Deterministic OR Methods for Data Scientists

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• transform LP problems from general form to standard form and vice versa (Section 2 (Simplex Method))
• describe complementary slackness conditions (Section 3 (Duality))
• describe the dual Simplex method (using dictionary) (Section 3 (Duality))
• apply branch and bound to solve simple IP (e.g. with no more than 3 variables) (Section 5 (Integer Programming))
• solve LP problems with two variables using the graphical method (Section 1 (Intro to MP))
• derive the range of the objective coefficient until the optimal basis start changing and the corresponding range of the optimal value (Section 4 (Sensitivity))
• formulate the knapsack problem using IP (Section 5 (Integer Programming))
• describe the motivation for using duality and formulate the dual problem given the primal
• describe graphical method, simplex methods (primal and dual) and apply them to solve relatively small LP problems
• describe how dual simplex can be used in the case additional constraint leads to infeasibility (Section 4 (Sensitivity))
• state LP duality theorem (Section 3 (Duality))
• apply these steps to the two examples in the handout (and similar problems) (Section 1 (Intro to MP))
• using duality and complementary slackness to verify optimality (Section 3 (Duality))
• derive the range of the RHS until the optimal basis start changing and the corresponding range of the optimal value (Section 4 (Sensitivity))
• describe economic interpretation of shadow price and perform sensitivity analysis on LP
• apply parametric programming to calculate the change of the objective value (and the optimal solutions) when more than one objective coefficient or the RHS are changed continuously (Section 4 (Sensitivity))
• describe the idea behind the Simplex method (Section 2 (Simplex Method))
• describe 5 steps for formulating a mathematical programming problem (Section 1 (Intro to MP))
• formulate various problems using LP, IP and NLP
• model either-or, set-up costs and other condition-based constraints using binary variables and constraints in the context of integer programming (Section 5 (Integer Programming))
• apply the dictionary and full tableau Simplex method to solve LP problems in standard form given a starting BFS (Section 2 (Simplex Method))
• describe general branch and bound method (Section 5 (Integer Programming))
• obtain (lower) bound on IP using rounding (Section 5 (Integer Programming))
• describe duality theorems and complementary slackness
• write down general form and standard form of LP problems (Section 2 (Simplex Method))
• derive (upper) bound on IP using LP relaxation (Section 5 (Integer Programming))
• describe the method for finding initial BFS and to avoid cycling (Section 2 (Simplex Method))
• write the general form of a mathematical programming problem (Section 1 (Intro to MP))
• define basic solution, basic feasible solution, optimal BSF of LP problems in standard form (Section 2 (Simplex Method))
• describe branch and bound methods
• interpret shadow price and primal-dual pair in terms of profit maximization vs. resource scarcity (Section 3 (Duality))
• describe relationships for primal-dual pair in terms of optimality, feasibility and boundedness (Section 3 (Duality))
• describe situations when sensitivity analysis is used (Section 4 (Sensitivity))
• distinguish between LP, IP, and NLP (Section 1 (Intro to MP))
• describe and prove week duality theorem (Section 3 (Duality))
• describe the motivation for using duality and formulate the dual problem given the primal (Section 3 (Duality))

Linear Programming.

• Model construction and modelling issues.
• Simplex method: two-phase algorithm, dual simplex method, network simplex method.
• Duality: motivation and definitions, duality theorem, complementary slackness, optimality
• testing.
• Sensitivity analysis: ranging for objective function coefficients and right hand-side constraints,
• economic interpretation and applications.
• Parametric programming. Integer Programming.
• Modelling techniques using zero-one variables.
• Branch and bound algorithm for integer programming.

Five 2-hour lectures

Two 1-hour class sessions

Two 2-hour computer sessions

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.