Scheduling State-Task Networks

Example (Kondili, et al., 1993)

A state-task network is a graphical representation of the activities in a multiproduct batch process. The representation includes the minimum details needed for short term scheduling of batch operations.

Shown below is a well-studied example due to Kondili (1993). Other examples are available in the references cited above.

Each circular node in the diagram designates material in a particular state. The materials are generally held in suitable vessels with a known capacity. The relevant information for each state is the initial inventory, storage capacity, and the unit price of the material in that state. The price of materials in intermediate states may be assigned penalities in order to minimize the amount of work in progress.

The rectangular nodes denote process tasks. When scheduled for execution, each task is assigned an appropriate piece of equipment, and assigned a batch of material according to the incoming arcs. Each incoming arc begins at a state where the associated label indicates the mass fraction of the batch coming from that particular state. Outgoing arcs indicate the disposition of the batch to product states. The outgoing are labels indicate the fraction of the batch assigned to each product state, and the time necessary to produce that product.

Not shown in the diagram is the process equipment used to execute the tasks. A separate list of process units is available, each characterized by a capacity and list of tasks which can be performed in that unit.

Encoding the STN data

The basic data structure specifies the states, tasks, and units comprising a state-task network. The intention is for all relevant problem data to be contained in a single JSON-like structure.

In [34]:
H = 12

Kondili = {
    'TIME':  range(0,H+1),
    'STATES': {
        'Feed_A'   : {'capacity': 500, 'initial': 500, 'price':  0},
        'Feed_B'   : {'capacity': 500, 'initial': 500, 'price':  0},
        'Feed_C'   : {'capacity': 500, 'initial': 500, 'price':  0},
        'Hot_A'    : {'capacity': 100, 'initial':   0, 'price': -1},
        'Int_AB'   : {'capacity': 200, 'initial':   0, 'price': -10},
        'Int_BC'   : {'capacity': 150, 'initial':   0, 'price': -1},
        'Impure_E' : {'capacity': 100, 'initial':   0, 'price': -1},
        'Product_1': {'capacity': 500, 'initial':   0, 'price': 10},
        'Product_2': {'capacity': 500, 'initial':   0, 'price': 10},
    },
    'ST_ARCS': {
        ('Feed_A',   'Heating')   : {'rho': 1.0},
        ('Feed_B',   'Reaction_1'): {'rho': 0.5},
        ('Feed_C',   'Reaction_1'): {'rho': 0.5},
        ('Feed_C',   'Reaction_3'): {'rho': 0.2},
        ('Hot_A',    'Reaction_2'): {'rho': 0.4},
        ('Int_AB',   'Reaction_3'): {'rho': 0.8},
        ('Int_BC',   'Reaction_2'): {'rho': 0.6},
        ('Impure_E', 'Separation'): {'rho': 1.0},
    },
    'TS_ARCS': {
        ('Heating',    'Hot_A')    : {'dur': 1, 'rho': 1.0},
        ('Reaction_2', 'Product_1'): {'dur': 2, 'rho': 0.4},
        ('Reaction_2', 'Int_AB')   : {'dur': 2, 'rho': 0.6},
        ('Reaction_1', 'Int_BC')   : {'dur': 2, 'rho': 1.0},
        ('Reaction_3', 'Impure_E') : {'dur': 1, 'rho': 1.0},
        ('Separation', 'Int_AB')   : {'dur': 2, 'rho': 0.1},
        ('Separation', 'Product_2'): {'dur': 1, 'rho': 0.9},
    },
    'UNIT_TASKS': {
        ('Heater',    'Heating')   : {'Bmin': 0, 'Bmax': 100, 'Cost': 1, 'vCost': 0, 'Tclean': 0},
        ('Reactor_1', 'Reaction_1'): {'Bmin': 0, 'Bmax':  80, 'Cost': 1, 'vCost': 0, 'Tclean': 0},
        ('Reactor_1', 'Reaction_2'): {'Bmin': 0, 'Bmax':  80, 'Cost': 1, 'vCost': 0, 'Tclean': 0},
        ('Reactor_1', 'Reaction_3'): {'Bmin': 0, 'Bmax':  80, 'Cost': 1, 'vCost': 0, 'Tclean': 0},
        ('Reactor_2', 'Reaction_1'): {'Bmin': 0, 'Bmax':  80, 'Cost': 1, 'vCost': 0, 'Tclean': 0},
        ('Reactor_2', 'Reaction_2'): {'Bmin': 0, 'Bmax':  80, 'Cost': 1, 'vCost': 0, 'Tclean': 0},
        ('Reactor_2', 'Reaction_3'): {'Bmin': 0, 'Bmax':  80, 'Cost': 1, 'vCost': 0, 'Tclean': 0},
        ('Reactor_3', 'Reaction_1'): {'Bmin': 0, 'Bmax': 120, 'Cost': 1, 'vCost': 0, 'Tclean': 0},
        ('Reactor_3', 'Reaction_2'): {'Bmin': 0, 'Bmax': 120, 'Cost': 1, 'vCost': 0, 'Tclean': 0},
        ('Reactor_3', 'Reaction_3'): {'Bmin': 0, 'Bmax': 120, 'Cost': 1, 'vCost': 0, 'Tclean': 0},
        ('Still',     'Separation'): {'Bmin': 0, 'Bmax': 200, 'Cost': 1, 'vCost': 0, 'Tclean': 0},
    },
}

Setting a Time Grid

The following computations can be done on any time grid, including real-valued time points. TIME is a list of time points commencing at 0.

