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Lecture 16: Network Models/ Scheduling Assignments

In this lecture, we explore the mathematical modeling of scheduling assignments using examples from Olympic swimming and sales representatives. We discuss how to effectively allocate resources, whether it’s positioning Michael Phelps in various swimming events or assigning sales reps to districts based on commission. The concepts focus on minimizing costs and meeting constraints for optimal resource utilization, including the transport of machines to tasks and umpire assignments in baseball. We also touch upon the significance of estimating opportunity costs and managing expertise in practical applications.

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Lecture 16: Network Models/ Scheduling Assignments

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  1. Lecture 16: Network Models/ Scheduling Assignments AGEC 352 Spring 2012 – March 26 R. Keeney

  2. Olympic Swimming • Michael Phelps is the world’s greatest swimmer • If you need to win a race, you pick him • If you need to win a relay race, you pick him but where do you use him? • Medley swimming • Backstroke, Breaststroke, Butterfly, Freestyle

  3. Medley Swimming (U.S. 2008)

  4. Sales Reps / Districts If these people are paid on commission is Seller A going to be happy about being the highest rated rep?

  5. Assignment Problems • Setup is identical to the transportation problem we have considered • Sources are people or things to be assigned • Destinations are jobs or roles to be filled • Other applications • Machines to tasks • E.g. Airplanes to routes • Sudoku (number to a cell)

  6. Example • Umpiring in American League Baseball • 14 teams • 7 umpiring crews assigned to 7 games • Minimum travel costs for crews going to games, other constraints • No afternoon games in city B if you worked a night game in city A the previous day • Day off required if leaving Pacific Time Zoneor Canada • Crew must not work more than one week straight on the same team’s games

  7. Case • Mathematical allocation of ‘n’ objects or agents to ‘n’ tasks • Agents/objects are indivisible, one task only • Autopower Company audit of assembly plants (destinations from transport) • Leipzig, Nancy, Liege, Tilburg • VP’s to manage audit • Finance, Marketing, Operations, Personnel

  8. Considerations on Costs • Expertise relative to problem areas of different plants • Time demand of VP • Language ability of VP

  9. Estimating the costs • Need something reliable for estimating the opportunity cost of each VP in each assignment • E.g. A Dutch speaker in the French plant may require a translator with him full-time • E.g. The finance VP may need an human resources specialist to assist her • Other measures: • Swimming times, skill tests (ASVAB)

  10. Solving • Simplex LP in Excel or by hand • For small problems, enumeration • An ‘n’ sized assignment problem has n! possible solutions • n! is called a factorial, multiply all the integers up to n together to find the factorial • E.g. 4! = 1*2*3*4 = 24

  11. Sales Reps / Districts If these people are paid on commission is Seller A going to be happy about being the highest rated rep?

  12. Setup • RHS values are always 1 • Sources (people) • The total jobs must be <= 1 • Destinations (jobs) • The number in the job must be >= 1 • Balanced: Need someone for each job, everyone needs a job

  13. Algebraic Form

  14. Notes • Decision variables will be zero or one. • Integers (but you don’t need integer constraints) • Transportation problem with supply at each source and demand at each destination equal to one.

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