qm b lego simplex. scenario you manufacture tables and chairs. tables and chairs are manufactured...
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QM B Lego Simplex
Scenario
• You manufacture tables and chairs.
• Tables and chairs are manufactured from small and large bricks.
Small brick
Large brick
Lego Simplex Data• Table
– 2 large bricks– 2 small bricks– $16 profit
• Chair– 1 large bricks– 2 small bricks– $10 profit
Lego My Simplex Resources
• You have 8 small bricks and 6 large bricks
The Goal• How many
tables and how many chairs should be produced to maximize profit?
Buzz Group Question 1 Groups of 3-4
One possible solution
Is this solution optimal? Profit = 3*16 = $48
Give up a table (-$16)
Make two chairs (+20)
Improves the solution by $4.
So the above solutions is not optimal.
Optimal Solution
2 tables
2 chairs
Profit: $52
Formulate as an LP
Buzz Group: Question 2
Be sure to:Define variablesWrite the objective functionWrite the constraintsInclude non-negativity constraints
Formulate as an LP
T – number of tables to produceC – number of chairs to produce
Max 16 T + 10 CSubject to:
2 T + C 6 Large bricks2 T + 2 C 8 Small bricksT 0, C 0 Non-negativity
Graphical Insight
Lego
0
1
2
3
4
5
6
7
0 1 2 3 4 5
Tables
Ch
air
s
Small Bricks
Large Bricks
2 T + C 62 T + 2 C 8
At the optimal solution:
• What if one more large brick becomes available?
Buzz Group: Question 3
What would you be willing to pay for the brick?
• Pay a maximum of $6. This is the shadow price for large bricks – the increase in the objective function if one more unit of the resource, large bricks, becomes available.
• Take a chair apart: (-10)
• Make a table: (+16)
How many large bricks should you buy at $6?
• If we have yet another additional large bricks (8 altogether), then we can take apart the second chair and make a table. ($6 improvement).
• If we add one more large brick (9 altogether), can we make another table?
Allowable increase
• The number of additional large bricks that are worth the shadow price of $6 is 2.
Sensitivity Analysis• Shadow Prices – increase in the
objective function if one more unit of the resource becomes available
• Allowable increases – amount that the resource can increase
and have the shadow price stay the same;
– if outside the allowable increase, change RHS and re-run.
Sensitivity ReportMicrosoft Excel 9.0 Sensitivity ReportWorksheet: [Lego.xls]LPReport Created: 9/13/2001 10:57:12 AM
Adjustable CellsFinal Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease$B$7 Decision Vars Table 2 0 16 4 6$C$7 Decision Vars Chair 2 0 10 6 2
ConstraintsFinal Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease$D$4 Small Brick Used 8 2 8 4 2$D$5 Large Brick Used 6 6 6 2 2
Think-pair-share: Stratton – Sensitivity Analysis• What is the optimal product mix?• What is the unused capacity of each
resource?• An additional labor hour is available. Where
should that labor hour be assigned? • An additional 8 labor hours are available.
Where should these labor hours be assigned?• The material manager has found an additional
2 pounds of additive mix for $1.20 per pound. Should he procure this additional mix?
Stratton Sensitivity Analysis• What is the optimal product mix
3 packages of Pipe 16 packages of Pipe 2
• What is the unused capacity of each resource:
Extrusion hours: 0 hoursPackaging hours: 0 hoursAdditive mix: 4 pounds
Stratton Sensitivity Analysis
• An additional labor hour is available, where should that labor hour be assigned?
Packaging – for every additional hour of packaging, profit increases $11 (up to a 2 hours increase)
Stratton Sensitivity Analysis
• An additional 8 labor hours are available. Where should these labor hours be assigned?
Need to re-run the model since 8 hours is larger than the allowable increase for both labor hour constraints.
Stratton Sensitivity Analysis
• The material manager has found an additional 2 pounds of additive mix for $1.20 per pound. Should he procure this additional mix?
No, the current optimal solution does not use all of the additive mix that is available.