Optimization and Decisions

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Optimization and Decisions

• Dolores Romero Morales
• University Reader in Operations Research
• University of Oxford

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Contents

• Taking decisions
• Optimization
• Linear programming
• Integer/multi-objective/stochastic/ programming
• Conclusions

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Taking decisions

• We are always confronted with decisions
• Shall I take my umbrella today?
• Shall I invest in emerging countries?
• Which route should the truck follow to deliver
  the groceries sold online?
• What is the optimal schedule for Major League
  Baseball?

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Optimization

• Whole battery of mathematical approaches for
  solving certain decision problems
• The number of alternatives is infinite or finite
  but big to apply enumeration
• Mathematical formulation of the problem
• Systematic way of solving decision problems

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Elements of optimization problems

• Decision
• Alternatives
• Constraints
• Objective
• Parameters

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Example

• Company makes
• products closets and tables,
• from
• raw materials plastics, wood, iron

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Data
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Elements for example

• Decisions amount of closets and tables
  (production planning)
• Alternatives (wait a second and you will see)
• Constraints limited availability of raws
• Objective maximize net profit

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Decision problem
maximize net profit
availability plastic
wood
iron
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How to solve this problem?

• There are many sophisticated packages Lingo,
  ILOG, Dash
• We will keep it simple and use the solver of
  EXCEL

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The Solver of EXCEL
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The optimal solution
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History of Linear Programming

• Kantorovitch rudimentary version (1939)
• Koopmans Game theory (1947)
• Dantzig Simplex method (1947)

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Applications

• Resource Allocation
• Portfolio Optimization
• Scheduling
• Routing
• Task Assignment
• Timetabling

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General form

• n products (j1,,n)
• m raw materials (i1,,m)
• Market price of raws ?i
• Available quantities of raws bi
• Revenue products ?j
• Net revenue products
• Decisions xj (units of product j produced)

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Linear program
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Another way to view problem

• Lost opportunity costs wi per unit of raw i
• Constraints
• Value at least market price (otherwise we sell to
  the market) wi ??i
• Competitors should not be able to buy our raws
  and produce the products cheaper
• Use yi wi - ?i

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Dual linear program
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Primal and Dual
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Sensitivity analysis

• What happens if c changes? (?)
• What happens if b changes? (?)
• What happens if A (aij ) changes? (?)

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Ranging

• What happens if a coefficient of b or c changes?
• Better said for what values of these changes
  does the current optimal basis remain optimal?

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Sensitivity analysis with Solver
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More than linear programming

• Linear Programming is the simplest we can find in
  optimization
• We also have
• Quadratic Programming
• Integer Programming
• Combinatorial Programming
• Multi-objective Programming
• Stochastic Programming

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Conclusions

• You may be confronted with a decision problem
  that can be written as an optimization problem
• There are commercial packages that can solved
  these problems very efficiently
• In some cases, you can derive sensitivity
  analysis results

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Exercise

• Steel company should allocate a weeks time
• Company produces bands and coils from slabs
• Objective maximize profit
• Constraints
• A week has 40 working hours
• Upper bound on production of bands and coils

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Data
Available time 40 hours
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Linear programming model
Hours dedicated to produce bands
Hours dedicated to produce coils