Linear Programming
We can make Product1 and Product2.
There are 3 resources; Resource1, Resource2, Resource3.
Product1 needs one hour of Resource1, nothing of Resource2, and three
hours of resource3.
Product2 needs nothing from Resource1, two hours of Resource2, and
two hours of resource3.
Available hours of resources 1, 2, 3 are 4, 12, 18, respectively.
Contribution Margin of product 1 and Product2 are $300 and $500,
respectively.
 Formulate the Problem
 Solve the problem using solver in excel
Problem Formulation
Objective Function
Z = 3 x1 +5 x2
Constraints
Resource 1
x1
4
Resource 2
2x2  12
Resource 3
3 x1 + 2 x2  18
Nonnegativity
x1  0, x2  0
Wyndor Example; Enter data
Organize the data for the model on the spreadsheet. Type in the
coefficients of the constraints and the objective function
Noncomputational Entries and Changing Cells
Changing cells. Assign a set of cells to represent the decision
variable in the model.
The problem starts with assuming a value of 0 in each decision variable
cell, and find the optimal solution.
SUMPRODUCT
Constraint cells. For each constraint, create a formula in a separate
cell that corresponds to the left-hand side (LHS) of the constraint.
SUMPRODUCT function is used to multiply element by element of two
tables and addup all values.
SUMPRODUCT: sums the products of individual cells in two ranges.
SUMPRODUCT(C6:E6, C4:E4) = C6*C4 + D6*D4+ E6*E4
The two specified ranges must be of the same size ( the same number of
rows and columns).
For linear programming you should try to always use the
SUMPRODUCT function (or SUM) for the objective function and
constraints. This is to remember that the equations are all linear.
In LP we do not have square, square-root, log, exponential, cos, etc.
SUMPRODUCT
F4
Copy Down; LHS, and the Objective Function
Target Cell: The Objective Function
Target cell. A cell containing a formula that corresponds to the
objective function.
Tools/Add-Ins/Solver
Defining the Target Cell ( The Objective Function)
The target cell contains an equation that defines the objective and
depends on the decision variables. You can only have one objective
function. Target cell must be a single cell.
In the Solver dialogue box select the “Set Target Cell” window, then
click on the cell that you have already defined it as the objective
function. This is the cell you wish to optimize. Then lick on the radio
button of either Max or Min.
Identifying the Changing Cells ( Decision Variables)
You next tell Excel which cells are decision variables, i.e., which cells
excel is allowed to change when trying to optimize. Move the cursor to
the “By Changing Cells” window, and drag the cursor across all cells
you wish to treat as decision variables
Adding Constraints
Click on the “Add” button to the right of the constraints window.
A new dialogue box will appear. The cursor will be in the “Cell
Reference” window within this dialogue box.
Click on the cell that contains the quantity you want to constrain.
The default inequality that first appears for a constraint is “<= ”.
To change this, click on the arrow beside the “<= ” sign.
After setting the inequality, move the cursor to the “Constraint”
window.
Click on the cell you want to use as the constraining value for that
constraint.
Adding Constraints
Adding Constraints
You may define a set of similar constraints (e.g., all <=
constraints, or all >= constraints) in one step if they are in
adjacent rows.
Simply select the range of cells for the set of constraints in both
the “Cell Reference” and “Constraint” window.
After you are satisfied with the constraint(s),
click the “Add” button if you want to add another
constraint, or
click the “OK” button if you want to go back to the original
dialogue box.
Notice that you may also force a decision variable to be an
integer or binary (i.e., either 0 or 1) using this window.
Important Options
The Solver dialogue box now contains the optimization
model, including the target cell (objective function),
changing cells (decision variables), and constraints.
Important Options
Click on the “Options” button in the Solver dialogue box,
and click in both the “Assume Linear Model” and the
“Assume Non-Negative” box.
The “Assume Linear Model” option tells Excel that it is a
linear program. This speeds the solution process, makes it
more accurate, and enables the more informative
sensitivity report.
The “Assume Non-Negative” box adds non-negativity
constraints to all of the decision variables.
Check the Options then Solve
The Solution
The Solution
When it is done, you will receive one of four messages:
Solver found a solution. All constraints and optimality conditions are
satisfied. This means that Solver has found the optimal solution.
Cell values did not converge. This means that the objective function can be
improved to infinity. You may have forgotten a constraint (perhaps the nonnegativity constraints) or made a mistake in a formula.
