0

Dear fellow programmers,

I want to know how to schedule activities based on availability (see input). There are people available on days where the number of available hours is more than zero. We want our schedule to be as close to zero as possible (efficient planning). This means that when we schedule a new project for a client that takes five activities, we need our algorithm to reduce the number of available hours by the duration of the activity for that particular project day. Some activities take 0,5 hours to complete and others 1 hour to complete. I tried to formulate an initial version of the model with the variables A, B, C, D, and E as the activities in the right sequence (yet) with only one employee (called '0' since I will add more employees in the future, but I do not want to start with a too complex situation in the beginning) and the day that the activity is scheduled. For the first and fifth activity, the patient is allowed to give a preference. For the second, third and fourth activity, the activity can be scheduled any time. The availability variable is here a list of the available hours of employee AW (see input).

days = range(1,32)
preferences = [4,11,18,25]

A = LpVariable.dicts(name = "Intake", indexs = [(0, v) for v in preferences], lowBound = 0, upBound = 1, cat = "Continuous")
B = LpVariable.dicts(name = "Write a transcript of intake", indexs = [(0, d) for d in days], lowBound = 0, upBound = 1, cat =  "Continuous")
C = LpVariable.dicts(name = "Create treatment plan", indexs = [(0, d) for d in days], lowBound = 0, upBound = 0.5, cat = "Continuous")
D = LpVariable.dicts(name = "Schedule a meeting", indexs = [(0, d) for d in days], lowBound = 0, upBound = 0.5, cat = "Continuous")
E = LpVariable.dicts(name = "Discuss treatment plan with patiënt", indexs = [(0, v) for v in preferences], lowBound = 0, upBound = 1, cat = "Continuous")

############################## OBJECTIVE ##############################################


# define the model to maximize the workday
schedule = LpProblem(name="Maximize_Schedule", sense=LpMaximize)

schedule += lpSum(A[(0,v)] for v in preferences) + lpSum(B[(0,d)] for d in days) + lpSum(C[(0,d)] for d in days) + lpSum(D[(0,d)] for d in days) + lpSum(E[(0,v)] for v in preferences)

############################## CONSTRAINTS ############################################

# Less than available hours

for v in preferences:
    schedule += A[0,v] + B[0,v] + C[0,v] + D[0,v] + E[0,v] <= availability[v-1]

for d in days:
    schedule += B[0,d] + C[0,d] + D[0,d] <= availability[d-1]


# There is no more than 4 hours to divide

schedule += lpSum(A[(0,v)] for v in preferences) + lpSum(B[(0,d)] for d in days) + lpSum(C[(0,d)] for d in days) + lpSum(D[(0,d)] for d in days) + lpSum(E[(0,v)] for v in preferences) <= 4

# Every activity must be in the solution

schedule += lpSum(A[(0,v)] for v in preferences) == 1

schedule += lpSum(B[(0,d)] for d in days) == 1

schedule += lpSum(C[(0,d)] for d in days) == 0.5

schedule += lpSum(D[(0,d)] for d in days) == 0.5

schedule += lpSum(E[(0,v)] for v in preferences) == 1


# An activity cannot be less than zero

for v in preferences:
    schedule += A[(0,v)] >= 0
    schedule += E[(0,v)] >= 0

for d in days:
    schedule += B[(0,d)] >= 0
    schedule += C[(0,d)] >= 0
    schedule += D[(0,d)] >= 0


status = schedule.solve()

The model provides me with the following output:

Create_treatment_plan_(0,_4) 0.5
Discuss_treatment_plan_with_patiënt_(0,_11) 1.0
Intake_(0,_11) 1.0
Schedule_a_meeting_(0,_4) 0.5
Write_a_transcript_of_intake_(0,_4) 1.0

Already a lot of constraints are satisfied, but we can see that there is no precedence in the outcome ('intake' is planned before 'Write a transcript of intake'). I have no idea how to implement a precedence constraint in my LP since I am not working with starting time and ending time but with availability. Could you please help me looking for a solution? On the bottom of this page you can find the full version of the my LP formulation in Pulp.

