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 variables Write the objective function Write the constraints Include non-negativity constraints

Formulate as an LP T – number of tables to produce C – number of chairs to produce Max 16 T + 10 C Subject to: 2 T + C  6Large bricks 2 T + 2 C  8Small bricks T  0, C  0Non-negativity

Graphical Insight 2 T + C  6 2 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 Report

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 1 6 packages of Pipe 2 What is the unused capacity of each resource: Extrusion hours:0 hours Packaging hours:0 hours Additive 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.