The Doctor Rostering Problem - Asser Fahrenholz

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Chapter 3. The model and design 9 ⎧ 1 if on day i, doctor k has a shift ⎪⎨ j outside the house. It follows that bijk = the doctor is not available for the surrounding shifts ⎪⎩ 0 otherwise ⎧ ⎪⎨ sij = ⎪⎩ ⎧ 1 if on day i, doctor k has ⎪⎨ requested shift j off. The doctor is then cijk = available for the surrounding shifts ⎪⎩ 0 otherwise l if on day i, shift j must be assigned 1+l doctors. This happens on noonshifts following weekends and holidays, and afternoonshifts preceding holidays 0 otherwise The index j defines: 1: Night shift, 2: Noon shift, 3: Afternoon shift, 4: Evening shift. Furthermore, shift j is said to be an assignment shift (as) if it doesn’t lie on a holiday, nor a weekend and is not a night- nor evening shift. Let n be the number of days in the problem, and m be the number of doctors: i ={1, 2 . . . n} k ={1, 2 . . . m} In the following, I assume the schedule starts on a Monday, that the index of this day is 1 and that the schedule ends on a Sunday and the index of this day is n. I also define the sets of days: 3.2 Hard constraints M : i%7 = 1 ∀i F : i%7 = 5 ∀i This section describes the constraints that must be satisfied for the solution to be feasible:
Chapter 3. The model and design 10 H0 No doctor may take two shifts in a row: xijk + x i(j+1)k ≤ 1 ∀ i, j = {1, 2, 3}, k (H0) H1 After an evening shift, a doctor cannot have a noon shift on the following day: bi4k + x (i+1)2k ≤ 1 ∀ i = {1 . . . n − 1}, k (H1) H2 Following a night shift, a doctor cannot have a noon shift nor an afternoon shift: bi1k + xi2k + xi3k ≤ 1 ∀ i, k (H2) H3a Doctors having a shift outside the house cannot have one in the house simultane- ously: H3b Requests for a shift off must be granted: H4 Demands must be met: xijk + bijk ≤ 1 ∀ i, j, k (H3a) xijk + cijk ≤ 1 ∀ i, j, k (H3b) m xijk = 1 + sij ∀ i, j (H4) k=1 H5 No afternoon shift before an evening shift: 3.3 Soft constraints xi3k + bi4k ≤ 1 ∀ i, k (H5) The soft constraints does not require validity for the solution to be feasible, though the quality of the solution improves the fewer soft constraints are invalid. Soft constraints are commonly modelled in the evaluation or objective function, as a sum of the penalties that arise when not satisfying the individual constraints. Each soft constraint has a weight δ attached, allowing the user to prioritise certain constraints over others. I now introduce the following variables:
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Appendix C. Tests 102 Test 3 RCL 1
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Bibliography 118 [10] Michael R. Ga
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The Doctor Rostering Problem - Asser Fahrenholz

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