You may also want to check out all available functions/classes of the module pyomo.opt, or try the search function . The first table is decomposition of the jobs into a series of tasks. As we see below, each additional set of three products takes an additionl 13 hours. Currently, it is set to be more than completion times of all the machines. 11 (1990): 1069-1072. How to use Pyomo - 10 common examples To help you get started, we've selected a few Pyomo examples, based on popular ways it is used in public projects. A task \((k,n) = \text{prec}_{j,m}\) that must be completed before task \((j,m)\). Use linear programming to determine how Dorian Auto can meet its advertising requirements at minimum cost. Examples Pyomo 5 6 9 documentation. Asking for help, clarification, or responding to other answers. The first step in the analysis is to decompose the process into a series of tasks. \[\begin{split} GAMS is a licensed software, for which we use a demo license in this course. flowshop.machines = Set(ordered=True) For example, if we charge the battery at maximum power rate (300MW), the battery will get 300 / 2 * 0.9 = 135MWh, where 15MWh were lost due to the conversion rate. $6,000,000 profit expected. The python code and the data are as follows. The python code and the data are as follows. The precedence variable corresponding to a pair of jobs can be found out using the indices of the jobs (for which the flowshop.jobs has to be an ordered set - flowshop.jobs = Set(ordered=True)). We can stick to Python! The 0/1 Knapsack Problem As a first example, consider the solution of the 0/1 knapsack problem: given a set \(I\)of items, each one with a weight \(w_i\)and estimated profit \(p_i\), one wants to select a subset with maximum Optimization by PYOMO in Python: A complete working example Jul 15, 2022 Keivan Tafakkori Step 1. Ipopt is a state-of-the-art optimization solver for nonlinear optimization problems. Transient Heat Conduction in Various Geometries, 6.2. Please note that some modules can be compiled to . In this problem, we need to find the optimum number of cars to be produced under given conditions. . This is an Abstract model. \begin{align*} \], \[ But for the purposes this notebook, we implement can implement a simple clean out policy with a single non-negative parameter \(t_{clean} \geq 0\) which, if specified, requires a period no less than \(t_{clean}\) between the finish of one task and the start of another on every piece of equipment. Each football commercial is seen by 2 million high-income women and 12 million high-income men. A rich set of features enables the modeling and analysis of complex systems. To learn more, see our tips on writing great answers. In this, we will try to optimize marketing budget allocations. We illustrate these techniques using Example II from Dunn (2013). OR-Library. Can leverage multiple solvers and libraries on python. To run, uncomment the the last lines in the cell. Because ROmodel and Pyomo are open-source, ROmodel can be extended to incorporate additional uncertainty set geometries and reformulations. In this example, there are three printed paper products that must pass through color printing presses in a particular order. The file jobshop1.txt contains 82 benchmark problems from a well-known collection of job shop scheduling problems in the OR-Library maintained by J. E. Beasley. The data on each line is a sequence of (machine number, time) pairs showing the order in which machines process each job. How do I solve this issue? Each task is indexed by an ordered pair \((j,m)\) where \(j\) is a job, and \(m\) is a machine. Dorian Auto is considering manufacturing three types of autos: compact, midsize, and large. Want to solve complex linear programming problems faster?Throw some Python at it!Linear programming is a part of the field of mathematical programming and is. points)) # create index objects b. dimensions = rangeset(0, ndim -1) b. simplices = rangeset(0, First, we will start with a linear programming example. Objective function maximizes or minimizes the given f(x) function under some constraints. Each task must be assigned to a single time slot, however, not every time slot is viable. In short: First we optimize F1 and F2 separately, just to know F2 values . to pyomo.