Solver could not find a feasible solution. This means that Solver could not
find a feasible solution to the constraints you entered. You may have made a
mistake in typing the constraints or in entering a formula in your spreadsheet.
Conditions for Assume Linear Model not satisfied. You may have included a
formula in your model that is nonlinear. There is also a slim chance that Solver
has made an error. (This bug shows up occasionally.)
The Solution
If Solver finds an optimal solution, you have some options.
First, you must choose whether you want Solver to keep the optimal
values in the spreadsheet (you usually want this one) or go back to the
original numbers you typed in.
Click the appropriate box to make you selection. you also get to choose
what kind of reports you want.
Once you have made your selections, click on “OK”.
You will often want to also have the “Sensitivity Report”.
To view the sensitivity report, click on the “Sensitivity Report” tab in
the lower-left-hand corner of the window.
Questions Answered by Excel
 What is the Optimal Solution?
 What is the value of the Objective Function for the Optimal Solution?
 If the net profit for a product changes, will the optimal solution
change?
 If more (or less) of a resource is available, how it affect our profit?
For example:
 How much do you pay to have one extra unit of Resource1
 How much do you pay to have one extra unit of Resource3
 How much do you pay to have one extra unit of Resource2
Sensitivity
Output from Computer Solution : Changing Cells
Adjustable Cells
Cell
Name
$B$9 Solution Product1
$C$9 Solution Product2
Final Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase Decrease
2
0
3
4.5
3
6
0
5
1E+30
3
Final Value
The value of the variable in the optimal
solution
Reduced Cost
Increase in the objective function value per
unit increase in the value of a zero-valued
variable (a product that the model has
decided not to produce).
Allowable
Increase/
Decrease
Defines the range of the cost coefficients in
the objective function for which the current
solution (value of the variables in the optimal
solution) will not change.
Output from Computer Solution : Constraints
Constraints
Cell
Name
$D$5 Resource1 LHS
$D$6 Resource2 LHS
$D$7 Resource3 LHS
Final Shadow Constraint Allowable Allowable
Value
Price
R.H. Side
Increase Decrease
2
0
4
1E+30
2
12
1.5
12
6
6
18
1
18
6
6
Final Value
The usage of the resource in the optimal solution.
Shadow price
The change in the value of the objective function
per unit increase in the right hand side of the
constraint: Z = (Shadow Price)(RHS)
(Only for change is within the allowable range)
Output from Computer Solution : Constraints
Constraints
Cell
Name
$D$5 Resource1 LHS
$D$6 Resource2 LHS
$D$7 Resource3 LHS
Final Shadow Constraint Allowable Allowable
Value
Price
R.H. Side
Increase Decrease
2
0
4
1E+30
2
12
1.5
12
6
6
18
1
18
6
6
Constraint
R.H. Side
The current value of the right hand side of the
constraint (the amount of the resource that
is available).
Allowable
Increase/
Decrease
The range of values of the RHS for which
the shadow price is valid and hence for which
the new objective function value can be
calculated. (NOT the range for which the
current solution will not change.)
Net Profit product 1 = $7
Change the profit to 7, Solver again, Solve again
Adjustable Cells
Cell
Name
$B$9 Solution Product1
$C$9 Solution Product2
Final Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase Decrease
2
0
3
4.5
3
6
0
5
1E+30
3
Net Profit product 1 = $8
Change the profit to 8, Solver again, Solve again
Adjustable Cells
Cell
Name
$B$9 Solution Product1
$C$9 Solution Product2
Final Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase Decrease
2
0
3
4.5
3
6
0
5
1E+30
3
Wyndor Optimal Solution
What is the optimal Objective function value for this problem?
What is the allowable range for changes in the objective coefficient for
Product 2
What is the allowable range for changes in the RHS for Resource 3.
If the coefficient of Product 2 in the objective function is changed to 7,
what will happen to the value of the objective function?
If the coefficient of Product 1 in the objective function is changed to 8,
what will happen to the value of the objective function?
If the RHS of Resource 2 is increased by 2 , what will happen to the
objective function.
If the RHS of Resource 1 is increased by 2, what will happen to the
objective function.
If the RHS of Resource 2 is decreased by 10, what will happen to the
objective function.