Creating a Pyomo Model

The following Pyomo model closely follows the development in Kondili, et al. (1993). In particular, the first step in the model is to process the STN data to create sets as given in Kondili. Two differences from Kondili are:

  • a natural time scale commencing at $t = 0$ and extending to $t = H$ (rather than from 1 to H+1).
  • an additional decision variable denoted by $Q_{j,t}$ indicating the amount of material in unit $j$ at time $t$. A material balance then reads

\begin{align*} Q_{jt} & = Q_{j(t-1)} + \sum_{i\in I_j}B_{ijt} - \sum_{i\in I_j}\sum_{\substack{s \in \bar{S}_i\\s\ni t-P_{is} \geq 0}}\bar{\rho}_{is}B_{ij(t-P_{is})} \qquad \forall j,t \end{align*}

Following Kondili's notation, $I_j$ is the set of tasks that can be performed in unit $j$, and $\bar{S}_i$ is the set of states fed by task $j$. We assume the units are empty at the beginning and end of production period, i.e.,

\begin{align*} Q_{j(-1)} & = 0 \qquad \forall j \\ Q_{j,H} & = 0 \qquad \forall j \end{align*}

The unit allocation constraints are written the full backward aggregation method described by Shah (1993). The allocation constraint reads

\begin{align*} \sum_{i \in I_j} \sum_{t'=t}^{t-p_i+1} W_{ijt'} & \leq 1 \qquad \forall j,t \end{align*}

Each processing unit $j$ is tagged with a minimum and maximum capacity, $B_{ij}^{min}$ and $B_{ij}^{max}$, respectively, denoting the minimum and maximum batch sizes for each task $i$. A minimum capacity may be needed to cover heat exchange coils in a reactor or mixing blades in a blender, for example. The capacity may depend on the nature of the task being performed. These constraints are written

\begin{align*} B_{ij}^{min}W_{ijt} & \leq B_{ijt} \leq B_{ij}^{max}W_{ijt} \qquad \forall j, \forall i\in I_j, \forall t \end{align*}

Characterization of Tasks

In [35]:
STN = Kondili

STATES = STN['STATES']
ST_ARCS = STN['ST_ARCS']
TS_ARCS = STN['TS_ARCS']
UNIT_TASKS = STN['UNIT_TASKS']
TIME = STN['TIME']
H = max(TIME)
In [36]:
TASKS = set([i for (j,i) in UNIT_TASKS])                         # set of all tasks 

S = {i: set() for i in TASKS}                                    # S[i] input set of states which feed task i
for (s,i) in ST_ARCS:
    S[i].add(s)

S_ = {i: set() for i in TASKS}                                   # S_[i] output set of states fed by task i
for (i,s) in TS_ARCS:
    S_[i].add(s)

rho = {(i,s): ST_ARCS[(s,i)]['rho'] for (s,i) in ST_ARCS}        # rho[(i,s)] input fraction of task i from state s

rho_ = {(i,s): TS_ARCS[(i,s)]['rho'] for (i,s) in TS_ARCS}       # rho_[(i,s)] output fraction of task i to state s

P = {(i,s): TS_ARCS[(i,s)]['dur'] for (i,s) in TS_ARCS}          # P[(i,s)] time for task i output to state s   

p = {i: max([P[(i,s)] for s in S_[i]]) for i in TASKS}           # p[i] completion time for task i

K = {i: set() for i in TASKS}                                    # K[i] set of units capable of task i
for (j,i) in UNIT_TASKS:
    K[i].add(j) 

Characterization of States

In [37]:
T = {s: set() for s in STATES}                                   # T[s] set of tasks receiving material from state s 
for (s,i) in ST_ARCS:
    T[s].add(i)

T_ = {s: set() for s in STATES}                                  # set of tasks producing material for state s
for (i,s) in TS_ARCS:
    T_[s].add(i)

C = {s: STATES[s]['capacity'] for s in STATES}                   # C[s] storage capacity for state s

Characterization of Units

In [38]:
UNITS = set([j for (j,i) in UNIT_TASKS])

I = {j: set() for j in UNITS}                                     # I[j] set of tasks performed with unit j
for (j,i) in UNIT_TASKS:
    I[j].add(i)

Bmax = {(i,j):UNIT_TASKS[(j,i)]['Bmax'] for (j,i) in UNIT_TASKS}  # Bmax[(i,j)] maximum capacity of unit j for task i
Bmin = {(i,j):UNIT_TASKS[(j,i)]['Bmin'] for (j,i) in UNIT_TASKS}  # Bmin[(i,j)] minimum capacity of unit j for task i

Pyomo Model

In [39]:
from pyomo.environ import *
import numpy as np

TIME = np.array(TIME)

model = ConcreteModel()

model.W = Var(TASKS, UNITS, TIME, domain=Boolean)             # W[i,j,t] 1 if task i starts in unit j at time t
model.B = Var(TASKS, UNITS, TIME, domain=NonNegativeReals)    # B[i,j,t,] size of batch assigned to task i in unit j at time t
model.S = Var(STATES.keys(), TIME, domain=NonNegativeReals)   # S[s,t] inventory of state s at time t
model.Q = Var(UNITS, TIME, domain=NonNegativeReals)           # Q[j,t] inventory of unit j at time t
model.Cost = Var(domain=NonNegativeReals)
model.Value = Var(domain=NonNegativeReals)

