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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Solve a Generalized Assignment Problem using Lagrangian relaxation
This tutorial includes data and information that you need to set up decision optimization engines and build mathematical programming models to solve a Generalized Assignment Problem using Lagrangian relaxation.
When you finish this tutorial, you'll have a foundational knowledge of Prescriptive Analytics.
This notebook is part of Prescriptive Analytics for Python
It requires either an installation of CPLEX Optimizers or it can be run on IBM Cloud Pak for Data as a Service (Sign up for a free IBM Cloud account
and you can start using IBM Cloud Pak for Data as a Service right away).
CPLEX is available on <i>IBM Cloud Pack for Data</i> and <i>IBM Cloud Pak for Data as a Service</i>
Step1:  If CPLEX is not installed, you can install CPLEX Community edition.
Step2:  2. Model the data
In this scenario, the data is simple. It is delivered as 3 input arrays
Step3:  3. Set up the prescriptive model
Start with viewing the environment information. This information should be updated when you run the notebook.
Step4:  We will firt create an optimization problem, composed of 2 basic constraints blocks, then we will resolve it using Lagrangian Relaxation on 1 of the constraints block.
3.1 Create the DOcplex model
The model contains the business constraints and the objective.
Step5:  3.2 Define the decision variables
Step6:  3.3 Define the business constraints
Step7:  3.4. Solve the model
Use the Decision Optimization to solve the model.
Step8:  3.5. Solve the model with Lagrangian Relaxation method
Let's consider for the demonstration of the Lagrangian Relaxation that this model was hard to solve for CPLEX.
We will approximate this problem by doing an iterative model, where the objective is modified at each iteration. 
(Wait a few seconds for the solution, due to a time limit parameter.)
We first remove the culprit constraints from the model
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Python Code:
import sys
try:
    import docplex.mp
except:
    raise Exception('Please install docplex. See https://pypi.org/project/docplex/')
Explanation: Solve a Generalized Assignment Problem using Lagrangian relaxation
This tutorial includes data and information that you need to set up decision optimization engines and build mathematical programming models to solve a Generalized Assignment Problem using Lagrangian relaxation.
When you finish this tutorial, you'll have a foundational knowledge of Prescriptive Analytics.
This notebook is part of Prescriptive Analytics for Python
It requires either an installation of CPLEX Optimizers or it can be run on IBM Cloud Pak for Data as a Service (Sign up for a free IBM Cloud account
and you can start using IBM Cloud Pak for Data as a Service right away).
CPLEX is available on <i>IBM Cloud Pack for Data</i> and <i>IBM Cloud Pak for Data as a Service</i>:
   - <i>IBM Cloud Pak for Data as a Service</i>: Depends on the runtime used:
        - <i>Python 3.x</i> runtime: Community edition
        - <i>Python 3.x + DO</i> runtime: full edition
   - <i>Cloud Pack for Data</i>: Community edition is installed by default. Please install DO addon in Watson Studio Premium for the full edition
Some familiarity with Python is recommended.
Table of contents
Describe the business problem
How Decision Optimization can help 
Use Decision Optimization to create and solve the model
Summary<br> 
Describe the business problem
This notebook illustrates how to solve an optimization model using Lagrangian relaxation technics. 
It solves a generalized assignment problem (GAP), as defined by Wolsey, using this relaxation technic.
The main aim is to show multiple optimization through modifications of different models existing in a single environment, not to show how to solve a GAP problem.
In the field of Mathematical Programming, this technic consists in the approximation of a difficult constrained problem by a simpler problem: you remove difficult constraints by integrating them in the objective function, penalizing it if the constraint is not respected.
The method penalizes violations of inequality constraints using a Lagrange multiplier, which imposes a cost on violations. These added costs are used instead of the strict inequality constraints in the optimization. In practice, this relaxed problem can often be solved more easily than the original problem.
For more information, see the following Wikipedia articles: Generalized assignment problem and Lagrangian relaxation.
This notebook first solves the standard problem (which is not important here), then shows how to reformulate it to meet the Lagrangian Relaxation features.
How  decision optimization can help
Prescriptive analytics (decision optimization) technology recommends actions that are based on desired outcomes. It considers specific scenarios, resources, and knowledge of past and current events. With this insight, your organization can make better decisions and have greater control over business outcomes.
Prescriptive analytics is the next step on the path to insight-based actions. It creates value through synergy with predictive analytics, which analyzes data to predict future outcomes. Prescriptive analytics takes that insight to the next level by suggesting the optimal way to handle a future situation. Organizations that act fast in dynamic conditions and make superior decisions in uncertain environments gain a strong competitive advantage.
With prescriptive analytics, you can:
Automate the complex decisions and trade-offs to better manage your limited resources.
Take advantage of a future opportunity or mitigate a future risk.
Proactively update recommendations based on changing events.
Meet operational goals, increase customer loyalty, prevent threats and fraud, and optimize business processes.
Use Decision Optimization
Perform the following steps to create and solve the model.
Import the library<br>
Model the Data<br>
Set up the prescriptive model<br>
      3.1 Define the decision variables<br>
      3.2 Express the business constraints<br>
      3.3 Express the objective<br>
      3.4 Solve the model<br>
      3.5 Solve the model with Lagrangian Relaxation<br>
Investigate the solution and run an example analysis<br>
1. Import the library
Run the following code to import the Decision Optimization CPLEX Modeling library.  The DOcplex library contains two modeling packages, mathematical programming (docplex.mp) package and constraint programming (docplex.cp) package.