@googlegroups.com. There are 3 jobs (chest, door and chair) and 3 machines (cutting, welding, packing in that order). This is an Abstract model Problem description There are 3 jobs (chest, door and chair) and 3 machines (cutting, welding, packing in that order). A number of power stations are committed to meeting the defined electricity load demands over a day. Data consists of two tables. After this, one can access any element by normal indexing - flowshop.machines[i], For the binary variables, one can declare them as -, Then, this variable can be used to decide the precedence between 2 jobs and to formulate the assignment constraints. Pyomo Optimization Modeling in Python William E Hart. There are 3 types of generating units available, 12 of type 1, 10 of type 2, and 5 of type 3. Solving the LA19 benchmark problem with NEOS, 4.3.12.2. How can we create psychedelic experiences for healthy people without drugs? The job shop scheduling problem is to schedule the jobs on the machines to minimize the time necessary to process all jobs (i.e, the makespan) or some other metric of productivity. Mathematical Model Formulated in Python Using Pyomo: The results of the algorithm can be seen below. Variable \(start_{j,m}\) denotes the time when task \((j,m)\) begins. Workshop Examples. Design of a Cold Weather Fuel for a Camping Stove, 2.6. This component can take a variety of arguments; this example illustrates use of the within option that is used by Pyomo to validate the data value that is assigned to the parameter. Associated with each task is data describing the time needed to perform the task, and a preceding task that must be completed before the index task can start. You could not without help going when ebook addition or library or borrowing from your connections to entry them. I am trying to formulate a flowshop scheduling problem in Pyomo. We can import our data to Python, conduct data analysis, machine learning, and optimization. x1 = Compact Cars, x2 = Midsize Cars, x3 = Large Cars, Maximize Z = 2000 * x1 + 3000 * x2 + 4000 * x3. In addition to meeting the estimated load demands, there must be sufficient generators working at any time to make it possible to meet an increase in load of up to 15%. For that, I tried flowshop.machines[-1], but it gives an error saying: 2000 midsize car to be manufactured. Enjoy our new release! Pyomo Installation Guide 4 Sandia National Laboratories Getting the books Pyomo Installation Guide 4 Sandia National Laboratories now is not type of inspiring means. The total amount of steel used for every car in total should not exceed 6,000 tons. Community Pyomo Forum Report a Bug. Scheduling with Disjunctive Constraints, 4.1. We have implemented 3 different problems in python using the Pyomo package. Decision variable examples: Constraints are logical conditions that a solution to an optimization problem must satisfy. Finally, for every task performed on machine \(m\), there can be no overlap among those tasks. Applegate, David, and William Cook. Robust optimization has been applied to many types of engineering problems. New Python content every day. Dorian would like the commercials to be seen by at least 28 million high-income women and 24 million high-income men. Different solvers work great at different tasks. Pyomo Gallery. This formulation is quite general, but can also specify situations with no feasible solutions. In most cases the time needed for clean out would be specific to the equipment and product. For marketing example, how much spent on radio or TV investment may be a decision variable. The given data consists of a flowsheet showing the order in which each job passes through the color presses. What does puncturing in cryptography mean, Math papers where the only issue is that someone else could've done it but didn't. The job shop scheduling problem is to schedule the jobs on the machines to minimize the time necessary to process all jobs (i.e, the makespan) or some other metric of productivity. Some of these problems can become quite larger, and therefore the gurobi solver has been set as a default. The constraints capture all the rules (not so realistic in this example!) If you dont have the gurobi solver then adjust the code to use the glpk solver, but know the calculations may take longer (and the benchmark problem will not solve at all). In the second problem, we will look at the car manufacturing case. You signed in with another tab or window. The following are 30 code examples of pyomo.environ.Constraint () . simplices) npoints = len( tri. Optimization algorithms aim to find the global optimum result. The notebooks in this collection were developed for instructional purposes at Notre Dame. . How to code efficiently, get familiarised with the techniques that will make your code scalable for large problems. Each task is a (job,machine) pair. The task data is stored in a Python dictionary indexed by (Job,Machine) tuples. