
Move a straight edge with this slope through the feasible region in the
direction of improving values of the objective function. Stop at the last instant
that the straight edge still passes through a point in the feasible region. This
line given by the straight edge is the optimal objective function line.
•
A feasible point on the optimal objective function line is an optimal solution.
McGraw-Hill/Irwin
2.31
© The McGraw-Hill Companies, Inc., 2008
A Production Problem
Weekly supply of raw materials:
8 Small Bricks
6 Large Bricks
Products:
Table
Profit = $20 / Table
McGraw-Hill/Irwin
Chair
Profit = $15 / Chair
2.32
© The McGraw-Hill Companies, Inc., 2008
Linear Programming
•
Linear programming uses a mathematical model to find the best allocation of
scarce resources to various activities so as to maximize profit or minimize
cost.
Let T = Number of tables to produce
C = Number of chairs to produce
Maximize Profit = ($20)T + ($15)C
subject to
2T + C ≤ 6 large bricks
2T + 2C ≤ 8 small bricks
and
T ≥ 0, C ≥ 0.
McGraw-Hill/Irwin
2.33
© The McGraw-Hill Companies, Inc., 2008
Graphical Representation
McGraw-Hill/Irwin
2.34
© The McGraw-Hill Companies, Inc., 2008
Components of a Linear Program
•
Data Cells
•
Changing Cells (“Decision Variables”)
•
Target Cell (“Objective Function”)
•
Constraints
McGraw-Hill/Irwin
2.35
© The McGraw-Hill Companies, Inc., 2008
Four Assumptions of Linear Programming
•
Linearity
•
Divisibility
•
Certainty
•
Nonnegativity
McGraw-Hill/Irwin
2.36
© The McGraw-Hill Companies, Inc., 2008
When is a Spreadsheet Model Linear?
•
All equations (output cells) must be of the form
= ax + by + cz + …
where a, b, c are constants (data cells) and x, y, z are changing cells.
•
Suppose C1:C6 are changing cells and D1:D6 are data cells.
Which of the following can be part of an LP?
–
–
–
–
–
–
–
–
–
SUMPRODUCT(D1:D6, C1:C6)
SUM(C1:C6)
C1 * SUM(C4:C6)
SUMPRODUCT(C1:C3, C4:C6)
IF(C1 > 3, 2*C3 + C4, 3*C3 + C5)
IF(D1 > 3, C1, C2)
MIN(C1, C2)
MIN(D1, D2) * C1
ROUND(C1)
McGraw-Hill/Irwin
2.37
© The McGraw-Hill Companies, Inc., 2008
Why Use Linear Programming?
•
Linear programs are easy (efficient) to solve
•
The best (optimal) solution is guaranteed to be found (if it exists)
•
Useful sensitivity analysis information is generated
•
Many problems are essentially linear
McGraw-Hill/Irwin
2.38
© The McGraw-Hill Companies, Inc., 2008
Developing a Spreadsheet Model
•
Step #1: Data Cells
– Enter all of the data for the problem on the spreadsheet.
– Make consistent use of rows and columns.
– It is a good idea to color code these “data cells” (e.g., light blue).
B
3
4
C
D
Tables
Profit
$20.00
E
F
G
Chairs
$15.00
5
6
Bill of Materials
Available
7
Large Bricks
2
1
6
8
Small Bricks
2
2
8
McGraw-Hill/Irwin
2.39
© The McGraw-Hill Companies, Inc., 2008
Developing a Spreadsheet Model
•
Step #2: Changing Cells
– Add a cell in the spreadsheet for every decision that needs to be made.
– If you don’t have any particular initial values, just enter 0 in each.
– It is a good idea to color code these “changing cells” (e.g., yellow with border).
B
3
C
D
Tables
4
Profit
$20.00
E
F
G
Chairs
$15.00
5
6
Bill of Materials
Available
7
Large Bricks
2
1
6
8
Small Bricks
2
2
8
Tables
Chairs
0
0
9
10
11
Production Quantity:
McGraw-Hill/Irwin
2.40
© The McGraw-Hill Companies, Inc., 2008
Developing a Spreadsheet Model
•
Step #3: Target Cell
– Develop an equation that defines the objective of the model.
