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Include time periods into arcs

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Anne Li van der veen
Gurobi-versary
First Comment
First Question

hi all, 

im trying to model the following network into python, as input for my mathematical model. however, i do not know how to include the different time periods to the possible connections between my nodes. for example: node CMP in period 1 flows to node DC in period 2. this would give ((CPM,t), (DC, t+1)). 

could anyone support in this matter? the second picture shows how i tried to model it so far, only references to t in set T do not work in a dictionary 

 

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4 comments

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  • Jonasz Staszek
    Community Moderator Community Moderator
    Gurobi-versary
    Thought Leader
    First Question

    Hi Anne,

    the code you posted, or at least its screenshot, won't work since you did not define \(t\) properly. (For future reference, it is much easier to help if you post the actual code and not a screenshot).

    If you wish to consider all \(t \in T\), you'd need to iterate over \(T\) (usually this is done by a for loop). I have too little information to help you with this matter further - you'd need to tell us more about the model you're trying to implement.

    As a side note, you could easily model your graph using NetworkX. Then, you can iterate over arcs in your graph to create a tupledict of variables in your Gurobi model.

    Hope this helps.

    Best regards
    Jonasz

     

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  • Anne Li van der veen
    Gurobi-versary
    First Comment
    First Question

    Hi Jonas,

    Thanks for your reply. I'm trying to model an optimzation of the optimal pallet mix for a company, in order to find the trade-off between costs and sustainability. Find below some of the code. Hope this helps you understand the problem, and look forward to your suggestions on modelling the time in the correct way. I also included the objective function, where i do refer to t in model T. CP is a parameter i did not define in below code, but which is defined in the actual code, including all other parameters. 

    thanks a lot in advance!

    #Import packages
    import pyomo.environ as pe
    import pyomo.opt as po

    from collections import defaultdict

     

    #Dictionary - Sets 

    #Pallet Management System  
    #1 = buy/sell, 2 = rental/lease

    P={1,2}

    #Materials 
    #3 = wood, 4 = plastic

    N={3,4} 

    #EOL Scenarios 
    #5 = repair, 6 = recycle, 7 = downcycle, 8 = landfill, 9 = incineration 

    D={5,6,7,8,9}

    #Time Periods in weeks 

    T=5

     

    #Nodes - Node (s,t) s = life cycle or use phase, t = period 
    V = {'G','RM', 'MP', 'CPM', 'DC', 'RET', 'RC', 'DEOL', 'LEOL', 'IEOL' }

    #Arcs - Arc ((s,t), (s',t')) - t' >= t 

    Arcs = {(('G', t-1), ('RM', t)),
            (('RM', t), ('MP', t)),
            (('MP', t), ('CPM', t)),
            (('CPM', t), ('DC', t+1)),
            (('DC', t+1), ('RET', t+2)),
            (('RET', t+2), ('RC', t+2)),
            (('RC', t+2), ('CPM', t+3)),
            (('RC', t+2), ('MP', t+3)),
            (('RC', t+2), ('RM', t+3)),
            (('RC', t+2), ('DEOL', t+3)),
            (('RC', t+2), ('LEOL', t+3)),
            (('RC', t+2), ('IEOL', t+3)),
            (('DC', t+2), ('CPM', t+3)),
            (('DC', t+2), ('MP', t+3)),
            (('DC', t+2), ('RM', t+3)),
            (('DC', t+2), ('DEOL', t+3)),
            (('DC', t+2), ('LEOL', t+3)),
            (('DC', t+2), ('IEOL', t+3))
    }

    #Making sets:
    model = pe.ConcreteModel()

    #Pallet types
    model.P = pe.Set(initialize = P, ordered = False)

    #Materials 
    model.N = pe.Set(initialize = N, ordered = False)

    #EOL scenarios 
    model.D = pe.Set(initialize = D, ordered = False)

    #Time Periods
    model.T = pe.RangeSet(1, T)

    #Nodes
    model.V = pe.Set(initialize = V, ordered = False)

    #Arcs
    model.A = pe.Set(initialize = A, ordered = False)

    #Decision Variables
    model.Y = pe.Var(model.P, model.N, model.A, model.T, domain = pe.NonNegativeReals)
    model.Inv = pe.Var(model.P, model.N, model.V, model.T, domain = pe.NonNegativeReals)
    model.X = pe.Var(model.P, model.N, model.V, domain = pe.Binary)