Kind regards,

Tom

Maximize_Schedule:
MAXIMIZE
1*Create_treatment_plan_(0,_1) + 1*Create_treatment_plan_(0,_10) + 1*Create_treatment_plan_(0,_11) + 1*Create_treatment_plan_(0,_12) + 1*Create_treatment_plan_(0,_13) + 1*Create_treatment_plan_(0,_14) + 1*Create_treatment_plan_(0,_15) + 1*Create_treatment_plan_(0,_16) + 1*Create_treatment_plan_(0,_17) + 1*Create_treatment_plan_(0,_18) + 1*Create_treatment_plan_(0,_19) + 1*Create_treatment_plan_(0,_2) + 1*Create_treatment_plan_(0,_20) + 1*Create_treatment_plan_(0,_21) + 1*Create_treatment_plan_(0,_22) + 1*Create_treatment_plan_(0,_23) + 1*Create_treatment_plan_(0,_24) + 1*Create_treatment_plan_(0,_25) + 1*Create_treatment_plan_(0,_26) + 1*Create_treatment_plan_(0,_27) + 1*Create_treatment_plan_(0,_28) + 1*Create_treatment_plan_(0,_29) + 1*Create_treatment_plan_(0,_3) + 1*Create_treatment_plan_(0,_30) + 1*Create_treatment_plan_(0,_31) + 1*Create_treatment_plan_(0,_4) + 1*Create_treatment_plan_(0,_5) + 1*Create_treatment_plan_(0,_6) + 1*Create_treatment_plan_(0,_7) + 1*Create_treatment_plan_(0,_8) + 1*Create_treatment_plan_(0,_9) + 1*Discuss_treatment_plan_with_patiënt_(0,_11) + 1*Discuss_treatment_plan_with_patiënt_(0,_18) + 1*Discuss_treatment_plan_with_patiënt_(0,_25) + 1*Discuss_treatment_plan_with_patiënt_(0,_4) + 1*Intake_(0,_11) + 1*Intake_(0,_18) + 1*Intake_(0,_25) + 1*Intake_(0,_4) + 1*Schedule_a_meeting_(0,_1) + 1*Schedule_a_meeting_(0,_10) + 1*Schedule_a_meeting_(0,_11) + 1*Schedule_a_meeting_(0,_12) + 1*Schedule_a_meeting_(0,_13) + 1*Schedule_a_meeting_(0,_14) + 1*Schedule_a_meeting_(0,_15) + 1*Schedule_a_meeting_(0,_16) + 1*Schedule_a_meeting_(0,_17) + 1*Schedule_a_meeting_(0,_18) + 1*Schedule_a_meeting_(0,_19) + 1*Schedule_a_meeting_(0,_2) + 1*Schedule_a_meeting_(0,_20) + 1*Schedule_a_meeting_(0,_21) + 1*Schedule_a_meeting_(0,_22) + 1*Schedule_a_meeting_(0,_23) + 1*Schedule_a_meeting_(0,_24) + 1*Schedule_a_meeting_(0,_25) + 1*Schedule_a_meeting_(0,_26) + 1*Schedule_a_meeting_(0,_27) + 1*Schedule_a_meeting_(0,_28) + 1*Schedule_a_meeting_(0,_29) + 1*Schedule_a_meeting_(0,_3) + 1*Schedule_a_meeting_(0,_30) + 1*Schedule_a_meeting_(0,_31) + 1*Schedule_a_meeting_(0,_4) + 1*Schedule_a_meeting_(0,_5) + 1*Schedule_a_meeting_(0,_6) + 1*Schedule_a_meeting_(0,_7) + 1*Schedule_a_meeting_(0,_8) + 1*Schedule_a_meeting_(0,_9) + 1*Write_a_transcript_of_intake_(0,_1) + 1*Write_a_transcript_of_intake_(0,_10) + 1*Write_a_transcript_of_intake_(0,_11) + 1*Write_a_transcript_of_intake_(0,_12) + 1*Write_a_transcript_of_intake_(0,_13) + 1*Write_a_transcript_of_intake_(0,_14) + 1*Write_a_transcript_of_intake_(0,_15) + 1*Write_a_transcript_of_intake_(0,_16) + 1*Write_a_transcript_of_intake_(0,_17) + 1*Write_a_transcript_of_intake_(0,_18) + 1*Write_a_transcript_of_intake_(0,_19) + 1*Write_a_transcript_of_intake_(0,_2) + 1*Write_a_transcript_of_intake_(0,_20) + 1*Write_a_transcript_of_intake_(0,_21) + 1*Write_a_transcript_of_intake_(0,_22) + 1*Write_a_transcript_of_intake_(0,_23) + 1*Write_a_transcript_of_intake_(0,_24) + 1*Write_a_transcript_of_intake_(0,_25) + 1*Write_a_transcript_of_intake_(0,_26) + 1*Write_a_transcript_of_intake_(0,_27) + 1*Write_a_transcript_of_intake_(0,_28) + 1*Write_a_transcript_of_intake_(0,_29) + 1*Write_a_transcript_of_intake_(0,_3) + 1*Write_a_transcript_of_intake_(0,_30) + 1*Write_a_transcript_of_intake_(0,_31) + 1*Write_a_transcript_of_intake_(0,_4) + 1*Write_a_transcript_of_intake_(0,_5) + 1*Write_a_transcript_of_intake_(0,_6) + 1*Write_a_transcript_of_intake_(0,_7) + 1*Write_a_transcript_of_intake_(0,_8) + 1*Write_a_transcript_of_intake_(0,_9) + 0
SUBJECT TO
_C1: Create_treatment_plan_(0,_4) + Discuss_treatment_plan_with_patiënt_(0,_4)
 + Intake_(0,_4) + Schedule_a_meeting_(0,_4)
 + Write_a_transcript_of_intake_(0,_4) <= 3

_C2: Create_treatment_plan_(0,_11)
 + Discuss_treatment_plan_with_patiënt_(0,_11) + Intake_(0,_11)
 + Schedule_a_meeting_(0,_11) + Write_a_transcript_of_intake_(0,_11) <= 4

_C3: Create_treatment_plan_(0,_18)
 + Discuss_treatment_plan_with_patiënt_(0,_18) + Intake_(0,_18)
 + Schedule_a_meeting_(0,_18) + Write_a_transcript_of_intake_(0,_18) <= 5

