MAN 327 2.0
Micro-computers & Their Applications
Solver
Linear Problem Solving
Installing Solver for Use
• File  Options Add-ins Solver-ins
• Data  Analysis  Solver
What is Linear Programming?
• A Linear Programming model seeks to
maximize or minimize a linear function,
subject to a set of linear constraints.
Introduction to Linear Programming
• The Importance of Linear Programming
– Many real world problems lend themselves to linear
programming modeling.
– Many real world problems can be approximated by linear
models.
– There are well-known successful applications in:
• Manufacturing
• Marketing
• Finance (investment)
• Advertising
• Agriculture
4
Setting up a Linear Problem
• The Objective Function
• The Decision Variables (optimal values will be
calculated by Solver)
• The Constraints
– Relationships ( >, <, =)
– Values
Output
• The Answer Report
• The Sensitivity Report
The Galaxy Linear Programming Model
Max 8X1 + 5X2
(Weekly profit)
subject to
2X1 + 1X2  1000 (Plastic)
3X1 + 4X2  2400 (Production Time)
X1 + X2  700 (Total production)
X1 - X2  350 (Mix)
Xj> = 0, j = 1,2
(Nonnegativity)
9
Using Excel Solver – Optimal Solution
GALAXY INDUSTRIES
Dozens
Profit
Plastic
Prod. Time
Total
Mix
Prod 1
320
8
2
3
1
1
Prod 2
360
5
1
4
1
-1
Total
4360
1000
2400
680
-40
Limit
<=
<=
<=
<=
1000
2400
700
350
10
Using Excel Solver –Answer Report
Microsoft Excel 9.0 Answer Report
Worksheet: [Galaxy.xls]Galaxy
Report Created: 11/12/2001 8:02:06 PM
Target Cell (Max)
Cell
Name
$D$6 Profit Total
Original Value
4360
Final Value
4360
Adjustable Cells
Cell
Name
Original Value
$B$4 Dozens Space Rays
320
$C$4 Dozens Zappers
360
Final Value
320
360
Constraints
Cell
Name
$D$7 Plastic Total
$D$8 Prod. Time Total
$D$9 Total Total
$D$10 Mix Total
Cell Value
1000
2400
680
-40
Formula
$D$7<=$F$7
$D$8<=$F$8
$D$9<=$F$9
$D$10<=$F$10
Status
Slack
Binding
0
Binding
0
Not Binding
20
Not Binding
390
11
Using Excel Solver –Sensitivity Report
Microsoft Excel Sensitivity Report
Worksheet: [Galaxy.xls]Sheet1
Report Created:
Adjustable Cells
Final Reduced Objective Allowable Allowable
Cell
Name
Value
Cost
Coefficient Increase
Decrease
$B$4 Dozens Space Rays
320
0
8
2
4.25
$C$4 Dozens Zappers
360
0
5 5.666666667
1
Constraints
Cell
$D$7
$D$8
$D$9
$D$10
Name
Plastic Total
Prod. Time Total
Total Total
Mix Total
Final Shadow Constraint Allowable Allowable
Value
Price
R.H. Side
Increase
Decrease
1000
3.4
1000
100
400
2400
0.4
2400
100
650
680
0
700
1E+30
20
-40
0
350
1E+30
390
12
The Role of Sensitivity Analysis
of the Optimal Solution
• Is the optimal solution sensitive to changes
in input parameters?
• Possible reasons for asking this question:
– Parameter values used were only best
estimates.
– Dynamic environment may cause changes.
– “What-if” analysis may provide economical
and operational information.
13
Solver – Unbounded solution
14
Solver – An Alternate Optimal Solution
• Solver does not alert the user to the existence
of alternate optimal solutions.
• Many times alternate optimal solutions exist
when the allowable increase or allowable
decrease is equal to zero.
• In these cases, we can find alternate optimal
solutions using Solver by the following
procedure:
15
Solver – An Alternate Optimal Solution
• Observe that for some variable Xj the
Allowable increase = 0, or
Allowable decrease = 0.
• Add a constraint of the form:
Objective function = Current optimal value.