– Typically this equation involves the data cells and the changing cells in order to
determine a quantity of interest (e.g., total profit or total cost).
– It is a good idea to color code this cell (e.g., orange with heavy border).
B
3
4
C
D
Tables
Profit
E
F
G
Chairs
$20.00
$15.00
5
6
Bill of Materials
Available
7
Large Bricks
2
1
6
8
Small Bricks
2
2
8
Tables
Chairs
Total Profit
1
0
$20.00
9
10
11
Production Quantity:
G
10
11
McGraw-Hill/Irwin
Total Profit
=SUMPRODUCT(C4:D4,C11:D11)
2.41
© The McGraw-Hill Companies, Inc., 2008
Developing a Spreadsheet Model
•
Step #4: Constraints
– For any resource that is restricted, calculate the amount of that resource used in a
cell on the spreadsheet (an output cell).
– Define the constraint in three consecutive cells. For example, if Quantity A ≤
Quantity B, put these three items (Quantity A, ≤, Quantity B) in consecutive cells.
– Note the use of relative and absolute addressing to make it easy to copy formulas
in column E.
B
3
4
C
D
Tables
Profit
E
F
G
Chairs
$20.00
$15.00
5
6
Bill of Materials
Total Us ed
Available
7
Large Bricks
2
1
3
<=
6
8
Small Bricks
2
2
4
<=
8
Tables
Chairs
Total Profit
1
1
$35.00
9
10
11
Production Quantity:
E
6
McGraw-Hill/Irwin
2.42
Total Used
7
=SUMPRODUCT(C7:D7,$C$11:$D$11)
8
=SUMPRODUCT(C8:D8,$C$11:$D$11)
© The McGraw-Hill Companies, Inc., 2008
Defining the Target Cell
•
•
•
Choose the “Solver” from the Tools menu.
Select the cell you wish to optimize in the “Set Target Cell” window.
Choose “Max” or “Min” depending on whether you want to maximize or
minimize the target cell.
B
C
3
4
D
Tables
Profit
$20.00
E
F
G
Chairs
$15.00
5
6
Bill of Materials
Total Us ed
Available
7
Large Bricks
2
1
3
<=
6
8
Small Bricks
2
2
4
<=
8
Tables
Chairs
Total Profit
1
1
$35.00
9
10
11
McGraw-Hill/Irwin
Production Quantity:
2.43
© The McGraw-Hill Companies, Inc., 2008
Identifying the Changing Cells
•
Enter all the changing cells in the “By Changing Cells” window.
– You may either drag the cursor across the cells or type the addresses.
– If there are multiple sets of changing cells, separate them by typing a comma.
B
3
4
C
D
Tables
Profit
$20.00
E
F
G
Chairs
$15.00
5
6
Bill of Materials
Total Us ed
Available
7
Large Bricks
2
1
3
<=
6
8
Small Bricks
2
2
4
<=
8
Tables
Chairs
Total Profit
1
1
$35.00
9
10
11
Production Quantity:
McGraw-Hill/Irwin
2.44
© The McGraw-Hill Companies, Inc., 2008
Adding Constraints
•
•
To begin entering constraints, click the “Add” button to the right of the
constraints window.
Fill in the entries in the resulting Add Constraint dialogue box.
B
3
4
C
D
Tables
Profit
$20.00
E
F
G
Chairs
$15.00
5
6
Bill of Materials
Total Us ed
Available
7
Large Bricks
2
1
3
<=
6
8
Small Bricks
2
2
4
<=
8
Tables
Chairs
Total Profit
1
1
$35.00
9
10
11
McGraw-Hill/Irwin
Production Quantity:
2.45
© The McGraw-Hill Companies, Inc., 2008
Some Important Options
•
Click on the “Options” button, and click in both the “Assume Linear Model”
and the “Assume Non-Negative” box.