Wyndor Optimal Solution
Practice: Given the following Sensitivity Analysis Report
Adjustable Cells
Cell
Name
$B$9 Solution Product1
$C$9 Solution Product2
$D$9 Solution Product3
Final
Value
0
500
375
Reduced
Cost
-1.25
0
0
Objective
Coefficient
15
20
25
Allowable
Increase
1.25
5
0.71
Allowable
Decrease
1E+30
0.42
5
Final
Value
400
350
150
Shadow
Price
0
12.5
100
Constraint
R.H. Side
500
350
150
Allowable
Increase
1E+30
50
25
Allowable
Decrease
100
50
18.75
Constraints
Cell
Name
$E$5 Resource1 LHS
$E$6 Resource2 LHS
$E$7 Resource3 LHS
Sensitivity Analysis
•
•
•
What is the optimal objective function value for this problem?
a.
It cannot be determined from the given information.
b.
$900.
c.
$987.5.
d.
$875.
e.
$19375.
What is the allowable range for the objective function coefficient for Product 3?
a.
0.71 ≤ P3 ≤ 5.
b.
20 ≤ P3 ≤ 25.71.
c.
0 .71≤ P3 ≤ 25.71.
d.
25 ≤ P3 ≤ 25.71.
e.
non of the above.
What is the allowable range of the right-hand-side for Resource1?
a.
500 ≤ RHS1 ≤ ∞.
b.
0 ≤ RHS1 ≤ 500.
c.
100 ≤ RHS1 ≤ ∞.
d.
400 ≤ RHS1 ≤ ∞.
e.
-∞ ≤ RHS1 ≤ 400.
Sensitivity Analysis
•
•
•
If the coefficient for Product2 in the objective function changes to $24, then the
objective function value:
a.
will increase by $24.
b.
will increase by $120,000.
c.
will increase by $2000.
d.
will remain the same.
e.
can only be discovered by resolving the problem.
If the coefficient for Product1 in the objective function changes to $5, then the objective
function value:
a.
will increase by $5.
b.
is $0.
c.
will increase by $10.
d.
will remain the same.
e.
can only be discovered by resolving the problem.
If the coefficient of Product2 in the objective function changes to $15, then:
a.
the original solution remains optimal.
b.
the problem must be resolved to find the optimal solution.
c.
the optimal solution will decrease by 2500.
d.
the shadow price will decrease by 5.
e.
the optimal solution will increase by 2500.
Sensitivity Analysis
• If the right-hand side of Resource1 increases, then the objective function value:
a.
will increase.
b. will decrease.
c.
will decrease then increase.
d. will remain the same.
e.
will increase then decrease.
• If the right-hand side of Resource2 changes to 370, then the objective function
value:
a.
will increase by $370.
b. will increase by $350.
c.
will increase by $250.
d. will remain the same.
e.
can only be discovered by resolving the problem.
• If the right-hand side of Resource3 changes to 130, then:
a.
the original solution remains optimal.
b. the problem must be resolved to find the optimal solution.
c.
the objective function will decrease by 130.
d. the objective function will decrease by 3000.
e.
the objective function will increase by 3000
More than one profit
OR More than one resource
• If the sum of the ratio of (Change)/(Change in the
Corresponding Direction) <=1
• Things remain the same.
• If we are talking about profit, the production plan
remains the same.
• If we are talking about RHS, the shadow prices
remain the same.
Sensitivity Analysis
•
•
•
If the objective coefficients of Product1 is increased by 1 and the objective coefficient of
product3 is decreased by 2, then:
a.
the objective function will decrease.
b.
the objective function will increase.
c.
the optimal solution will remain the same.
d.
the shadow prices will remain the same.
e.
can only be discovered by resolving the problem.
If the right-hand side of all three resources, each increases by 10 units:
a.
the optimal solution remains the same. The objective function value will increase.
b.
the optimal solution will change. The objective function value will increase.
c.
the optimal solution and the shadow prices will remain the same.
d.
the optimal solution and the shadow prices both will change.
e.
can only be discovered by resolving the problem.
If the right-hand side of resouses1 and 2 each decreases by 20 and the right and side of
resource 3 increases by 5:
a.
the objective function value will increase by 250.
b.
the objective function value will decrease by 250.
c.
the objective function value will remain the same.
d.
the optimal solution will remain the same.
e.
can only be discovered by resolving the problem.