End of explanation
try:
    import cplex
except:
    raise Exception('Please install CPLEX. See https://pypi.org/project/cplex/')
Explanation: If CPLEX is not installed, you can install CPLEX Community edition.
End of explanation
B = [15, 15, 15]
C = [
    [ 6, 10, 1],
    [12, 12, 5],
    [15,  4, 3],
    [10,  3, 9],
    [8,   9, 5]
]
A = [
    [ 5,  7,  2],
    [14,  8,  7],
    [10,  6, 12],
    [ 8,  4, 15],
    [ 6, 12,  5]
]
Explanation: 2. Model the data
In this scenario, the data is simple. It is delivered as 3 input arrays: A, B, and C. The data does not need changing or refactoring.
End of explanation
from docplex.mp.environment import Environment
env = Environment()
env.print_information()
Explanation: 3. Set up the prescriptive model
Start with viewing the environment information. This information should be updated when you run the notebook.
End of explanation
from docplex.mp.model import Model
mdl = Model("GAP per Wolsey")
Explanation: We will firt create an optimization problem, composed of 2 basic constraints blocks, then we will resolve it using Lagrangian Relaxation on 1 of the constraints block.
3.1 Create the DOcplex model
The model contains the business constraints and the objective.
End of explanation
print("#As={}, #Bs={}, #Cs={}".format(len(A), len(B), len(C)))
number_of_cs = len(C)
# variables
x_vars = [mdl.binary_var_list(c, name=None) for c in C]
Explanation: 3.2 Define the decision variables
End of explanation
# constraints
cts = mdl.add_constraints(mdl.sum(xv) <= 1 for xv in x_vars)
mdl.add_constraints(mdl.sum(x_vars[ii][j] * A[ii][j] for ii in range(number_of_cs)) <= bs for j, bs in enumerate(B))
# objective
total_profit = mdl.sum(mdl.scal_prod(x_i, c_i) for c_i, x_i in zip(C, x_vars))
mdl.maximize(total_profit)
mdl.print_information()
Explanation: 3.3 Define the business constraints
End of explanation
s = mdl.solve()
assert s is not None
obj = s.objective_value
print("* GAP with no relaxation run OK, best objective is: {:g}".format(obj))
Explanation: 3.4. Solve the model
Use the Decision Optimization to solve the model.
End of explanation
for ct in cts:
    mdl.remove_constraint(ct)
#p_vars are the penalties attached to violating the constraints
p_vars = mdl.continuous_var_list(C, name='p')  # new for relaxation
# new version of the approximated constraint where we apply the penalties
mdl.add_constraints(mdl.sum(xv) == 1 - pv for xv, pv in zip(x_vars, p_vars))
;
#Define the maximum number of iterations
max_iters = 10
number_of_cs = len(C)
c_range = range(number_of_cs)
# Langrangian relaxation loop 
eps = 1e-6
loop_count = 0
best = 0
initial_multiplier = 1
multipliers = [initial_multiplier] * len(C)
# Objective function
# I'd write the key perfromance indicator (kpi) as
# total_profit = mdl.sum(mdl.sum(x_vars[task][worker] * C[task][worker]) for task, worker in zip(tasks, workers))
total_profit = mdl.sum(mdl.scal_prod(x_i, c_i) for c_i, x_i in zip(C, x_vars))
mdl.add_kpi(total_profit, "Total profit")
print("starting the loop")
while loop_count <= max_iters:
    loop_count += 1
    # Rebuilt at each loop iteration
    total_penalty = mdl.scal_prod(p_vars, multipliers)
    
    mdl.maximize(total_profit + total_penalty)
    s = mdl.solve()
    if not s:
        print("*** solve fails, stopping at iteration: %d" % loop_count)
        break
    best = s.objective_value
    penalties = [pv.solution_value for pv in p_vars]
    print('%d> new lagrangian iteration:\n\t obj=%g, m=%s, p=%s' % (loop_count, best, str(multipliers), str(penalties)))
    do_stop = True
    justifier = 0
    for k in c_range:
        penalized_violation = penalties[k] * multipliers[k]
        if penalized_violation >= eps:
            do_stop = False
            justifier = penalized_violation
            break
    if do_stop:
        print("* Lagrangian relaxation succeeds, best={:g}, penalty={:g}, #iterations={}"
                .format(best, total_penalty.solution_value, loop_count))
        break
    else:
        # Update multipliers and start the loop again.
        scale_factor = 1.0 / float(loop_count)
        multipliers = [max(multipliers[i] - scale_factor * penalties[i], 0.) for i in c_range]
        print('{0}> -- loop continues, m={1!s}, justifier={2:g}'.format(loop_count, multipliers, justifier))
print(best)
Explanation: 3.5. Solve the model with Lagrangian Relaxation method
Let's consider for the demonstration of the Lagrangian Relaxation that this model was hard to solve for CPLEX.
We will approximate this problem by doing an iterative model, where the objective is modified at each iteration. 
(Wait a few seconds for the solution, due to a time limit parameter.)
We first remove the culprit constraints from the model
End of explanation
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