– “Assume Linear Model” tells the Solver that this is a linear programming model.
– “Assume Non-Negative” adds nonnegativity constraints to all the changing cells.
McGraw-Hill/Irwin
2.46
© The McGraw-Hill Companies, Inc., 2008
The Solution
•
After clicking “Solve”, you will receive one of four messages:
– “Solver found a solution. All constraints and optimality conditions are satisfied.”
– “Set cell values did not converge.”
– “Solver could not find a feasible solution.”
– “Conditions for Assume Linear Model are not satisfied.”
B
3
C
D
Tables
4
Profit
$20.00
E
F
G
Chairs
$15.00
5
6
Bill of Materials
Total Us ed
Available
7
Large Bricks
2
1
6
<=
6
8
Small Bricks
2
2
8
<=
8
Tables
Chairs
Total Profit
2
2
$70.00
9
10
11
Production Quantity:
McGraw-Hill/Irwin
2.47
© The McGraw-Hill Companies, Inc., 2008
The Graphical Method for Solving LP’s
•
Formulate the problem as a linear program
•
Plot the constraints
•
Identify the feasible region
•
Draw an imaginary line parallel to the objective function (Z = a)
•
Find the optimal solution
McGraw-Hill/Irwin
2.48
© The McGraw-Hill Companies, Inc., 2008
Example #1
Maximize Z = 3x1 + 5x2
subject to
x1 ≤ 4
2x2 ≤ 12
3x1 + 2x2 ≤ 18
and
x1 ≥ 0, x2 ≥ 0.
x2
10
9
8
7
6
5
4
3
2
1
1
McGraw-Hill/Irwin
2.49
2
3
4
5
6
7
8
9
10
x1
© The McGraw-Hill Companies, Inc., 2008
Example #2
Minimize Z = 15x1 + 20x2
subject to
x1 +2x2 ≥ 10
2x1 – 3x2 ≤ 6
x1 + x2 ≥ 6
and
x1 ≥ 0, x2 ≥ 0.
x2
10
9
8
7
6
5
4
3
2
1
1
McGraw-Hill/Irwin
2.50
2
3
4
5
6
7
8
9
10
x1
© The McGraw-Hill Companies, Inc., 2008
Example #3
Maximize Z = x1 + x2
subject to
x1 +2x2 = 8
x1 – x2 ≤ 0
and
x1 ≥ 0, x2 ≥ 0.
x2
10
9
8
7
6
5
4
3
2
1
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Properties of Linear Programming Solutions
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An optimal solution must lie on the boundary of the feasible region.
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There are exactly four possible outcomes of linear programming:
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A unique optimal solution is found.
An infinite number of optimal solutions exist.
No feasible solutions exist.
The objective function is unbounded (there is no optimal solution).
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If an LP model has one optimal solution, it must be at a corner point.
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If an LP model has many optimal solutions, at least two of these optimal
solutions are at corner points.
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Example #4 (Multiple Optimal Solutions)
Minimize Z = 6x1 + 4x2
subject to
x1 ≤ 4
2x2 ≤ 12
3x1 + 2x2 ≤ 18
and
x1 ≥ 0, x2 ≥ 0.
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Example #5 (No Feasible Solution)
Maximize Z = 3x1 + 5x2
subject to
x1 ≥ 5
x2 ≥ 4
3x1 + 2x2 ≤ 18
and
x1 ≥ 0, x2 ≥ 0.
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Example #6 (Unbounded Solution)
Maximize Z = 5x1 + 12x2
subject to
x1 ≤ 5
2x1 –x2 ≤ 2
and
x1 ≥ 0, x2 ≥ 0.
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The Simplex Method Algorithm
1. Start at a feasible corner point (often the origin).
2. Check if adjacent corner points improve the objective function:
a) If so, move to adjacent corner and repeat step 2.
b) If not, current corner point is optimal. Stop.
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