Assignment
• The following 11 Questions refer to the following sensitivity report.
Adjustable Cells
Cell
$B$6
$C$6
$D$6
Name
Solution Activity 1
Solution Activity 2
Solution Activity 3
Final
Value
0
27.5
0
Reduced
Cost
425
0.0
250
Objective
Coefficient
500
300
400
Allowable
Increase
Allowable
Decrease
1E+30
500
1E+30
425
300
250
Allowable
Increase
Allowable
Decrease
Constraints
Cell
$E$2
$E$3
$E$4
Name
Benefit A Totals
Benefit B Totals
Benefit C Totals
Final
Value
110
110
137.5
Shadow
Price
0
75
0
Constraint
R.H. Side
60
110
80
50
1E+30
57.5
1E+30
46
1E+30
Assignment ( Taken from The management Sciences Hillier and Hillier)
• What is the optimal objective function value for this problem?
a.
it cannot be determined from the given information.
b. $1,200.
c.
$975.
d. $8,250.
e.
$500.
• What is the allowable range for the objective function coefficient for Activity 3?
a.
150 ≤ A3 ≤ ∞.
b. 0 ≤ A3 ≤ 650.
c.
0 ≤ A3 ≤ 250.
d. 400 ≤ A3 ≤ ∞.
e.
300 ≤ A3 ≤ 500.
• What is the allowable range of the right-hand-side for Resource A?
a.
0 ≤ RHSA ≤ 60.
b. 0 ≤ RHSA ≤ 110.
c.
60 ≤ RHSA ≤ 110.
d. 110 ≤ RHSA ≤ 160.
e.
0 ≤ RHSA ≤ 160.
Assignment ( Taken from The management Sciences Hillier and Hillier)
• If the coefficient for Activity 2 in the objective function changes to $400, then the
objective function value:
a.
will increase by $7,500.
b. will increase by $2,750.
c.
will increase by $100.
d. will remain the same.
e.
can only be discovered by resolving the problem.
• If the coefficient for Activity 1 in the objective function changes to $50, then the
objective function value:
a.
will decrease by $450.
b. is $0.
c.
will decrease by $2750.
d. will remain the same.
e.
can only be discovered by resolving the problem.
• If the coefficient of Activity 2 in the objective function changes to $100, then:
a.
the original solution remains optimal. The objective function value decreases.
b. the problem must be resolved to find the optimal solution.
c.
the shadow prices will remain the same.
d. the original solution and the objective function value remain the same.
e.
none of the above.
Assignment ( Taken from The management Sciences Hillier and Hillier)
•
•
•
If the right-hand side of Resource B changes to 80, then the objective function value:
a.
will decrease by $750.
b.
will decrease by $1500.
c.
will decrease by $2250.
d.
will remain the same.
e.
can only be discovered by resolving the problem.
If the right-hand side of Resource C changes to 140, then the objective function value:
a.
will increase by $137.50.
b.
will increase by $57.50.
c.
will increase by $80.
d.
will remain the same.
e.
can only be discovered by resolving the problem.
If the right-hand side of Resource C changes to 130, then:
a.
the original solution remains optimal. The objective function value will increase.
b.
the problem must be resolved to find the optimal solution.
c.
the shadow prices remain the same. The objective function value will decrease.
d.
the shadow prices remain the same. the objective function value remains the same.
e.
the shadow prices will change. The objective function value will increase.
Assignment ( Taken from The management Sciences Hillier and Hillier)
•
•
•
If the objective coefficients of Activity 2 and Activity 3 are both decreased by $100, then:
a.
the objective function will decrease by 2750.
b.
the objective function will decrease by less than 2750.
c.
the objective function will decrease by more than 2750.
d.
The objective function will remain the same.
e.
can only be discovered by resolving the problem.
If the right-hand side of Resource C is increased by 40, and the right-hand side of Resource B is
decreased by 20, then:
a.
the optimal solution remains the same.
b.
the objective function value increases by 1500.
c.
the shadow prices remain the same.
d.
can only be discovered by resolving the problem.
e.
the objective function value decreases by 1500.
If the right-hand side of Resource A is increased by 25, and the right-hand side of Resource B is
decreased by 20, then:
a.
the optimal solution remains the same.
b.
the objective function value increases by 1500.
c.
the objective function value remains the same.
d.
can only be discovered by resolving the problem.