Service Optimization

Abstract

This chapter provides an overview of Operations Research and its mathematical models for planning problems arising in the area of services. For a better understanding, a basic introduction into the field of Operations Research is given. Different examples from service areas are presented. Several methods for solving the mathematical problems are discussed. In addition, the optimization software called IBM ILOG CPLEX Optimization Studio is presented that can be used to determine an optimal solution for a mathematical problem. The use of simulation in the area of Operations Research is also discussed and the software AnyLogic is used to provide examples.

7.1 Review Section

• 1. (Foundations) Explain the idealized planning process in your own words and use an example to illustrate the steps.

• 2. (Foundations) Give a generic formulation for a linear and a mixed integer linear program.

• 3. (Foundations) Explain the difference between an exact approach and a heuristic and give examples for both.

• 4. (Bin Packing) Find a sequence of items such that the two heuristics First Fit and Best Fit yield different bin assignments for those items. Try to use as few items as possible.

• 5. (Queuing) Consider a doctor’s practice operating under the walk-in policy just as in the example. Let us assume the following parameters: arrival rate λ = 5 per hour (exponentially distributed inter-arrival times), service rate μ = 6 per hour (exponentially distributed service times) and waiting room capacity K − 1 = 4. Is the waiting room capacity high enough if the probability that a patient is rejected should be less than 0. 05? If this is not the case what capacity should the waiting room have at least?

  Derive the formulas for L _q and W _q in the M∕M∕1∕K case. Hint: To compute L _q use the probabilities $p_{i},i \in \{ 0,\ldots,K\}$ already calculated. Then apply Little’s Law to derive the formula for W _q .

• 6. (Location Problems) Find examples for the location problems either from literature or your everyday life. Explain why you think the model fits to the problem. If possible, choose different service domains.

• 7. (TSP) Apply the savings heuristic to the home healthcare example in Sect. 7.6.2.

• 8. (VRPTW) Add the following two constraints to the VRPTW formulation:

  • Not more than P customers with $P \geq \frac{\vert N\vert } {\vert K\vert }$ shall be assigned to each vehicle k ∈ K.

  • Each route should end at the returning depot before the closing time l _n+1.

• 9. (OPL) Implement all the formulations of this chapter using either the data given in the examples or come up with new examples. Separate the data from the model by using a.mod-file for the model and a.dat-file for the data.

• 10. (Simulation) Implement a simple queueing system in AnyLogic with one server and a limited queue capacity. Use the following parameters: exponential inter-arrival times with rate λ = 1 per minute, exponential service times with rate μ = 1. 2 per minute and a queue capacity of 3. Collect statistics on the average queue length and the average waiting time over a long period of time. Compare these values to the results you obtain using the formulas from the section Queueing Theory.

7.2 Key Terms

Operations Research :

Operations Research is the discipline of applying advanced analytical methods, i.e., optimization methods, to help make better decisions.

Linear Programming :

Linear Programming deals with solving linear programs, i.e, optimization problems where decision variables take real values and the objective function and the constraints are linear.

Integer Programming :

Integer Programming is concerned with solving integer and mixed integer programs, i.e., optimization models where some or all of the decision variables are restricted to integer values.

Simplex Algorithm :

The simplex algorithm is one of the most important algorithms to determine an optimal solution to a linear program.

Branch-and-Bound Algorithm :

The Branch-and-Bound algorithm is one of the most important algorithms to determine an optimal solution to an integer (linear) program.

Heuristic :

A heuristic is an algorithm that finds a solution to an optimization problem which is in general not optimal.

Simulation :

Simulation is a methodology to imitate the most important real world processes of a system in order to investigate complex dynamic systems and to support decisions.

7.3 Further Reading

George Nemhauser and Laurence Wolsey. Integer and Combinatorial Optimization. Wiley, 1999.

Frederick Hillier and Gerald Lieberman. Introduction to Operations Research. McGraw Hill, 2001.

Andrei Borshchev. The Big Book of Simulation Modeling: Multimethod Modeling with AnyLogic6. AnyLogic North America, 2013.