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Week 11b - QMA 2006S2 (Linear Programming)

# Week 11b - QMA 2006S2 (Linear Programming) - Week 10 11...

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Jessica Chung & Derek Hui Week 10 & 11 QMA PASS | 2006 S2 | Tues, 3-4pm | QUAD G042 Questions? Email us: [email protected] 1/3 G RAPHING I NEQUALITIES Strong inequalities Weak inequalities ax + by > c ax + by < c ax + by ° c ax + by ± c - The union of two inequalities is the area belonging to EITHER of the inequalities - The intersection of two inequalities is the area belonging to BOTH of the inequalities (i.e. only the overlapping regions) L INEAR P ROGRAMMING Linear programming is a method of maximising or minimising a function subject to a number of constraints. Its most common application is to decide on how to efficiently allocate levels of resources to various uses. Steps in Linear Programming PART ONE 1. Formulate the OBJECTIVE FUNCTION mathematically 2. Identify whether the problem asks you to maximise or minimise the objective function 3. Formulate the CONSTRAINTS mathematically (don’t forget the assumptions that x ° 0 and y ° 0) 4. Graph the constraints and determine/shade the FEASIBLE REGION PART TWO – METHOD A A1. Graph the ISO-OBJECTIVE LINE by letting the objective function equal any arbitrary number A2. ‘Slide’ the iso-objective line until it intersects with a corner point of the feasible region Maximisation ° corner point furthest from the origin Minimisation ° corner point closest to the origin A3. Solve the values of x and y at this corner point by simultaneously solving the constraint lines which intersect to produce this point A4. Substitute the values of x and y obtained to find the optimal value PART TWO – METHOD B B1. Locate all the corner points of the feasible region B2. Solve for the x and y values (the coordinates) of these corner points by solving simultaneously the constraint lines that intersect to produce each corner point B3. Substitute each coordinate obtained into the objective function to find the optimal value (identify the maximum/minimum value depending on the question) - NOTE: Multiple optimal solutions can exist. This occurs when the slope/gradient of the iso-objective line equals the slope of one of the constraint lines that form the feasible region. No solution is also possible where a given constraint precludes all other possibilities and thus there is no feasible region.

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Week 11b - QMA 2006S2 (Linear Programming) - Week 10 11...

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