_C4: Create_treatment_plan_(0,_25)
 + Discuss_treatment_plan_with_patiënt_(0,_25) + Intake_(0,_25)
 + Schedule_a_meeting_(0,_25) + Write_a_transcript_of_intake_(0,_25) <= 6

_C5: Create_treatment_plan_(0,_1) + Schedule_a_meeting_(0,_1)
 + Write_a_transcript_of_intake_(0,_1) <= 0

_C6: Create_treatment_plan_(0,_2) + Schedule_a_meeting_(0,_2)
 + Write_a_transcript_of_intake_(0,_2) <= 5

_C7: Create_treatment_plan_(0,_3) + Schedule_a_meeting_(0,_3)
 + Write_a_transcript_of_intake_(0,_3) <= 7

_C8: Create_treatment_plan_(0,_4) + Schedule_a_meeting_(0,_4)
 + Write_a_transcript_of_intake_(0,_4) <= 3

_C9: Create_treatment_plan_(0,_5) + Schedule_a_meeting_(0,_5)
 + Write_a_transcript_of_intake_(0,_5) <= 0

_C10: Create_treatment_plan_(0,_6) + Schedule_a_meeting_(0,_6)
 + Write_a_transcript_of_intake_(0,_6) <= 0

_C11: Create_treatment_plan_(0,_7) + Schedule_a_meeting_(0,_7)
 + Write_a_transcript_of_intake_(0,_7) <= 6

_C12: Create_treatment_plan_(0,_8) + Schedule_a_meeting_(0,_8)
 + Write_a_transcript_of_intake_(0,_8) <= 0

_C13: Create_treatment_plan_(0,_9) + Schedule_a_meeting_(0,_9)
 + Write_a_transcript_of_intake_(0,_9) <= 6

_C14: Create_treatment_plan_(0,_10) + Schedule_a_meeting_(0,_10)
 + Write_a_transcript_of_intake_(0,_10) <= 1

_C15: Create_treatment_plan_(0,_11) + Schedule_a_meeting_(0,_11)
 + Write_a_transcript_of_intake_(0,_11) <= 4

_C16: Create_treatment_plan_(0,_12) + Schedule_a_meeting_(0,_12)
 + Write_a_transcript_of_intake_(0,_12) <= 0

_C17: Create_treatment_plan_(0,_13) + Schedule_a_meeting_(0,_13)
 + Write_a_transcript_of_intake_(0,_13) <= 0

_C18: Create_treatment_plan_(0,_14) + Schedule_a_meeting_(0,_14)
 + Write_a_transcript_of_intake_(0,_14) <= 4

_C19: Create_treatment_plan_(0,_15) + Schedule_a_meeting_(0,_15)
 + Write_a_transcript_of_intake_(0,_15) <= 0

_C20: Create_treatment_plan_(0,_16) + Schedule_a_meeting_(0,_16)
 + Write_a_transcript_of_intake_(0,_16) <= 4

_C21: Create_treatment_plan_(0,_17) + Schedule_a_meeting_(0,_17)
 + Write_a_transcript_of_intake_(0,_17) <= 7

_C22: Create_treatment_plan_(0,_18) + Schedule_a_meeting_(0,_18)
 + Write_a_transcript_of_intake_(0,_18) <= 5

_C23: Create_treatment_plan_(0,_19) + Schedule_a_meeting_(0,_19)
 + Write_a_transcript_of_intake_(0,_19) <= 0

_C24: Create_treatment_plan_(0,_20) + Schedule_a_meeting_(0,_20)
 + Write_a_transcript_of_intake_(0,_20) <= 0

_C25: Create_treatment_plan_(0,_21) + Schedule_a_meeting_(0,_21)
 + Write_a_transcript_of_intake_(0,_21) <= 4

_C26: Create_treatment_plan_(0,_22) + Schedule_a_meeting_(0,_22)
 + Write_a_transcript_of_intake_(0,_22) <= 0

_C27: Create_treatment_plan_(0,_23) + Schedule_a_meeting_(0,_23)
 + Write_a_transcript_of_intake_(0,_23) <= 4

_C28: Create_treatment_plan_(0,_24) + Schedule_a_meeting_(0,_24)
 + Write_a_transcript_of_intake_(0,_24) <= 7

_C29: Create_treatment_plan_(0,_25) + Schedule_a_meeting_(0,_25)
 + Write_a_transcript_of_intake_(0,_25) <= 6

_C30: Create_treatment_plan_(0,_26) + Schedule_a_meeting_(0,_26)
 + Write_a_transcript_of_intake_(0,_26) <= 0

_C31: Create_treatment_plan_(0,_27) + Schedule_a_meeting_(0,_27)
 + Write_a_transcript_of_intake_(0,_27) <= 0

_C32: Create_treatment_plan_(0,_28) + Schedule_a_meeting_(0,_28)
 + Write_a_transcript_of_intake_(0,_28) <= 4

_C33: Create_treatment_plan_(0,_29) + Schedule_a_meeting_(0,_29)
 + Write_a_transcript_of_intake_(0,_29) <= 0

_C34: Create_treatment_plan_(0,_30) + Schedule_a_meeting_(0,_30)
 + Write_a_transcript_of_intake_(0,_30) <= 5

_C35: Create_treatment_plan_(0,_31) + Schedule_a_meeting_(0,_31)
 + Write_a_transcript_of_intake_(0,_31) <= 7