# Objective is to maximize the value of the final state (see Kondili, Sec. 5)
model.Obj = Objective(expr = model.Value - model.Cost, sense = maximize)

# Constraints
model.cons = ConstraintList()
model.cons.add(model.Value == sum([STATES[s]['price']*model.S[s,H] for s in STATES]))
model.cons.add(model.Cost == sum([UNIT_TASKS[(j,i)]['Cost']*model.W[i,j,t] +
        UNIT_TASKS[(j,i)]['vCost']*model.B[i,j,t] for i in TASKS for j in K[i] for t in TIME])) 

# unit constraints
for j in UNITS:
    rhs = 0
    for t in TIME:
        # a unit can only be allocated to one task 
        lhs = 0
        for i in I[j]:
            for tprime in TIME:
                if tprime >= (t-p[i]+1-UNIT_TASKS[(j,i)]['Tclean']) and tprime <= t:
                    lhs += model.W[i,j,tprime]
        model.cons.add(lhs <= 1)

        # capacity constraints (see Konkili, Sec. 3.1.2)
        for i in I[j]:
            model.cons.add(model.W[i,j,t]*Bmin[i,j] <= model.B[i,j,t])
            model.cons.add(model.B[i,j,t] <= model.W[i,j,t]*Bmax[i,j])

        # unit mass balance
        rhs += sum([model.B[i,j,t] for i in I[j]])
        for i in I[j]:
            for s in S_[i]:
                if t >= P[(i,s)]:
                    rhs -= rho_[(i,s)]*model.B[i,j,max(TIME[TIME <= t-P[(i,s)]])]
        model.cons.add(model.Q[j,t] == rhs)
        rhs = model.Q[j,t]
        
    # terminal condition  
    model.cons.add(model.Q[j,H] == 0)

# state constraints
for s in STATES.keys():
    rhs = STATES[s]['initial']
    for t in TIME:
        # state capacity constraint
        model.cons.add(model.S[s,t] <= C[s])
        
        # state mass balanace
        for i in T_[s]:
            for j in K[i]:
                if t >= P[(i,s)]: 
                    rhs += rho_[(i,s)]*model.B[i,j,max(TIME[TIME <= t-P[(i,s)]])]             
        for i in T[s]:
            rhs -= rho[(i,s)]*sum([model.B[i,j,t] for j in K[i]])
        model.cons.add(model.S[s,t] == rhs)
        rhs = model.S[s,t] 
        
# additional production constraints      
model.cons.add(model.S['Product_2',H] >= 250)


SolverFactory('glpk').solve(model).write()
# ==========================================================
# = Solver Results                                         =
# ==========================================================
# ----------------------------------------------------------
#   Problem Information
# ----------------------------------------------------------
Problem: 
- Name: unknown
  Lower bound: 6992.91666666667
  Upper bound: 6992.91666666667
  Number of objectives: 1
  Number of constraints: 659
  Number of variables: 471
  Number of nonzeros: 1991
  Sense: maximize
# ----------------------------------------------------------
#   Solver Information
# ----------------------------------------------------------
Solver: 
- Status: ok
  Termination condition: optimal
  Statistics: 
    Branch and bound: 
      Number of bounded subproblems: 12747
      Number of created subproblems: 12747
  Error rc: 0
  Time: 7.91475510597229
# ----------------------------------------------------------
#   Solution Information
# ----------------------------------------------------------
Solution: 
- number of solutions: 0
  number of solutions displayed: 0

Analysis

Profitability

In [40]:
print("Value of State Inventories = {0:12.2f}".format(model.Value()))
print("  Cost of Unit Assignments = {0:12.2f}".format(model.Cost()))
print("             Net Objective = {0:12.2f}".format(model.Value() - model.Cost()))
Value of State Inventories =      7017.92
  Cost of Unit Assignments =        25.00
             Net Objective =      6992.92

State Inventories

In [41]:
%matplotlib inline
import matplotlib.pyplot as plt
import pandas as pd
from IPython.display import display, HTML