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. For marketing example, how much spent on radio or TV investment may be a decision variable. The choice of decision variables for this problem are key to modeling. One needs to provide ordered=True argument in the while declaring the set - Earlier we found it tood 11.5 hours to produce one batch of product A. As an example, we implemented uncertainty sets based on (warped) Gaussian processes for black-box constrained problems. If you post a toy example I can work on an example implementation. Why is there no passive form of the present/past/future perfect continuous? Cross-Platform Installation of Pyomo and Solvers, 2.1. \(makespan\) is a candidate objective function. The constraints include lower bounda on the start and an upper bound on the completion of each task \((j,m)\). The solution may take several minutes, and depends on the current length of the NEOS job queue. The following cell specifies the solver to used in the subsequent calculations. The following cell may take many minutes to hours to run, depending on the choice of solver and hardware. 2 (1991): 149-156. pdf available, Beasley, John E. OR-Library: distributing test problems by electronic mail. Journal of the operational research society 41, no. Before going further, we create a function to streamline the generation of the TASKS dictionary. The order of the names is important. Keywords: job shop, scheduling, cbc usage, neos usage, cplex, gdp, disjunctive programming, batch processes. Do US public school students have a First Amendment right to be able to perform sacred music? Pyomo Anaconda Cloud. A 1-minute comedy ad costs 50,000 dollars and a 1-minute football ad costs 100,000 dollars. You may also want to check out all available functions/classes of the module pyomo.environ , or try the search function . Stack Overflow for Teams is moving to its own domain! Each job is a series of tasks that require use of particular machines for known duration, and which must be completed in specified order. In this post, we will shortly look at the components of optimization. \end{split}\], \[ Step-by-step modeling and optimization of a workforce design and assignation problem using Python and Pyomo. # This is a toy example for scheduling a sequence of reactions taking # place in a single reactor. Model Predictive Control of a Double Integrator, 4. Introduction to Modeling Optimization Problems in Python. For that, I believe, I have to get the last index of the machines set. The data format for each example consists of a single line for each job. next step on music theory as a guitar player. Code snippet is below. In pymoo, each objective function is supposed to be minimized, and each constraint needs to be provided in the form of \ (\leq 0\). Why so many wires in my old light fixture? A tag already exists with the provided branch name. In this Optimization course you will learn: How to formulate your problem and implement it in Python (Pyomo) and make optimal decisions in your real-life problems. This increase would have to be accomplished by adjusting the output generators already operating within their permitted limits. Model Variables Visualizing the results Step 7. To reach these groups, Dorian Auto has embarked on an ambitious TV advertising campaign and has decided to purchase 1-minute commercial spots on two types of programs: comedy shows and football games. The following cell shows how to solve a model using CPLEX, a high performance commericial solver, on NEOS. Developers Pyomo GitHub Site Acknowledgements No large and compact cars will be manufactured according to the result. The second table list task pairs where the first task must be completed before the second task can be started. So there is considerable efficiency gained by scheduling over longer intervals whenever possible. How to design an action block with a clearly defined conversion goal. The first table is decomposition of the jobs into a series of . Copyright 2022. Maximizing Concentration of an Intermediate in a Batch Reactor, 8.2. Find centralized, trusted content and collaborate around the technologies you use most. As we see here, we can produce a second batch with only 5.0 additional hours because some of the tasks overlap. We are going to solve this problem using open-source Pyomo optimization module. and a table of data showing, in minutes, the amount of time each job requires on each machine. That states we have reached the optimal result. Pyomo Tutorial Examples. \begin{align} Modeling Examples This chapter includes commented examples on modeling and solving optimization problems with Python-MIP. Using the optimization interface Step 3. What can I do if my pomade tin is 0.1 oz over the TSA limit? gdp import * m = ConcreteModel () When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. dae import * from pyomo. "" " b = block( concrete = true) ndim = len( xvars) nsimplices = len( tri. Defining or feeding datasets Step 5. Pyomo: Python Optimization Modeling Objects, cog-imperial / suspect / tests / polynomial / test_rules.py, test_division_rule_with_nonconstant_denominator, Pyomo / pyomo / pyomo / contrib / satsolver / test_satsolver.py, coin-or / rbfopt / tests / test_rbfopt_degree1_models.py, """Test the create_min_rbf_model function. Sensitivity analysis In C, why limit || and && to evaluate to booleans? If you do have the gurobi solver, edit the location of the executable to match the location on your computer. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. exercises_w_solns.zip. They reflect real-world limits. Defining the optimization model Step 2. Then we will mathematically formulate and solve 3 different problems (marketing budget allocation, car manufacturing, and energy optimization) using Python with Pyomo library. that ensure the solution returned by the model constitutes a feasible theatre schedule. Archived Linear optimization in Python Part 1 Solve. The same problem above is solved with sets. from pyomo. As the error says Cannot index unordered sets, the set flowshop.machines is not ordered. Read More Installation The easiest way to install Pyomo is to use pip. Introduction to Disjunctive Programming, 4.5. Cannot index unordered set machines. A job shop consists of a set of distinct machines that process jobs. The COVID-19 global recession is the deepest since the end of World War II (Figure 1). \begin{align} For example, in an optimization model for labor scheduling, the number of workers to employ during the morning shift in a factory may be a decision variable. Whatever names are given here will continue to be used to refer to the stages in the rest of the file. with Pyomo and Coopr. \end{align} The following are 10 code examples of pyomo.opt.SolverFactory(). Clean out operations are often slow and time consuming. Further connect your project with Snyk to gain real-time vulnerability One file for all these things! 2022 Moderator Election Q&A Question Collection, source error message="Index '('d1', 'i1')' is not valid for indexed component 'NFix'", How to constrain optimization based on number of negative values of variable in pyomo, Pyomo scheduling optimization problem with non-continuous objective function, Pyomo: TypeError: unhashable type: 'OrderedScalarSet', Constraint issue with pyomo involving a scalar, Job Shop Scheduling with Machine Constraint. Objective is to minimise the makespan. \end{align} What is the minimum amount of time (i.e, what is the makespan) for this set of jobs? At least 1000 cars have to be produced or shouldnt be produced at all to be economically feasible. Demand for electricity at different timestamps. The overlapping of tasks is the key to gaining efficiency in batch processing facilities. Constraints Next, we add our constraints. Each task lists a job name, name of the required machine, and task duration. Pyomo is an open-source package in Python, which we use Google Colaboratory to run. Computational impact of a zero-wait policy, A computational study of the job-shop scheduling problem., OR-Library: distributing test problems by electronic mail., Applications of optimization with Xpress-MP.. If f, h, or g has a quadratic nonlinear term(s), then it will be a special case: In the figure below we can see the difference between local and global minimum. \begin{align*} Pyomo Documentation Read the Docs. The number of generators used & started at different time periods are also listed below. The implementation consists of of a function JobShopModel(TASKS) that accepts a dictionary of tasks and returns a Pyomo model. Pyomo-Workshop-Summer-2018.pdf. With Pyomo, one can embed within Python an optimization model consisting of decision variables, constraints, and an optimization objective. Connect and share knowledge within a single location that is structured and easy to search. Currently, 6,000 tons of steel and 60,000 hours of labor are available. Pyomo and GLPK. The example optimizes the memory that stores elapsed time, the interval of time between two events. Are you sure you want to create this branch? This is an entirely simple means to specifically acquire lead by . Applications of optimization with Xpress-MP. contract (1999): 00034. Repeat the benchmark problem calculation, but with a zero-wait policy. I havent added all the constraints yet, I plan to add them after this issue gets fixed. Problem description Learn more about bidirectional Unicode characters. We have seen how to create mathematical models from the text. Further connect your project with Snyk to gain real-time vulnerability scanning and remediation. We introduce \(makespan\) as the time needed to complete all tasks. Decision variables are declared with the