_C36: Create_treatment_plan_(0,_1) + Create_treatment_plan_(0,_10)
 + Create_treatment_plan_(0,_11) + Create_treatment_plan_(0,_12)
 + Create_treatment_plan_(0,_13) + Create_treatment_plan_(0,_14)
 + Create_treatment_plan_(0,_15) + Create_treatment_plan_(0,_16)
 + Create_treatment_plan_(0,_17) + Create_treatment_plan_(0,_18)
 + Create_treatment_plan_(0,_19) + Create_treatment_plan_(0,_2)
 + Create_treatment_plan_(0,_20) + Create_treatment_plan_(0,_21)
 + Create_treatment_plan_(0,_22) + Create_treatment_plan_(0,_23)
 + Create_treatment_plan_(0,_24) + Create_treatment_plan_(0,_25)
 + Create_treatment_plan_(0,_26) + Create_treatment_plan_(0,_27)
 + Create_treatment_plan_(0,_28) + Create_treatment_plan_(0,_29)
 + Create_treatment_plan_(0,_3) + Create_treatment_plan_(0,_30)
 + Create_treatment_plan_(0,_31) + Create_treatment_plan_(0,_4)
 + Create_treatment_plan_(0,_5) + Create_treatment_plan_(0,_6)
 + Create_treatment_plan_(0,_7) + Create_treatment_plan_(0,_8)
 + Create_treatment_plan_(0,_9) + Discuss_treatment_plan_with_patiënt_(0,_11)
 + Discuss_treatment_plan_with_patiënt_(0,_18)
 + Discuss_treatment_plan_with_patiënt_(0,_25)
 + Discuss_treatment_plan_with_patiënt_(0,_4) + Intake_(0,_11)
 + Intake_(0,_18) + Intake_(0,_25) + Intake_(0,_4) + Schedule_a_meeting_(0,_1)
 + Schedule_a_meeting_(0,_10) + Schedule_a_meeting_(0,_11)
 + Schedule_a_meeting_(0,_12) + Schedule_a_meeting_(0,_13)
 + Schedule_a_meeting_(0,_14) + Schedule_a_meeting_(0,_15)
 + Schedule_a_meeting_(0,_16) + Schedule_a_meeting_(0,_17)
 + Schedule_a_meeting_(0,_18) + Schedule_a_meeting_(0,_19)
 + Schedule_a_meeting_(0,_2) + Schedule_a_meeting_(0,_20)
 + Schedule_a_meeting_(0,_21) + Schedule_a_meeting_(0,_22)
 + Schedule_a_meeting_(0,_23) + Schedule_a_meeting_(0,_24)
 + Schedule_a_meeting_(0,_25) + Schedule_a_meeting_(0,_26)
 + Schedule_a_meeting_(0,_27) + Schedule_a_meeting_(0,_28)
 + Schedule_a_meeting_(0,_29) + Schedule_a_meeting_(0,_3)
 + Schedule_a_meeting_(0,_30) + Schedule_a_meeting_(0,_31)
 + Schedule_a_meeting_(0,_4) + Schedule_a_meeting_(0,_5)
 + Schedule_a_meeting_(0,_6) + Schedule_a_meeting_(0,_7)
 + Schedule_a_meeting_(0,_8) + Schedule_a_meeting_(0,_9)
 + Write_a_transcript_of_intake_(0,_1) + Write_a_transcript_of_intake_(0,_10)
 + Write_a_transcript_of_intake_(0,_11) + Write_a_transcript_of_intake_(0,_12)
 + Write_a_transcript_of_intake_(0,_13) + Write_a_transcript_of_intake_(0,_14)
 + Write_a_transcript_of_intake_(0,_15) + Write_a_transcript_of_intake_(0,_16)
 + Write_a_transcript_of_intake_(0,_17) + Write_a_transcript_of_intake_(0,_18)
 + Write_a_transcript_of_intake_(0,_19) + Write_a_transcript_of_intake_(0,_2)
 + Write_a_transcript_of_intake_(0,_20) + Write_a_transcript_of_intake_(0,_21)
 + Write_a_transcript_of_intake_(0,_22) + Write_a_transcript_of_intake_(0,_23)
 + Write_a_transcript_of_intake_(0,_24) + Write_a_transcript_of_intake_(0,_25)
 + Write_a_transcript_of_intake_(0,_26) + Write_a_transcript_of_intake_(0,_27)
 + Write_a_transcript_of_intake_(0,_28) + Write_a_transcript_of_intake_(0,_29)
 + Write_a_transcript_of_intake_(0,_3) + Write_a_transcript_of_intake_(0,_30)
 + Write_a_transcript_of_intake_(0,_31) + Write_a_transcript_of_intake_(0,_4)
 + Write_a_transcript_of_intake_(0,_5) + Write_a_transcript_of_intake_(0,_6)
 + Write_a_transcript_of_intake_(0,_7) + Write_a_transcript_of_intake_(0,_8)
 + Write_a_transcript_of_intake_(0,_9) <= 4