pd.DataFrame([[model.S[s,t]() for s in STATES.keys()] for t in TIME], columns = STATES.keys(), index = TIME)
Out[41]:
Feed_A Feed_B Feed_C Hot_A Int_AB Int_BC Impure_E Product_1 Product_2
0 424.000000 360.0 360.00 0.000000 0.0 0.0 0.00 0.000000 0.0
1 324.000000 360.0 360.00 76.000000 0.0 0.0 0.00 0.000000 0.0
2 324.000000 360.0 360.00 64.000000 0.0 112.0 0.00 0.000000 0.0
3 324.000000 360.0 360.00 64.000000 0.0 112.0 0.00 0.000000 0.0
4 324.000000 360.0 320.00 32.000000 8.0 64.0 0.00 112.000000 0.0
5 324.000000 300.0 260.00 0.000000 8.0 16.0 0.00 112.000000 0.0
6 233.333333 300.0 246.00 0.000000 0.0 16.0 0.00 144.000000 180.0
7 233.333333 300.0 230.00 10.666667 4.0 16.0 70.00 176.000000 180.0
8 233.333333 300.0 230.00 10.666667 4.0 16.0 0.00 176.000000 180.0
9 233.333333 300.0 199.00 0.000000 0.0 0.0 0.00 256.000000 315.0
10 233.333333 300.0 199.00 0.000000 15.0 0.0 0.00 256.000000 315.0
11 233.333333 300.0 191.25 0.000000 0.0 0.0 0.00 266.666667 454.5
12 233.333333 300.0 191.25 0.000000 15.5 0.0 38.75 266.666667 454.5
In [42]:
plt.figure(figsize=(10,6))
for (s,idx) in zip(STATES.keys(),range(0,len(STATES.keys()))):
    plt.subplot(ceil(len(STATES.keys())/3),3,idx+1)
    tlast,ylast = 0,STATES[s]['initial']
    for (t,y) in zip(list(TIME),[model.S[s,t]() for t in TIME]):
        plt.plot([tlast,t,t],[ylast,ylast,y],'b')
        #plt.plot([tlast,t],[ylast,y],'b.',ms=10)
        tlast,ylast = t,y
    plt.ylim(0,1.1*C[s])
    plt.plot([0,H],[C[s],C[s]],'r--')
    plt.title(s)
plt.tight_layout()

Unit Assignment

In [43]:
UnitAssignment = pd.DataFrame({j:[None for t in TIME] for j in UNITS},index=TIME)

for t in TIME:
    for j in UNITS:
        for i in I[j]:
            for s in S_[i]:
                if t-p[i] >= 0:
                    if model.W[i,j,max(TIME[TIME <= t-p[i]])]() > 0:
                        UnitAssignment.loc[t,j] = None               
        for i in I[j]:
            if model.W[i,j,t]() > 0:
                UnitAssignment.loc[t,j] = (i,model.B[i,j,t]())

UnitAssignment
Out[43]:
Heater Reactor_1 Reactor_2 Reactor_3 Still
0 (Heating, 76.0) (Reaction_1, 80.0) (Reaction_1, 80.0) (Reaction_1, 120.0) None
1 (Heating, 100.0) None None None None
2 None (Reaction_2, 80.0) (Reaction_2, 80.0) (Reaction_2, 120.0) None
3 None None None None None
4 None (Reaction_3, 80.0) (Reaction_2, 80.0) (Reaction_3, 120.0) None
5 None (Reaction_2, 80.0) None (Reaction_1, 120.0) (Separation, 200.0)
6 (Heating, 90.6666666666667) None (Reaction_3, 70.0) None None
7 None (Reaction_2, 80.0) (Reaction_3, 80.0) (Reaction_2, 120.0) None
8 None None None None (Separation, 150.0)
9 None (Reaction_3, 80.0) (Reaction_3, 75.0) (Reaction_2, 26.6666666666667) None
10 None None None None (Separation, 155.0)
11 None None None (Reaction_3, 38.75) None
12 None None None None None

Unit Batch Inventories

In [44]:
pd.DataFrame([[model.Q[j,t]() for j in UNITS] for t in TIME], columns = UNITS, index = TIME)
Out[44]:
Still Reactor_2 Reactor_3 Heater Reactor_1
0 0.0 80.0 120.000000 76.000000 80.0
1 0.0 80.0 120.000000 100.000000 80.0
2 0.0 80.0 120.000000 0.000000 80.0
3 0.0 80.0 120.000000 0.000000 80.0
4 0.0 80.0 120.000000 0.000000 80.0
5 200.0 80.0 120.000000 0.000000 80.0
6 20.0 70.0 120.000000 90.666667 80.0
7 0.0 80.0 120.000000 0.000000 80.0
8 150.0 0.0 120.000000 0.000000 80.0
9 15.0 75.0 26.666667 0.000000 80.0
10 155.0 0.0 26.666667 0.000000 0.0
11 15.5 0.0 38.750000 0.000000 0.0
12 0.0 0.0 0.000000 0.000000 0.0

Gannt Chart

In [45]:
%matplotlib inline
import matplotlib.pyplot as plt

plt.figure(figsize=(12,6))

gap = H/500
idx = 1
lbls = []
ticks = []
for j in sorted(UNITS):
    idx -= 1
    for i in sorted(I[j]):
        idx -= 1
        ticks.append(idx)
        lbls.append("{0:s} -> {1:s}".format(j,i))
        plt.plot([0,H],[idx,idx],lw=20,alpha=.3,color='y')
        for t in TIME:
            if model.W[i,j,t]() > 0:
                plt.plot([t+gap,t+p[i]-gap], [idx,idx],'b', lw=20, solid_capstyle='butt')
                txt = "{0:.2f}".format(model.B[i,j,t]())
                plt.text(t+p[i]/2, idx, txt, color='white', weight='bold', ha='center', va='center')
plt.xlim(0,H)
plt.gca().set_yticks(ticks)
plt.gca().set_yticklabels(lbls);

