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.