var () method, the objective and constraint are added with the += operator, and the solve () method is used to perform optimization. # This software is distributed under the 3-clause BSD License. model = pyo.AbstractModel() We declare the parameters m and n using the Pyomo Param component. Bethany. Not the answer you're looking for? A decision variable is a quantity that the decision-maker controls. # This is a toy example for scheduling a sequence of reactions taking, # place in a single reactor. # This is an optional code path that allows the script to be # run outside of Pyomo command-line. In this example, we used "FirstStage" and "SecondStage" but we could have used "EtapPrimero" and "ZweiteEtage" if we had wanted to. Data consists of two tables. When the problem variable & constraints size grows, sets must be used because constraint size grows exponentially. Optimization modelling, most of the time used as simply 'optimization', is a part of broader research field called Operations Research. The implementation consists of of a function JobShopModel (TASKS) that accepts a dictionary of tasks and returns a Pyomo model.. scanning and remediation. Nov 2, 2016, 10:55:56 AM. An alternative to solving on a laptop is to submit the job to NEOS, a free internet-based service for solving optimization problems hosted by the University of Wisconsin and utilizing high performance servers at locations across the globe. A computational study of the job-shop scheduling problem. ORSA Journal on computing 3, no. ND Pyomo Cookbook is a collection of notebooks showing the use Pyomo to solve modeling and optimization problems. pyomo.ConcreteModel object. Some solvers stuck on the local minimum or couldnt find the global minimum in reasonable time and computational resources. The job shop scheduling problem is implemented below in Pyomo. \], \[ You can vote up the ones you like or vote down the ones you don't like, and go to the original project or source file by following the links above each example. \begin{align} How do I simplify/combine these two methods for finding the smallest and largest int in an array? Pyomo is a python-based modelling language that allows us to formulate linear programming models by defining a set of decision variables, parameters, operational . Pyomo is a Python-based, open-source optimization modeling language with a diverse set of optimization capabilities. Dorian Auto manufactures luxury cars and trucks. A common feature of batch unit operations is a requirement that equipment be cleaned prior to reuse. Here we have 6 rules: 4. \end{align} returns: a pyomo block object containing variables and constraints that define the piecewise function. Simple Scheduling Problem Schedule 10 tasks. As we see here, by scheduling the production simultaneously, we can get all three batches done in just 15 hours. This results in minimizing \ (-f_2 (x)\) instead of maximizing \ (f_2 (x)\). Application to the scheduling of batch processes, 4.3.10. \text{start}_{k,n}+\text{dur}_{k,n}\leq\text{start}_{j,m}\ \ \ \ \text{for } (k,n) =\text{prec}_{j,m} Follow to join 500k+ monthly readers. Get exclusive access to writing opportunities and advice in our community Discord. The job shop sheduling problem is an old operation research problem. \text{start}_{k,n}+\text{Dur}_{k,n} = \text{start}_{j,m}\ \ \ \ \text{for } (k,n) =\text{Prec}_{j,m}\text{ and ZW is True} While this could be implemented on an equipment or product specific basis, here we add an optional ZW flag to the JobShop function that, by default, is set to False. The company believes that its most likely customers are high-income women and men. Decision variable examples: - Temperature of a Factory - Sales price Job shop scheduling is one of the classic problems in Operations Research. For example: python transport.py # This replicates what the Pyomo command-line tools does from pyomo.opt import SolverFactory opt = SolverFactory("glpk") results = opt.solve(model) # save results model.solutions.load_from(results) x = model.x._data . Home / About / Download / Documentation / Blog / Files for the Pyomo Workshop: exercises_wo_solns.zip. Does a creature have to see to be affected by the Fear spell initially since it is an illusion? We want to minimize the number of timeslots used. One of the issues in the use of job shop scheduling for chemical process operations are situations where there it is not possible to store intermediate materials. Provision & Configure Web Server on AWS EC2 Instance using Ansible, Learnings using Phoenix LiveView for Internal Web Applications, Using the CMI Tech EF-45N Iris Scanner with M2SYS eGovs Custom Identity Management Solutions, Quick Answer Is Windows Subsystem For Linux