Trace of Events and States

In [46]:
sep = '\n--------------------------------------------------------------------------------------------\n'
print(sep)
print("Starting Conditions")
print("    Initial Inventories:")            
for s in STATES.keys():
        print("        {0:10s}  {1:6.1f} kg".format(s,STATES[s]['initial']))
        
units = {j:{'assignment':'None', 't':0} for j in UNITS}

for t in TIME:
    print(sep)
    print("Time =",t,"hr")
    print("    Instructions:")
    for j in UNITS:
        units[j]['t'] += 1
        # transfer from unit to states
        for i in I[j]:  
            for s in S_[i]:
                if t-P[(i,s)] >= 0:
                    amt = rho_[(i,s)]*model.B[i,j,max(TIME[TIME <= t - P[(i,s)]])]()
                    if amt > 0:
                        print("        Transfer", amt, "kg from", j, "to", s)
    for j in UNITS:
        # release units from tasks
        for i in I[j]:
            if t-p[i] >= 0:
                if model.W[i,j,max(TIME[TIME <= t-p[i]])]() > 0:
                    print("        Release", j, "from", i)
                    units[j]['assignment'] = 'None'
                    units[j]['t'] = 0
        # assign units to tasks             
        for i in I[j]:
            if model.W[i,j,t]() > 0:
                print("        Assign", j, "with capacity", Bmax[(i,j)], "kg to task",i,"for",p[i],"hours")
                units[j]['assignment'] = i
                units[j]['t'] = 1
        # transfer from states to starting tasks
        for i in I[j]:
            for s in S[i]:
                amt = rho[(i,s)]*model.B[i,j,t]()
                if amt > 0:
                    print("        Transfer", amt,"kg from", s, "to", j)
    print("\n    Inventories are now:")            
    for s in STATES.keys():
        print("        {0:10s}  {1:6.1f} kg".format(s,model.S[s,t]()))
    print("\n    Unit Assignments are now:")
    for j in UNITS:
        if units[j]['assignment'] != 'None':
            fmt = "        {0:s} performs the {1:s} task with a {2:.2f} kg batch for hour {3:f} of {4:f}"
            i = units[j]['assignment']
            print(fmt.format(j,i,model.Q[j,t](),units[j]['t'],p[i]))
            
print(sep)
print('Final Conditions')
print("    Final Inventories:")            
for s in STATES.keys():
        print("        {0:10s}  {1:6.1f} kg".format(s,model.S[s,H]()))
--------------------------------------------------------------------------------------------

Starting Conditions
    Initial Inventories:
        Feed_A       500.0 kg
        Feed_B       500.0 kg
        Feed_C       500.0 kg
        Hot_A          0.0 kg
        Int_AB         0.0 kg
        Int_BC         0.0 kg
        Impure_E       0.0 kg
        Product_1      0.0 kg
        Product_2      0.0 kg

--------------------------------------------------------------------------------------------

Time = 0 hr
    Instructions:
        Assign Reactor_2 with capacity 80 kg to task Reaction_1 for 2 hours
        Transfer 40.0 kg from Feed_B to Reactor_2
        Transfer 40.0 kg from Feed_C to Reactor_2
        Assign Reactor_3 with capacity 120 kg to task Reaction_1 for 2 hours
        Transfer 60.0 kg from Feed_B to Reactor_3
        Transfer 60.0 kg from Feed_C to Reactor_3
        Assign Heater with capacity 100 kg to task Heating for 1 hours
        Transfer 76.0 kg from Feed_A to Heater
        Assign Reactor_1 with capacity 80 kg to task Reaction_1 for 2 hours
        Transfer 40.0 kg from Feed_B to Reactor_1
        Transfer 40.0 kg from Feed_C to Reactor_1

    Inventories are now:
        Feed_A       424.0 kg
        Feed_B       360.0 kg
        Feed_C       360.0 kg
        Hot_A          0.0 kg
        Int_AB         0.0 kg
        Int_BC         0.0 kg
        Impure_E       0.0 kg
        Product_1      0.0 kg
        Product_2      0.0 kg

    Unit Assignments are now:
        Reactor_2 performs the Reaction_1 task with a 80.00 kg batch for hour 1.000000 of 2.000000
        Reactor_3 performs the Reaction_1 task with a 120.00 kg batch for hour 1.000000 of 2.000000
        Heater performs the Heating task with a 76.00 kg batch for hour 1.000000 of 1.000000
        Reactor_1 performs the Reaction_1 task with a 80.00 kg batch for hour 1.000000 of 2.000000

--------------------------------------------------------------------------------------------

Time = 1 hr
    Instructions:
        Transfer 76.0 kg from Heater to Hot_A
        Release Heater from Heating
        Assign Heater with capacity 100 kg to task Heating for 1 hours
        Transfer 100.0 kg from Feed_A to Heater

    Inventories are now:
        Feed_A       324.0 kg
        Feed_B       360.0 kg
        Feed_C       360.0 kg
        Hot_A         76.0 kg
        Int_AB         0.0 kg
        Int_BC         0.0 kg
        Impure_E       0.0 kg
        Product_1      0.0 kg
        Product_2      0.0 kg

    Unit Assignments are now:
        Reactor_2 performs the Reaction_1 task with a 80.00 kg batch for hour 2.000000 of 2.000000
        Reactor_3 performs the Reaction_1 task with a 120.00 kg batch for hour 2.000000 of 2.000000
        Heater performs the Heating task with a 100.00 kg batch for hour 1.000000 of 1.000000
        Reactor_1 performs the Reaction_1 task with a 80.00 kg batch for hour 2.000000 of 2.000000

--------------------------------------------------------------------------------------------