    #1 - Objective function - minimize costs 

    obj_expr = sum(model.Y[p,n,i,j,t,t]*CP[p,n] for p in model.P for n in model.N for (i,j) in model.A for t in model.T) \
    + sum(model.Y[p,n,i,j,t,t+1]*CP[p,n]for p in model.P for n in model.N for (i,j) in model.A for t in model.T)                                                                                                                                                                                                      
    model.obj=pe.Objective(expr=obj_expr, sense=pe.minimize)

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  • Jonasz Staszek
    Community Moderator Community Moderator
    Gurobi-versary
    Thought Leader
    First Question

    You need to define your arcs somewhat like this:

    Arcs = []

    for t in range(1, T+1):
        if t-1 >= 1 and t+3 <= T:

            Arcs += [(('G', t-1), ('RM', t)),
                    (('RM', t), ('MP', t)),
                    (('MP', t), ('CPM', t)),
                    (('CPM', t), ('DC', t+1)),
                    (('DC', t+1), ('RET', t+2)),
                    (('RET', t+2), ('RC', t+2)),
                    (('RC', t+2), ('CPM', t+3)),
                    (('RC', t+2), ('MP', t+3)),
                    (('RC', t+2), ('RM', t+3)),
                    (('RC', t+2), ('DEOL', t+3)),
                    (('RC', t+2), ('LEOL', t+3)),
                    (('RC', t+2), ('IEOL', t+3)),
                    (('DC', t+2), ('CPM', t+3)),
                    (('DC', t+2), ('MP', t+3)),
                    (('DC', t+2), ('RM', t+3)),
                    (('DC', t+2), ('DEOL', t+3)),
                    (('DC', t+2), ('LEOL', t+3)),
                    (('DC', t+2), ('IEOL', t+3))
            ]

    Is this the snippet you were looking for?

    Best regards
    Jonasz

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  • Anne Li van der veen
    Gurobi-versary
    First Comment
    First Question

    Hi Jonasz,

    thanks a lot . i implemented it like this and it seems to work. unfortunately i ran into the next problem while trying to define the objective function. i used the code described above and added your part on modelling the arcs. when i try to code the objective function, it keeps giving me a key error for the CT[p,n,a] * L[a] part. furthermore, i need to use a to call an element from model A and am not able to use (i,j). would you know how to fix this? at some point in my code i will need to call only node i from the pair (i,j) in model.A, but at the moment i am only able to call the full arc using a in model A

     

    thanks a lot in advance!

     

    #CO(p,n) - Purchase/leasing cost of pallet type p of material m - €/PAL
    CO = {(1, 3): 5, (1, 4): 5, \
          (2, 3): 5, (2, 4): 5}


    #CT(p,n) - Transportation cost per pallet type p per km - €/km-PAL
    #CT = {}
    #for a in model.A:
        #CT = {(1, 3, a): 0.00015, (1, 4, a): 0.00015, (2, 3, a): 0.00015, (2, 4, a): 0.00015}

    CT = {}
    for t in range(1, T+1):
        if t-1 >= 1 and t+3 <= T:
            CT = {((1, 3), (('CPM', t), ('DC', t+1))): 0.00015,\
                  ((1, 4), (('DC', t+1), ('RET', t+2))): 0.00015,\
                  ((2, 3), (('RET', t+2), ('RC', t+2))): 0.00015, \
                  ((2, 4), (('RC', t+2), ('CPM', t+3))): 0.00015,\
                  ((1, 3), (('DC', t+2), ('CPM', t+3))): 0.00015
                 }

     
    #S(p,n) - Credit for selling or returning pallet type p - €/PAL
    S = {(1, 3): 5, (1, 4): 15, \
          (2, 3): 8, (2, 4): 20}

    #L(i,j) - Distance travelled between nodes i and j - km 

    #L = {('CPM', 'DC'): 100, ('DC', 'RET'): 150, ('DC', 'CPM'): 50, ('RET', 'RC'): 75, ('RC', 'CPM'): 125}

    L = {} 
    for t in range(1, T+1):
        if t-1 >= 1 and t+3 <= T:
            L = {(('CPM', t), ('DC', t+1)):100, (('DC', t+1), ('RET', t+2)):150, (('DC', t+2), ('CPM', t+3)):50,\
                 (('RET', t+2), ('RC', t+2)):75, (('RC', t+2), ('CPM', t+3)):125}

     

    #OBjective funciton

    objexpr = sum(model.Y[p,n,a] * (CO[p,n] + CT[p,n,a] * L[a]- S[p,n])  for p in model.P for n in model.N for a in model.A)
    model.obj=pe.Objective(expr=objexpr,sense=pe.minimize)
                   

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