ISYE 6644 - Summer 2026 - Final Exam
Questions and Answers with complete
solution
Course
ISYE 6644
1. Linear Programming Fundamentals
Know how to formulate an optimization problem from a word description.
General form
Maximize/Minimize
Z=c1x1+c2x2+⋯+cnxnZ=c_1x_1+c_2x_2+\cdots+c_nx_n
subject to
a11x1+⋯+a1nxn≤b1a_{11}x_1+\cdots+a_{1n}x_n\le b_1 a21x1+⋯+a2nxn≤b2a_{21}x_1+\
cdots+a_{2n}x_n\le b_2 xi≥0x_i\ge0
Be able to identify:
Decision variables
Objective function
Constraints
Nonnegativity restrictions
Feasible region
Optimal solution
Binding vs. nonbinding constraints
Key idea
A feasible solution satisfies all constraints. The optimal solution is the feasible solution
producing the best objective value.
2. Graphical Linear Programming
,For two-variable problems, know how to:
1. Convert constraints into boundary equations.
2. Plot the constraints.
3. Identify the feasible region.
4. Find corner points.
5. Evaluate the objective at each corner.
6. Select the best point.
Important theorem
For a linear program with a nonempty bounded feasible region, an optimal solution occurs at an
extreme point/corner point.
Example
Maximize
Z=3x+5yZ=3x+5y
subject to
x+y≤8x+y\le 8 2x+y≤102x+y\le10 x,y≥0x,y\ge0
Candidate corner points include:
(0,0),(5,0),(0,8)(0,0),\quad(5,0),\quad(0,8)
and the intersection of the two constraint boundaries.
Always check all feasible corners.
3. Simplex Method
Understand the logic behind the simplex algorithm rather than memorizing isolated steps.
Core concepts
Basic variables
Nonbasic variables
Basic feasible solution
, Entering variable
Leaving variable
Pivot
Reduced cost
Optimality condition
The simplex method moves from one basic feasible solution to another while improving the
objective until no improving move remains.
Important warning
A variable entering the basis does not automatically mean the current variable leaves. The ratio
test determines the leaving variable.
4. Slack, Surplus, and Artificial Variables
Slack variable
For
2x+y≤102x+y\le10
write
2x+y+s=102x+y+s=10
where
s≥0.s\ge0.
The slack represents unused capacity.
Surplus variable
For
2x+y≥102x+y\ge10
write
2x+y−s=10.2x+y-s=10.
Artificial variables
Questions and Answers with complete
solution
Course
ISYE 6644
1. Linear Programming Fundamentals
Know how to formulate an optimization problem from a word description.
General form
Maximize/Minimize
Z=c1x1+c2x2+⋯+cnxnZ=c_1x_1+c_2x_2+\cdots+c_nx_n
subject to
a11x1+⋯+a1nxn≤b1a_{11}x_1+\cdots+a_{1n}x_n\le b_1 a21x1+⋯+a2nxn≤b2a_{21}x_1+\
cdots+a_{2n}x_n\le b_2 xi≥0x_i\ge0
Be able to identify:
Decision variables
Objective function
Constraints
Nonnegativity restrictions
Feasible region
Optimal solution
Binding vs. nonbinding constraints
Key idea
A feasible solution satisfies all constraints. The optimal solution is the feasible solution
producing the best objective value.
2. Graphical Linear Programming
,For two-variable problems, know how to:
1. Convert constraints into boundary equations.
2. Plot the constraints.
3. Identify the feasible region.
4. Find corner points.
5. Evaluate the objective at each corner.
6. Select the best point.
Important theorem
For a linear program with a nonempty bounded feasible region, an optimal solution occurs at an
extreme point/corner point.
Example
Maximize
Z=3x+5yZ=3x+5y
subject to
x+y≤8x+y\le 8 2x+y≤102x+y\le10 x,y≥0x,y\ge0
Candidate corner points include:
(0,0),(5,0),(0,8)(0,0),\quad(5,0),\quad(0,8)
and the intersection of the two constraint boundaries.
Always check all feasible corners.
3. Simplex Method
Understand the logic behind the simplex algorithm rather than memorizing isolated steps.
Core concepts
Basic variables
Nonbasic variables
Basic feasible solution
, Entering variable
Leaving variable
Pivot
Reduced cost
Optimality condition
The simplex method moves from one basic feasible solution to another while improving the
objective until no improving move remains.
Important warning
A variable entering the basis does not automatically mean the current variable leaves. The ratio
test determines the leaving variable.
4. Slack, Surplus, and Artificial Variables
Slack variable
For
2x+y≤102x+y\le10
write
2x+y+s=102x+y+s=10
where
s≥0.s\ge0.
The slack represents unused capacity.
Surplus variable
For
2x+y≥102x+y\ge10
write
2x+y−s=10.2x+y-s=10.
Artificial variables