• If Allowable increase = 0, change the objective to
Maximize Xj
• If Allowable decrease = 0, change the objective to
Minimize Xj
16
Setup
Variant
Number
Assembly
Polish
Pack
Profit
1
2
3
2
1.5
2
4
2
3
2.5
3
3
3
2
3
4
7
4
5
4.5
Total
0
0
0
Limit
100000
50000
60000
=$B$2*C2+$B$3*C3+$B$4*C4+$B$5*C5
=$B$2*F2+$B$3*F3+$B$4*F4+$B$5*F5
0
Solution
Variant
Number
Assembly
Polish
Pack
Profit
1
0
2
3
2
1.5
2
16000
4
2
3
2.5
3
6000
3
3
2
3
4
0
7
4
5
4.5
Total
82000
50000
60000
Limit
100000
50000
60000
58000
Note : Analysis below
ONLY applies for a
single change
Sensitivity Analysis of
Objective Function Coefficients.
•
Range of Optimality
– The optimal solution will remain unchanged as long as
• An objective function coefficient lies within its range of
optimality
• There are no changes in any other input parameters.
– The value of the objective function will change if the
coefficient multiplies a variable whose value is nonzero.
21
What they mean ?
• Changing the objective function coefficient
for a variable
• Optimal LP solution will remain unchanged
– Decision to produce 16000 of variant 2 and 6000
of variant 3 remains optimal even if the profit per
unit on variant 2 is not actually 2.5. It lies in the
range 2.3571 ( 2.5 - 0.1429 ) to 4.50 ( 2.5 + 2).
What they mean ?
• Forcing a variable which is currently zero to be nonzero
• Reduced Cost column gives us, for each variable which
is currently zero (X1 and X4), an estimate of how much
the objective function will change if we make (force)
that variable to be non-zero.
• Often called the 'opportunity cost' for the variable.
– profit per unit on variant 1 would need to increase by 1.5
before it would be profitable to produce any of variant 1.
Shadow Prices
• Assuming there are no other changes to the
input parameters, the change to the objective
function value per unit increase to a right
hand side of a constraint is called the “Shadow
Price”
24
What they mean ?
• Changing the right-hand side of a constraint
• Shadow Price tells us exactly how much the objective
function will change if we change the right-hand side of
the corresponding constraint within the limits given in
the Allowable Increase/Decrease columns
• Often called the 'marginal value' or 'dual value' for that
constraint.
– For the polish constraint, the objective function change
will be exactly 0.80.
Sensitivity Analysis of
Right-Hand Side Values
• In sensitivity analysis of right-hand sides of
constraints we are interested in the
following questions:
– Keeping all other factors the same, how much
would the optimal value of the objective
function (for example, the profit) change if the
right-hand side of a constraint changed by one
unit?
– For how many additional or fewer units will this
per unit change be valid?
26
Sensitivity Analysis of
Right-Hand Side Values
• Any change to the right hand side of a
binding constraint will change the optimal
solution.
• Any change to the right-hand side of a nonbinding constraint that is less than its slack
or surplus, will cause no change in the
optimal solution.
27
• To decide whether the objective function will go up or
down use:
– constraint more (less) restrictive after change in right-hand side
implies objective function worse (better)
– if objective is maximise (minimise) then worse means down
(up), better means up (down)
• Hence
– if you had an extra 100 hours to which operation would you
assign it?
– if you had to take 50 hours away from polishing or packing
which one would you choose?
– what would the new objective function value be in these two
cases?
Minimizing Problem
• Nutritional facts of foods (per 100 g)
Bread (500cal) – P 10 g, F 30g, C 60g,
Rice (800cal) - P 10 g, F 10g, C 80g,
Dahl (700cal) - P 70 g, F 10g, C 20g,
Egg (1100cal) - P 80 g, F 15g, C 5g,
• Suppose we need to minimize our calories (2000cal), while
keeping the minimum level of nutritionals.
Protein – 300 cal(75g)
Fat – 600 cal(67g)
Carbohydrate – 1100 cal(275g)
Solver Help
• www.solver.com.