_C37: Intake_(0,_11) + Intake_(0,_18) + Intake_(0,_25) + Intake_(0,_4) = 1

_C38: Write_a_transcript_of_intake_(0,_1)
 + Write_a_transcript_of_intake_(0,_10) + Write_a_transcript_of_intake_(0,_11)
 + Write_a_transcript_of_intake_(0,_12) + Write_a_transcript_of_intake_(0,_13)
 + Write_a_transcript_of_intake_(0,_14) + Write_a_transcript_of_intake_(0,_15)
 + Write_a_transcript_of_intake_(0,_16) + Write_a_transcript_of_intake_(0,_17)
 + Write_a_transcript_of_intake_(0,_18) + Write_a_transcript_of_intake_(0,_19)
 + Write_a_transcript_of_intake_(0,_2) + Write_a_transcript_of_intake_(0,_20)
 + Write_a_transcript_of_intake_(0,_21) + Write_a_transcript_of_intake_(0,_22)
 + Write_a_transcript_of_intake_(0,_23) + Write_a_transcript_of_intake_(0,_24)
 + Write_a_transcript_of_intake_(0,_25) + Write_a_transcript_of_intake_(0,_26)
 + Write_a_transcript_of_intake_(0,_27) + Write_a_transcript_of_intake_(0,_28)
 + Write_a_transcript_of_intake_(0,_29) + Write_a_transcript_of_intake_(0,_3)
 + Write_a_transcript_of_intake_(0,_30) + Write_a_transcript_of_intake_(0,_31)
 + Write_a_transcript_of_intake_(0,_4) + Write_a_transcript_of_intake_(0,_5)
 + Write_a_transcript_of_intake_(0,_6) + Write_a_transcript_of_intake_(0,_7)
 + Write_a_transcript_of_intake_(0,_8) + Write_a_transcript_of_intake_(0,_9)
 = 1

_C39: Create_treatment_plan_(0,_1) + Create_treatment_plan_(0,_10)
 + Create_treatment_plan_(0,_11) + Create_treatment_plan_(0,_12)
 + Create_treatment_plan_(0,_13) + Create_treatment_plan_(0,_14)
 + Create_treatment_plan_(0,_15) + Create_treatment_plan_(0,_16)
 + Create_treatment_plan_(0,_17) + Create_treatment_plan_(0,_18)
 + Create_treatment_plan_(0,_19) + Create_treatment_plan_(0,_2)
 + Create_treatment_plan_(0,_20) + Create_treatment_plan_(0,_21)
 + Create_treatment_plan_(0,_22) + Create_treatment_plan_(0,_23)
 + Create_treatment_plan_(0,_24) + Create_treatment_plan_(0,_25)
 + Create_treatment_plan_(0,_26) + Create_treatment_plan_(0,_27)
 + Create_treatment_plan_(0,_28) + Create_treatment_plan_(0,_29)
 + Create_treatment_plan_(0,_3) + Create_treatment_plan_(0,_30)
 + Create_treatment_plan_(0,_31) + Create_treatment_plan_(0,_4)
 + Create_treatment_plan_(0,_5) + Create_treatment_plan_(0,_6)
 + Create_treatment_plan_(0,_7) + Create_treatment_plan_(0,_8)
 + Create_treatment_plan_(0,_9) = 0.5

_C40: Schedule_a_meeting_(0,_1) + Schedule_a_meeting_(0,_10)
 + Schedule_a_meeting_(0,_11) + Schedule_a_meeting_(0,_12)
 + Schedule_a_meeting_(0,_13) + Schedule_a_meeting_(0,_14)
 + Schedule_a_meeting_(0,_15) + Schedule_a_meeting_(0,_16)
 + Schedule_a_meeting_(0,_17) + Schedule_a_meeting_(0,_18)
 + Schedule_a_meeting_(0,_19) + Schedule_a_meeting_(0,_2)
 + Schedule_a_meeting_(0,_20) + Schedule_a_meeting_(0,_21)
 + Schedule_a_meeting_(0,_22) + Schedule_a_meeting_(0,_23)
 + Schedule_a_meeting_(0,_24) + Schedule_a_meeting_(0,_25)
 + Schedule_a_meeting_(0,_26) + Schedule_a_meeting_(0,_27)
 + Schedule_a_meeting_(0,_28) + Schedule_a_meeting_(0,_29)
 + Schedule_a_meeting_(0,_3) + Schedule_a_meeting_(0,_30)
 + Schedule_a_meeting_(0,_31) + Schedule_a_meeting_(0,_4)
 + Schedule_a_meeting_(0,_5) + Schedule_a_meeting_(0,_6)
 + Schedule_a_meeting_(0,_7) + Schedule_a_meeting_(0,_8)
 + Schedule_a_meeting_(0,_9) = 0.5

_C41: Discuss_treatment_plan_with_patiënt_(0,_11)
 + Discuss_treatment_plan_with_patiënt_(0,_18)
 + Discuss_treatment_plan_with_patiënt_(0,_25)
 + Discuss_treatment_plan_with_patiënt_(0,_4) = 1

