Deterministic OR Methods for Data Scientists

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

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

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.