The Doctor Rostering Problem - Asser Fahrenholz

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Chapter 3. The model and design 13 3.4 The objective function The constraints described in the two previous sections is the complete list of constraints as the medical practice has described them. The objective function value of a schedule s is the collective sum of the above described soft constraints and is denoted z(s): min z(s) = δB0 + δB2 i ⎛ yik0 ⎝max k ⎛ + δB3 ⎝ + δB5 + δB6b i,j i i + δB1 i,j∈sa i yik1 xijk − min k ⎞ ⎠ + δB4 yijk3 yik5 yik6b + δB6a i ∀k i i,j∈sa yik4 yik6a xijk ⎞ ⎠ (Z(S)) An objective function value of 0 means that we have found an optimal solution for any DRP problem. If the objective function value is positive, it can still be optimal, but under any circumstances means that one or more soft constraints are invalid. Leaving the weight, δ, of all the soft constraints equal to 1 means that we can derive from an objective function value of i.e. 4, that soft constraints are invalid in 4 cases. If we choose to alter the weights of the rules, the same still applies, but the transparency of the objective function value decreases. The objective function thus determines the value of solutions for the DRP and is used by optimal solvers such as GAMS (see chapter 5) and the heuristics in the next chapter, which will describe how solutions can be found for the DRP. 3.4.1 Problem size The amount of variables with 4 shifts per day, 6 doctors and 100 working days:
Chapter 3. The model and design 14 |V ars| = |Shifts| · |Doctors| · |Schedule| = 4 · m · n = 4 · 6 · 100 = 2400 Given 10 constraints, the total number of constraints for this problem becomes: |Equations| = |Shifts| · |Doctors| · |Schedule| · |Constraints| = 4 · m · n · 10 = 4 · 6 · 100 · 10 = 24000 Commonly, problems are big if they have more than 2500 variables and 10000 equations. This is not ruling out that any larger problems can not be solved, as techniques (i.e. Cutting-plane method, Branch and cut) exist that, using the constraints, can limit the search region of the problem. A topic not addressed in this project.
Page 1 and 2: TECHNICAL UNIVERSITY OF DENMARK The
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Page 5 and 6: Acknowledgements Foremost, I would
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Bibliography 118 [10] Michael R. Ga
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