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IE 312 Optimization

IE 312 Optimization. Siggi Olafsson 3018 Black olafsson@iastate.edu. Optimization. In this class you will learn to solve industrial engineering problems by modeling them as optimization problems You will understand common optimization algorithms for solving such problems

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IE 312 Optimization

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  1. IE 312 Optimization Siggi Olafsson 3018 Black olafsson@iastate.edu

  2. Optimization • In this class you will learn to solve industrial engineering problems by modeling them as optimization problems • You will understand common optimization algorithms for solving such problems • You will learn the use of software for solving complex problems, and you will work as part of a team to address complex ill-structured problems with multiple solutions

  3. Optimization Formulation Decision variables: Integer programming problem: This is a variant of what is called the knapsack problem

  4. How about the traveling salesman? What is the shortest route that visits each city exactly once?

  5. The Process Problem Problem Formulation Model Analysis Conclusions

  6. Problem Formulation • What is the objective function? • Maximize profit, • Minimize inventory, ... • What are the decision variables? • Capacity, routing, production and stock levels • What are the constraints? • Capacity is limited by capital • Production is limited by capacity

  7. Mathematical Programs Decision variables

  8. Analysis • Optimization Algorithm • Computer Implementation • Excel (or other spreadsheet) • Optimization software (e.g., LINDO) • Modeling software (e.g., LINGO) Increasing Complexity

  9. Optimization Algorithms • Find an initial solution • Loop: • Look at “neighbors” of current solution • Select one of those neighbors • Decide if to move to selected solution • Check stopping criterion

  10. In This Class You Will … • Learn problem formulation (modeling) • Learn selecting appropriate algorithms • Learn using those algorithms

  11. Academic Honesty You are expected to be honest in all of your actions and communications in this class. Students suspected of committing academic dishonesty will be referred to the Dean of Students Office as per University policy. For more information regarding Academic Misconduct see http://www.dso.iastate.edu/ja/academic/misconduct.html

  12. Professionalism You are expected to behave in a professional manner during this class.

  13. The OR Process Problem (System) Problem Formulation Model Analysis Conclusions

  14. Problem Formulation • Capture the essence of the system • Variables • Relationships • Ask ourselves: • What is the objective? • What are the decision variables? • What are the constraints?

  15. Mortimer Middleman • Wholesale diamond business • sale price $900/carat • average order 55 carats/week • International market • purchase price $700/carat • minimum order 100 carats/trip • trip takes one week and costs $2000

  16. Inventory Problem • Cost of keeping inventory • insurance • tied up capital • 0.5% of wholesale value/week • Cost of not keeping inventory • lost sales (no backordering)

  17. The Current Situation • Holding cost of $38,409 in past year • Unrealized profits of $31,600 • Resupply travel cost $24,000 • Total of $94,009 • Can we do better? • How do we start answering that question?

  18. Problem Formulation • What are the decision variables? • When should we order? • Reorder point r (quantity that trigger order) • How much should we order? • Order quantity q • Note that this grossly simplifies the reality of Mortimer’s life!

  19. Problem Formulation • What are the constraints? • What is the objective? • Minimize cost • Holding cost • Replenishment cost • Lost-sales cost

  20. Relationships(System Dynamics) • Assumptions • Constant-rate demand • Is this a strong or weak assumption? • Is this assumption realistic?

  21. System Dynamics With safety stock No safety stock or lost sales With lost sales r r r 1 week 1 week 1 week Why might we get lost sales despite our planning? Can we ignore this?

  22. Assuming No Lost Sales • No lost sales implies r  55 • Cycle length • Average inventory

  23. Optimization Model Minimize Subject to

  24. Solving the Problem (Analysis) • Feasible solution • Any set of values that satisfies the constraints • Optimal solution • A feasible solution that has the best possible objective function value • Algorithms: • Find a feasible solution • Try to improve on it

  25. A Solution for Mortimer • What are some feasible solutions? • The smallest feasible value of r is 55 • What happens if we change it to 56? • Increased holding cost! • ‘Clearly’ the optimal replenishment point is