Time = 2 hr
    Instructions:
        Transfer 80.0 kg from Reactor_2 to Int_BC
        Transfer 120.0 kg from Reactor_3 to Int_BC
        Transfer 100.0 kg from Heater to Hot_A
        Transfer 80.0 kg from Reactor_1 to Int_BC
        Release Reactor_2 from Reaction_1
        Assign Reactor_2 with capacity 80 kg to task Reaction_2 for 2 hours
        Transfer 32.0 kg from Hot_A to Reactor_2
        Transfer 48.0 kg from Int_BC to Reactor_2
        Release Reactor_3 from Reaction_1
        Assign Reactor_3 with capacity 120 kg to task Reaction_2 for 2 hours
        Transfer 48.0 kg from Hot_A to Reactor_3
        Transfer 72.0 kg from Int_BC to Reactor_3
        Release Heater from Heating
        Release Reactor_1 from Reaction_1
        Assign Reactor_1 with capacity 80 kg to task Reaction_2 for 2 hours
        Transfer 32.0 kg from Hot_A to Reactor_1
        Transfer 48.0 kg from Int_BC to Reactor_1

    Inventories are now:
        Feed_A       324.0 kg
        Feed_B       360.0 kg
        Feed_C       360.0 kg
        Hot_A         64.0 kg
        Int_AB         0.0 kg
        Int_BC       112.0 kg
        Impure_E       0.0 kg
        Product_1      0.0 kg
        Product_2      0.0 kg

    Unit Assignments are now:
        Reactor_2 performs the Reaction_2 task with a 80.00 kg batch for hour 1.000000 of 2.000000
        Reactor_3 performs the Reaction_2 task with a 120.00 kg batch for hour 1.000000 of 2.000000
        Reactor_1 performs the Reaction_2 task with a 80.00 kg batch for hour 1.000000 of 2.000000

--------------------------------------------------------------------------------------------

Time = 3 hr
    Instructions:

    Inventories are now:
        Feed_A       324.0 kg
        Feed_B       360.0 kg
        Feed_C       360.0 kg
        Hot_A         64.0 kg
        Int_AB         0.0 kg
        Int_BC       112.0 kg
        Impure_E       0.0 kg
        Product_1      0.0 kg
        Product_2      0.0 kg

    Unit Assignments are now:
        Reactor_2 performs the Reaction_2 task with a 80.00 kg batch for hour 2.000000 of 2.000000
        Reactor_3 performs the Reaction_2 task with a 120.00 kg batch for hour 2.000000 of 2.000000
        Reactor_1 performs the Reaction_2 task with a 80.00 kg batch for hour 2.000000 of 2.000000

--------------------------------------------------------------------------------------------

Time = 4 hr
    Instructions:
        Transfer 48.0 kg from Reactor_2 to Int_AB
        Transfer 32.0 kg from Reactor_2 to Product_1
        Transfer 72.0 kg from Reactor_3 to Int_AB
        Transfer 48.0 kg from Reactor_3 to Product_1
        Transfer 48.0 kg from Reactor_1 to Int_AB
        Transfer 32.0 kg from Reactor_1 to Product_1
        Release Reactor_2 from Reaction_2
        Assign Reactor_2 with capacity 80 kg to task Reaction_2 for 2 hours
        Transfer 32.0 kg from Hot_A to Reactor_2
        Transfer 48.0 kg from Int_BC to Reactor_2
        Release Reactor_3 from Reaction_2
        Assign Reactor_3 with capacity 120 kg to task Reaction_3 for 1 hours
        Transfer 96.0 kg from Int_AB to Reactor_3
        Transfer 24.0 kg from Feed_C to Reactor_3
        Release Reactor_1 from Reaction_2
        Assign Reactor_1 with capacity 80 kg to task Reaction_3 for 1 hours
        Transfer 64.0 kg from Int_AB to Reactor_1
        Transfer 16.0 kg from Feed_C to Reactor_1

    Inventories are now:
        Feed_A       324.0 kg
        Feed_B       360.0 kg
        Feed_C       320.0 kg
        Hot_A         32.0 kg
        Int_AB         8.0 kg
        Int_BC        64.0 kg
        Impure_E       0.0 kg
        Product_1    112.0 kg
        Product_2      0.0 kg

    Unit Assignments are now:
        Reactor_2 performs the Reaction_2 task with a 80.00 kg batch for hour 1.000000 of 2.000000
        Reactor_3 performs the Reaction_3 task with a 120.00 kg batch for hour 1.000000 of 1.000000
        Reactor_1 performs the Reaction_3 task with a 80.00 kg batch for hour 1.000000 of 1.000000

--------------------------------------------------------------------------------------------

Time = 5 hr
    Instructions:
        Transfer 120.0 kg from Reactor_3 to Impure_E
        Transfer 80.0 kg from Reactor_1 to Impure_E
        Assign Still with capacity 200 kg to task Separation for 2 hours
        Transfer 200.0 kg from Impure_E to Still
        Release Reactor_3 from Reaction_3
        Assign Reactor_3 with capacity 120 kg to task Reaction_1 for 2 hours
        Transfer 60.0 kg from Feed_B to Reactor_3
        Transfer 60.0 kg from Feed_C to Reactor_3
        Release Reactor_1 from Reaction_3
        Assign Reactor_1 with capacity 80 kg to task Reaction_2 for 2 hours
        Transfer 32.0 kg from Hot_A to Reactor_1
        Transfer 48.0 kg from Int_BC to Reactor_1