_C42: Intake_(0,_4) >= 0

_C43: Discuss_treatment_plan_with_patiënt_(0,_4) >= 0

_C44: Intake_(0,_11) >= 0

_C45: Discuss_treatment_plan_with_patiënt_(0,_11) >= 0

_C46: Intake_(0,_18) >= 0

_C47: Discuss_treatment_plan_with_patiënt_(0,_18) >= 0

_C48: Intake_(0,_25) >= 0

_C49: Discuss_treatment_plan_with_patiënt_(0,_25) >= 0

_C50: Write_a_transcript_of_intake_(0,_1) >= 0

_C51: Create_treatment_plan_(0,_1) >= 0

_C52: Schedule_a_meeting_(0,_1) >= 0

_C53: Write_a_transcript_of_intake_(0,_2) >= 0

_C54: Create_treatment_plan_(0,_2) >= 0

_C55: Schedule_a_meeting_(0,_2) >= 0

_C56: Write_a_transcript_of_intake_(0,_3) >= 0

_C57: Create_treatment_plan_(0,_3) >= 0

_C58: Schedule_a_meeting_(0,_3) >= 0

_C59: Write_a_transcript_of_intake_(0,_4) >= 0

_C60: Create_treatment_plan_(0,_4) >= 0

_C61: Schedule_a_meeting_(0,_4) >= 0

_C62: Write_a_transcript_of_intake_(0,_5) >= 0

_C63: Create_treatment_plan_(0,_5) >= 0

_C64: Schedule_a_meeting_(0,_5) >= 0

_C65: Write_a_transcript_of_intake_(0,_6) >= 0

_C66: Create_treatment_plan_(0,_6) >= 0

_C67: Schedule_a_meeting_(0,_6) >= 0

_C68: Write_a_transcript_of_intake_(0,_7) >= 0

_C69: Create_treatment_plan_(0,_7) >= 0

_C70: Schedule_a_meeting_(0,_7) >= 0

_C71: Write_a_transcript_of_intake_(0,_8) >= 0

_C72: Create_treatment_plan_(0,_8) >= 0

_C73: Schedule_a_meeting_(0,_8) >= 0

_C74: Write_a_transcript_of_intake_(0,_9) >= 0

_C75: Create_treatment_plan_(0,_9) >= 0

_C76: Schedule_a_meeting_(0,_9) >= 0

_C77: Write_a_transcript_of_intake_(0,_10) >= 0

_C78: Create_treatment_plan_(0,_10) >= 0

_C79: Schedule_a_meeting_(0,_10) >= 0

_C80: Write_a_transcript_of_intake_(0,_11) >= 0

_C81: Create_treatment_plan_(0,_11) >= 0

_C82: Schedule_a_meeting_(0,_11) >= 0

_C83: Write_a_transcript_of_intake_(0,_12) >= 0

_C84: Create_treatment_plan_(0,_12) >= 0

_C85: Schedule_a_meeting_(0,_12) >= 0

_C86: Write_a_transcript_of_intake_(0,_13) >= 0

_C87: Create_treatment_plan_(0,_13) >= 0

_C88: Schedule_a_meeting_(0,_13) >= 0

_C89: Write_a_transcript_of_intake_(0,_14) >= 0

_C90: Create_treatment_plan_(0,_14) >= 0

_C91: Schedule_a_meeting_(0,_14) >= 0

_C92: Write_a_transcript_of_intake_(0,_15) >= 0

_C93: Create_treatment_plan_(0,_15) >= 0

_C94: Schedule_a_meeting_(0,_15) >= 0

_C95: Write_a_transcript_of_intake_(0,_16) >= 0

_C96: Create_treatment_plan_(0,_16) >= 0

_C97: Schedule_a_meeting_(0,_16) >= 0

_C98: Write_a_transcript_of_intake_(0,_17) >= 0

_C99: Create_treatment_plan_(0,_17) >= 0

_C100: Schedule_a_meeting_(0,_17) >= 0

_C101: Write_a_transcript_of_intake_(0,_18) >= 0

_C102: Create_treatment_plan_(0,_18) >= 0

_C103: Schedule_a_meeting_(0,_18) >= 0

_C104: Write_a_transcript_of_intake_(0,_19) >= 0

_C105: Create_treatment_plan_(0,_19) >= 0

_C106: Schedule_a_meeting_(0,_19) >= 0

_C107: Write_a_transcript_of_intake_(0,_20) >= 0

_C108: Create_treatment_plan_(0,_20) >= 0

_C109: Schedule_a_meeting_(0,_20) >= 0

_C110: Write_a_transcript_of_intake_(0,_21) >= 0

_C111: Create_treatment_plan_(0,_21) >= 0

_C112: Schedule_a_meeting_(0,_21) >= 0

_C113: Write_a_transcript_of_intake_(0,_22) >= 0

_C114: Create_treatment_plan_(0,_22) >= 0

_C115: Schedule_a_meeting_(0,_22) >= 0

_C116: Write_a_transcript_of_intake_(0,_23) >= 0

_C117: Create_treatment_plan_(0,_23) >= 0

_C118: Schedule_a_meeting_(0,_23) >= 0

_C119: Write_a_transcript_of_intake_(0,_24) >= 0

_C120: Create_treatment_plan_(0,_24) >= 0

_C121: Schedule_a_meeting_(0,_24) >= 0

_C122: Write_a_transcript_of_intake_(0,_25) >= 0

_C123: Create_treatment_plan_(0,_25) >= 0

_C124: Schedule_a_meeting_(0,_25) >= 0

_C125: Write_a_transcript_of_intake_(0,_26) >= 0

_C126: Create_treatment_plan_(0,_26) >= 0

_C127: Schedule_a_meeting_(0,_26) >= 0

_C128: Write_a_transcript_of_intake_(0,_27) >= 0

_C129: Create_treatment_plan_(0,_27) >= 0

_C130: Schedule_a_meeting_(0,_27) >= 0

_C131: Write_a_transcript_of_intake_(0,_28) >= 0

_C132: Create_treatment_plan_(0,_28) >= 0

_C133: Schedule_a_meeting_(0,_28) >= 0

_C134: Write_a_transcript_of_intake_(0,_29) >= 0

_C135: Create_treatment_plan_(0,_29) >= 0

_C136: Schedule_a_meeting_(0,_29) >= 0

_C137: Write_a_transcript_of_intake_(0,_30) >= 0

_C138: Create_treatment_plan_(0,_30) >= 0

_C139: Schedule_a_meeting_(0,_30) >= 0

_C140: Write_a_transcript_of_intake_(0,_31) >= 0

_C141: Create_treatment_plan_(0,_31) >= 0

_C142: Schedule_a_meeting_(0,_31) >= 0

VARIABLES
Create_treatment_plan_(0,_1) <= 0.5 Continuous
Create_treatment_plan_(0,_10) <= 0.5 Continuous
Create_treatment_plan_(0,_11) <= 0.5 Continuous
Create_treatment_plan_(0,_12) <= 0.5 Continuous
Create_treatment_plan_(0,_13) <= 0.5 Continuous