  26. New Optimization Problem Minimize Subject to Differentiate the objective function and set equal to zero:

  27. Optimal Solution Optimum

  28. Economic Order Quantity (EOQ) Cost of replenishment Classical result in inventory theory: Holding cost Lead time (replenishment) Weekly demand

  29. Discussion • Sensitivity Analysis • Exploring how the results change if parameters change • Why is this important? • Closed-Form Solutions • Final result a simple formula in terms of the input variables • Very fast computationally • Makes sensitivity analysis very easy

  30. Evaluating the Model • Tractability of Model • Ease by which we can analyze the model • Validity • The degree by which inferences drawn from model also hold for actual system • Trade-Off! • A “Good” Model is Tractable and Valid

  31. Model Validity • A model is valid for a specific purpose • It only has to answer the questions we ask correctly! • Recipe for a Good Model • Start with a simple model • Evaluate assumptions • Does adding complexity change the outcome? • Relax assumption/add constraint

  32. Validity of Our Model • Customer demand (average 55)

  33. Simulation Analysis • Using the same conceptual model, we can simulate the performance using the historical data • Check if Mortimer is due with a shipment • Check if a new trip is warranted • Reduce inventory by actual demand • Simulation of our policy q=251, r=55 implies a cost of $108,621

  34. Evaluation of Cost • Our predicted cost is • Mortimer’s current cost is $94,009 • The simulated cost is $108,621

  35. Simulation Validity • Which model should we trust: • The simulation model that predicts a performance of $108,621, or • the EOQ model that predicts a performance of $45,630? • Examine the assumptions made: • EOQ model: constant demand • Simulation: future identical to past • In general, simulation models have a high degree of validity

  36. Descriptive vs Prescriptive • How about tractability? • What does the simulation tell us? How easy is it to do sensitivity analysis? • Descriptive models • Only evaluate an alternative or solution • Prescriptive models • Suggest a good (or optimal) alternative

  37. Numerical Search • We can use our descriptive simulation model to look for a better solution • Algorithm: • Start with an initial (good) solution • Check ‘similar’ solution (neighbors) • Select one of the neighbors • Repeat until a stopping criterion is satisfied

  38. Search for Reorder Point $64,242 $63,054 $63,254 $108,421 $108,621

  39. Search for Order Quantity $63,054 $72,781 $95,193 Best Point Found

  40. Evaluation • We have found a solution q=251, r=85 with cost $63,054 • Better than current ($94,009) and previously obtained solution ($108,621) • Is this the best solution? • We don’t know! • What if we started with a different initial solution?

  41. A New Search 59,539 Initial Solution 56,904 58,467 54,193 56,900 59,732 Best Point Found

  42. Heuristic vs Optimal • Optimal Solution • Solution that is gives the best objective function value • Heuristic Solution • A ‘good’ feasible solution • Should we demand optimality? • Inaccurate search vs approximation in model

  43. Deterministic vs Stochastic Models • Our deterministic simulation model assumed future identical to past • Not true! Demand is random • Stochastic simulation fits a random distribution to the historical data • The world is stochastic • Why not always use stochastic models? • Tractability versus validity

  44. Mathematical Programming • Deterministic models • Assume all data known with certainty • Validity • Often produce valid results • Tractability • Easier than stochastic models • Known as mathematical programming

  45. Problem Formulation • Decision variables • Constraints • Variable-type constraints • Main constraints • Objective function

  46. Two Crude Petroleum $20 9000 barrels/day 2000 barrels Gasoline Saudi Arabia 1500 barrels Jet Fuel Refinery Venezuela 500 barrels Lubricants $15 6000 barrels/day

  47. Oil Processing Data • Barrel of Saudi Crude • 0.3 barrels of gasoline • 0.4 barrels of jet fuel • 0.2 barrels of lubricant • Barrel of Venezuela Crude • 0.4 barrels of gasoline • 0.2 barrels of jet fuel • 0.3 barrels of lubricant

  48. Decision Variables • What can we control or decide upon? • How much of each crude • Thus, define • Clearly define what you mean!

  49. Constraints • Variable-type constraints • Domain of decision variables (most often a range) • Very simple here:

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