    Inventories are now:
        Feed_A       324.0 kg
        Feed_B       300.0 kg
        Feed_C       260.0 kg
        Hot_A          0.0 kg
        Int_AB         8.0 kg
        Int_BC        16.0 kg
        Impure_E       0.0 kg
        Product_1    112.0 kg
        Product_2      0.0 kg

    Unit Assignments are now:
        Still performs the Separation task with a 200.00 kg batch for hour 1.000000 of 2.000000
        Reactor_2 performs the Reaction_2 task with a 80.00 kg batch for hour 2.000000 of 2.000000
        Reactor_3 performs the Reaction_1 task with a 120.00 kg batch for hour 1.000000 of 2.000000
        Reactor_1 performs the Reaction_2 task with a 80.00 kg batch for hour 1.000000 of 2.000000

--------------------------------------------------------------------------------------------

Time = 6 hr
    Instructions:
        Transfer 180.0 kg from Still to Product_2
        Transfer 48.0 kg from Reactor_2 to Int_AB
        Transfer 32.0 kg from Reactor_2 to Product_1
        Release Reactor_2 from Reaction_2
        Assign Reactor_2 with capacity 80 kg to task Reaction_3 for 1 hours
        Transfer 56.0 kg from Int_AB to Reactor_2
        Transfer 14.0 kg from Feed_C to Reactor_2
        Assign Heater with capacity 100 kg to task Heating for 1 hours
        Transfer 90.6666666666667 kg from Feed_A to Heater

    Inventories are now:
        Feed_A       233.3 kg
        Feed_B       300.0 kg
        Feed_C       246.0 kg
        Hot_A          0.0 kg
        Int_AB         0.0 kg
        Int_BC        16.0 kg
        Impure_E       0.0 kg
        Product_1    144.0 kg
        Product_2    180.0 kg

    Unit Assignments are now:
        Still performs the Separation task with a 20.00 kg batch for hour 2.000000 of 2.000000
        Reactor_2 performs the Reaction_3 task with a 70.00 kg batch for hour 1.000000 of 1.000000
        Reactor_3 performs the Reaction_1 task with a 120.00 kg batch for hour 2.000000 of 2.000000
        Heater performs the Heating task with a 90.67 kg batch for hour 1.000000 of 1.000000
        Reactor_1 performs the Reaction_2 task with a 80.00 kg batch for hour 2.000000 of 2.000000

--------------------------------------------------------------------------------------------

Time = 7 hr
    Instructions:
        Transfer 20.0 kg from Still to Int_AB
        Transfer 70.0 kg from Reactor_2 to Impure_E
        Transfer 120.0 kg from Reactor_3 to Int_BC
        Transfer 90.6666666666667 kg from Heater to Hot_A
        Transfer 48.0 kg from Reactor_1 to Int_AB
        Transfer 32.0 kg from Reactor_1 to Product_1
        Release Still from Separation
        Release Reactor_2 from Reaction_3
        Assign Reactor_2 with capacity 80 kg to task Reaction_3 for 1 hours
        Transfer 64.0 kg from Int_AB to Reactor_2
        Transfer 16.0 kg from Feed_C to Reactor_2
        Release Reactor_3 from Reaction_1
        Assign Reactor_3 with capacity 120 kg to task Reaction_2 for 2 hours
        Transfer 48.0 kg from Hot_A to Reactor_3
        Transfer 72.0 kg from Int_BC to Reactor_3
        Release Heater from Heating
        Release Reactor_1 from Reaction_2
        Assign Reactor_1 with capacity 80 kg to task Reaction_2 for 2 hours
        Transfer 32.0 kg from Hot_A to Reactor_1
        Transfer 48.0 kg from Int_BC to Reactor_1

    Inventories are now:
        Feed_A       233.3 kg
        Feed_B       300.0 kg
        Feed_C       230.0 kg
        Hot_A         10.7 kg
        Int_AB         4.0 kg
        Int_BC        16.0 kg
        Impure_E      70.0 kg
        Product_1    176.0 kg
        Product_2    180.0 kg

    Unit Assignments are now:
        Reactor_2 performs the Reaction_3 task with a 80.00 kg batch for hour 1.000000 of 1.000000
        Reactor_3 performs the Reaction_2 task with a 120.00 kg batch for hour 1.000000 of 2.000000
        Reactor_1 performs the Reaction_2 task with a 80.00 kg batch for hour 1.000000 of 2.000000

--------------------------------------------------------------------------------------------

Time = 8 hr
    Instructions:
        Transfer 80.0 kg from Reactor_2 to Impure_E
        Assign Still with capacity 200 kg to task Separation for 2 hours
        Transfer 150.0 kg from Impure_E to Still
        Release Reactor_2 from Reaction_3

    Inventories are now:
        Feed_A       233.3 kg
        Feed_B       300.0 kg
        Feed_C       230.0 kg
        Hot_A         10.7 kg
        Int_AB         4.0 kg
        Int_BC        16.0 kg
        Impure_E       0.0 kg
        Product_1    176.0 kg
        Product_2    180.0 kg

    Unit Assignments are now:
        Still performs the Separation task with a 150.00 kg batch for hour 1.000000 of 2.000000
        Reactor_3 performs the Reaction_2 task with a 120.00 kg batch for hour 2.000000 of 2.000000
        Reactor_1 performs the Reaction_2 task with a 80.00 kg batch for hour 2.000000 of 2.000000