Create_treatment_plan_(0,_14) <= 0.5 Continuous
Create_treatment_plan_(0,_15) <= 0.5 Continuous
Create_treatment_plan_(0,_16) <= 0.5 Continuous
Create_treatment_plan_(0,_17) <= 0.5 Continuous
Create_treatment_plan_(0,_18) <= 0.5 Continuous
Create_treatment_plan_(0,_19) <= 0.5 Continuous
Create_treatment_plan_(0,_2) <= 0.5 Continuous
Create_treatment_plan_(0,_20) <= 0.5 Continuous
Create_treatment_plan_(0,_21) <= 0.5 Continuous
Create_treatment_plan_(0,_22) <= 0.5 Continuous
Create_treatment_plan_(0,_23) <= 0.5 Continuous
Create_treatment_plan_(0,_24) <= 0.5 Continuous
Create_treatment_plan_(0,_25) <= 0.5 Continuous
Create_treatment_plan_(0,_26) <= 0.5 Continuous
Create_treatment_plan_(0,_27) <= 0.5 Continuous
Create_treatment_plan_(0,_28) <= 0.5 Continuous
Create_treatment_plan_(0,_29) <= 0.5 Continuous
Create_treatment_plan_(0,_3) <= 0.5 Continuous
Create_treatment_plan_(0,_30) <= 0.5 Continuous
Create_treatment_plan_(0,_31) <= 0.5 Continuous
Create_treatment_plan_(0,_4) <= 0.5 Continuous
Create_treatment_plan_(0,_5) <= 0.5 Continuous
Create_treatment_plan_(0,_6) <= 0.5 Continuous
Create_treatment_plan_(0,_7) <= 0.5 Continuous
Create_treatment_plan_(0,_8) <= 0.5 Continuous
Create_treatment_plan_(0,_9) <= 0.5 Continuous
Discuss_treatment_plan_with_patiënt_(0,_11) <= 1 Continuous
Discuss_treatment_plan_with_patiënt_(0,_18) <= 1 Continuous
Discuss_treatment_plan_with_patiënt_(0,_25) <= 1 Continuous
Discuss_treatment_plan_with_patiënt_(0,_4) <= 1 Continuous
Intake_(0,_11) <= 1 Continuous
Intake_(0,_18) <= 1 Continuous
Intake_(0,_25) <= 1 Continuous
Intake_(0,_4) <= 1 Continuous
Schedule_a_meeting_(0,_1) <= 0.5 Continuous
Schedule_a_meeting_(0,_10) <= 0.5 Continuous
Schedule_a_meeting_(0,_11) <= 0.5 Continuous
Schedule_a_meeting_(0,_12) <= 0.5 Continuous
Schedule_a_meeting_(0,_13) <= 0.5 Continuous
Schedule_a_meeting_(0,_14) <= 0.5 Continuous
Schedule_a_meeting_(0,_15) <= 0.5 Continuous
Schedule_a_meeting_(0,_16) <= 0.5 Continuous
Schedule_a_meeting_(0,_17) <= 0.5 Continuous
Schedule_a_meeting_(0,_18) <= 0.5 Continuous
Schedule_a_meeting_(0,_19) <= 0.5 Continuous
Schedule_a_meeting_(0,_2) <= 0.5 Continuous
Schedule_a_meeting_(0,_20) <= 0.5 Continuous
Schedule_a_meeting_(0,_21) <= 0.5 Continuous
Schedule_a_meeting_(0,_22) <= 0.5 Continuous
Schedule_a_meeting_(0,_23) <= 0.5 Continuous
Schedule_a_meeting_(0,_24) <= 0.5 Continuous
Schedule_a_meeting_(0,_25) <= 0.5 Continuous
Schedule_a_meeting_(0,_26) <= 0.5 Continuous
Schedule_a_meeting_(0,_27) <= 0.5 Continuous
Schedule_a_meeting_(0,_28) <= 0.5 Continuous
Schedule_a_meeting_(0,_29) <= 0.5 Continuous
Schedule_a_meeting_(0,_3) <= 0.5 Continuous
Schedule_a_meeting_(0,_30) <= 0.5 Continuous
Schedule_a_meeting_(0,_31) <= 0.5 Continuous
Schedule_a_meeting_(0,_4) <= 0.5 Continuous
Schedule_a_meeting_(0,_5) <= 0.5 Continuous
Schedule_a_meeting_(0,_6) <= 0.5 Continuous
Schedule_a_meeting_(0,_7) <= 0.5 Continuous
Schedule_a_meeting_(0,_8) <= 0.5 Continuous
Schedule_a_meeting_(0,_9) <= 0.5 Continuous
Write_a_transcript_of_intake_(0,_1) <= 1 Continuous
Write_a_transcript_of_intake_(0,_10) <= 1 Continuous
Write_a_transcript_of_intake_(0,_11) <= 1 Continuous
Write_a_transcript_of_intake_(0,_12) <= 1 Continuous
Write_a_transcript_of_intake_(0,_13) <= 1 Continuous
Write_a_transcript_of_intake_(0,_14) <= 1 Continuous
Write_a_transcript_of_intake_(0,_15) <= 1 Continuous
Write_a_transcript_of_intake_(0,_16) <= 1 Continuous
Write_a_transcript_of_intake_(0,_17) <= 1 Continuous
Write_a_transcript_of_intake_(0,_18) <= 1 Continuous
Write_a_transcript_of_intake_(0,_19) <= 1 Continuous
Write_a_transcript_of_intake_(0,_2) <= 1 Continuous
Write_a_transcript_of_intake_(0,_20) <= 1 Continuous
Write_a_transcript_of_intake_(0,_21) <= 1 Continuous
Write_a_transcript_of_intake_(0,_22) <= 1 Continuous
Write_a_transcript_of_intake_(0,_23) <= 1 Continuous
Write_a_transcript_of_intake_(0,_24) <= 1 Continuous
Write_a_transcript_of_intake_(0,_25) <= 1 Continuous
Write_a_transcript_of_intake_(0,_26) <= 1 Continuous
Write_a_transcript_of_intake_(0,_27) <= 1 Continuous
Write_a_transcript_of_intake_(0,_28) <= 1 Continuous
Write_a_transcript_of_intake_(0,_29) <= 1 Continuous
Write_a_transcript_of_intake_(0,_3) <= 1 Continuous
Write_a_transcript_of_intake_(0,_30) <= 1 Continuous
Write_a_transcript_of_intake_(0,_31) <= 1 Continuous
Write_a_transcript_of_intake_(0,_4) <= 1 Continuous
Write_a_transcript_of_intake_(0,_5) <= 1 Continuous
Write_a_transcript_of_intake_(0,_6) <= 1 Continuous
Write_a_transcript_of_intake_(0,_7) <= 1 Continuous
Write_a_transcript_of_intake_(0,_8) <= 1 Continuous
Write_a_transcript_of_intake_(0,_9) <= 1 Continuous
Tom van Eijk
  • 21
  • 1

1 Answers1

1

With a time-indexed variable x[t], y[t] ∈ {0,1}, I often also keep scalar variables xt, yt ≥ 0 around which hold the time for x,y:

 xt = sum{ t*x[t] }
 yt = sum{ t*y[t] }

Of course this assumes we have:

 sum{ x[t] } = 1
 sum{ y[t] } = 1

i.e. x,y happen at exactly one time (it also works with "at most one time" as long as the first t is greater than zero).

With this you can easily impose a precedence constraint:

 xt ≤ yt-1
Erwin Kalvelagen
  • 15,677
  • 2
  • 14
  • 39
  • Hi Erwin! Thank you very much for your help! This solution indeed works for some of the values such as A[(0,v)], B[(0,d)] and E[(0,v)] since these take exactly one hour, but it does not yet work for the activities C[(0,d)] and D[(0,d)] since these take 0,5 hours. Do you know to make sure that C[t] ∈ {0,0.5} comes after B[t] ∈ {0,1}? Pulp only accepts integers and continuous variables, which makes it more difficult to put a constraint on a float such as 0.5 – Tom van Eijk Feb 03 '22 at 11:17
  • I think there is some confusion here. We deal with binary variables, so never something like `C[t] ∈ {0,0.5}`. What would `C[1]=0.5` even mean? – Erwin Kalvelagen Feb 03 '22 at 11:35
  • We have 5 variables (A, B, C, D and E) where C[(m,d)] is activity c assigned to employee m (I use only one employee which is referred to as 0 everywhere to make the problem less complex) on day d. Two out of the five activities take each half a hour to complete and three out of five activities take each 1 hour to complete. The third activity takes half a hour to complete meaning that ```C[(m,d)] = 0.5 ```. So we have a time-indexed variable with time-index days and the variable itself represents the number of hours (0.5 or 1) that will be subtracted from the availability hours in the input – Tom van Eijk Feb 03 '22 at 12:15
  • Sorry, that does not make sense to me. That is not how a time-indexed scheduling model works. – Erwin Kalvelagen Feb 03 '22 at 12:16