--------------------------------------------------------------------------------------------

Time = 9 hr
    Instructions:
        Transfer 135.0 kg from Still to Product_2
        Transfer 72.0 kg from Reactor_3 to Int_AB
        Transfer 48.0 kg from Reactor_3 to Product_1
        Transfer 48.0 kg from Reactor_1 to Int_AB
        Transfer 32.0 kg from Reactor_1 to Product_1
        Assign Reactor_2 with capacity 80 kg to task Reaction_3 for 1 hours
        Transfer 60.0 kg from Int_AB to Reactor_2
        Transfer 15.0 kg from Feed_C to Reactor_2
        Release Reactor_3 from Reaction_2
        Assign Reactor_3 with capacity 120 kg to task Reaction_2 for 2 hours
        Transfer 10.66666666666668 kg from Hot_A to Reactor_3
        Transfer 16.000000000000018 kg from Int_BC to Reactor_3
        Release Reactor_1 from Reaction_2
        Assign Reactor_1 with capacity 80 kg to task Reaction_3 for 1 hours
        Transfer 64.0 kg from Int_AB to Reactor_1
        Transfer 16.0 kg from Feed_C to Reactor_1

    Inventories are now:
        Feed_A       233.3 kg
        Feed_B       300.0 kg
        Feed_C       199.0 kg
        Hot_A          0.0 kg
        Int_AB         0.0 kg
        Int_BC         0.0 kg
        Impure_E       0.0 kg
        Product_1    256.0 kg
        Product_2    315.0 kg

    Unit Assignments are now:
        Still performs the Separation task with a 15.00 kg batch for hour 2.000000 of 2.000000
        Reactor_2 performs the Reaction_3 task with a 75.00 kg batch for hour 1.000000 of 1.000000
        Reactor_3 performs the Reaction_2 task with a 26.67 kg batch for hour 1.000000 of 2.000000
        Reactor_1 performs the Reaction_3 task with a 80.00 kg batch for hour 1.000000 of 1.000000

--------------------------------------------------------------------------------------------

Time = 10 hr
    Instructions:
        Transfer 15.0 kg from Still to Int_AB
        Transfer 75.0 kg from Reactor_2 to Impure_E
        Transfer 80.0 kg from Reactor_1 to Impure_E
        Release Still from Separation
        Assign Still with capacity 200 kg to task Separation for 2 hours
        Transfer 155.0 kg from Impure_E to Still
        Release Reactor_2 from Reaction_3
        Release Reactor_1 from Reaction_3

    Inventories are now:
        Feed_A       233.3 kg
        Feed_B       300.0 kg
        Feed_C       199.0 kg
        Hot_A          0.0 kg
        Int_AB        15.0 kg
        Int_BC         0.0 kg
        Impure_E       0.0 kg
        Product_1    256.0 kg
        Product_2    315.0 kg

    Unit Assignments are now:
        Still performs the Separation task with a 155.00 kg batch for hour 1.000000 of 2.000000
        Reactor_3 performs the Reaction_2 task with a 26.67 kg batch for hour 2.000000 of 2.000000

--------------------------------------------------------------------------------------------

Time = 11 hr
    Instructions:
        Transfer 139.5 kg from Still to Product_2
        Transfer 16.000000000000018 kg from Reactor_3 to Int_AB
        Transfer 10.66666666666668 kg from Reactor_3 to Product_1
        Release Reactor_3 from Reaction_2
        Assign Reactor_3 with capacity 120 kg to task Reaction_3 for 1 hours
        Transfer 31.0 kg from Int_AB to Reactor_3
        Transfer 7.75 kg from Feed_C to Reactor_3

    Inventories are now:
        Feed_A       233.3 kg
        Feed_B       300.0 kg
        Feed_C       191.2 kg
        Hot_A          0.0 kg
        Int_AB         0.0 kg
        Int_BC         0.0 kg
        Impure_E       0.0 kg
        Product_1    266.7 kg
        Product_2    454.5 kg

    Unit Assignments are now:
        Still performs the Separation task with a 15.50 kg batch for hour 2.000000 of 2.000000
        Reactor_3 performs the Reaction_3 task with a 38.75 kg batch for hour 1.000000 of 1.000000

--------------------------------------------------------------------------------------------

Time = 12 hr
    Instructions:
        Transfer 15.5 kg from Still to Int_AB
        Transfer 38.75 kg from Reactor_3 to Impure_E
        Release Still from Separation
        Release Reactor_3 from Reaction_3

    Inventories are now:
        Feed_A       233.3 kg
        Feed_B       300.0 kg
        Feed_C       191.2 kg
        Hot_A          0.0 kg
        Int_AB        15.5 kg
        Int_BC         0.0 kg
        Impure_E      38.8 kg
        Product_1    266.7 kg
        Product_2    454.5 kg

    Unit Assignments are now:

--------------------------------------------------------------------------------------------

Final Conditions
    Final Inventories:
        Feed_A       233.3 kg
        Feed_B       300.0 kg
        Feed_C       191.2 kg
        Hot_A          0.0 kg
        Int_AB        15.5 kg
        Int_BC         0.0 kg
        Impure_E      38.8 kg
        Product_1    266.7 kg
        Product_2    454.5 kg