Good, # ==========================================================, https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf, http://edge.rit.edu/content/P18751/public/Google%20drive%20backup/Pyomo%20-%20Optimization%20Modeling%20in%20Python%2C%20Second%20Edition.pdf, https://www.udemy.com/course/mathematical-optimization-with-gams-and-pyomo-python/, Non-quadratic but still nonlinear terms are 1*2, 1^(0.5), The special case for NLP is Quadratically Constrained Program (QCP), The special case for MINLP is Mixed Integer Quadratically Constrained Program (MIQCP), Objective Function = Minimize Total Advertising Cost (in thousands of dollars), Total advertising cost = cost of comedy ads + cost of football ads = (cost per comedy ad) * (# comedy ads purchased) + (cost per football ad) * (# football ads purchased) =, Objective Function -> Z= 50 * x1 + 100 * x2, Constraint 1: Commercials must reach at least 28 million high-income women, Eq1: (HIW per comedy ad) x (# comedy ads purchased) + (HIW per football ad) x (# football ads purchased) >= 28, Constraint 2: Commercials must reach at least 24 million high-income men, Eq2: (HIM per comedy ad) x (# comedy ads purchased) + (HIM per football ad) x (# football ads purchased) >= 24, Status: ok and termination condition: optimal. Pyomo PyPI. In the table below, we can see the list of the solvers. Response of a First Order System to Step and Square Wave Inputs, 5.3. Which generators should be working in which periods of the day to minimized total cost? We will now turn our attention to the application of the job shop scheduling problem to the short term scheduling of batch processes. For example: python transport.py # This replicates what the Pyomo command-line tools does from pyomo.opt import SolverFactory opt = SolverFactory("glpk") results = opt.solve(model) # save results model.solutions.load_from(results) x = model.x._data . In pyomo we can do this when initializing the variable: 0 S O C ( t) b a t t e r y C a p a c i t y (4) # variables (all indexed by time) m.SOC = en.Var(m.Time, bounds= (0,batt.capacity), initialize=0) Integer constraints Here is where the "mixed integer" part of the formulation comes in. Lets now consider an optimal scheduling problem where we are wish to make two batches of Product A. Total cost of power generation = $ 988540. . Unfortunately, the only interface available is C/C++ or Fortran. This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. Additional Pyomo tutorials and examples can be found at the following links: Prof. Jeffrey Kantor's Pyomo Cookbook. With Pyomo, one can embed within Python an optimization model consisting of decision variables, constraints, and an optimization objective. Water leaving the house when water cut off. . \end{align*} + Follow. Read more Docs and Examples Pyomo documentation and examples are available online. Cannot retrieve contributors at this time. Scheduling Multipurpose Batch Processes using State-Task Networks, 5.1. By Jeffrey C. Kantor Implementing the model Step 6. A single linear objective function for our Pyomo model to maximise. \text{start}_{j,m} & \geq 0\\ No need to use a specific software language. \left[\text{start}_{j,m}+\text{dur}_{j,m} \leq \text{start}_{k,m}\right] \vee \left[\text{start}_{k,m}+\text{dur}_{k,m} \leq \text{start}_{j,m}\right] Published Feb 9, 2022. Pyomo includes a rich set of features enables the modeling and analysis of complex systems. Fix for free Installing a Pyomo/Python Development Environment, 1.5. Any preceding tasks must be completed before task \((j,m)\) can start. For this exercise, create a data format to include task-specific clean out times, and model the job shop model to accomodate this additional informaton. \], \[ A zero-wait policy requires subsequent processing machines to be available immediately upon completion of any task. Manne, Alan S. On the job-shop scheduling problem. Operations Research 8, no. The task data conists of a dictionary with duration (dur) and (Job,Machine) pair for any prerequisite task. Running Pyomo on the Notre Dame CRC Cluster, 1.6. Each comedy commercial is seen by 7 million high-income women and 2 million high-income men. For the production of a type of car to be economically feasible, at least 1,000 cars of that type must be produced. Our framework offers state of the art single- and multi-objective optimization algorithms and many more features related to multi-objective optimization such as visualization and decision making. \end{align*} \end{align} """, self.assertIsInstance(model, pyomo.environ.ConcreteModel), IDAES / idaes-pse / idaes / core / util / testing.py, # Attribute to switch flow basis for testing, cog-imperial / suspect / tests / convexity / test_rules.py, self, visitor, base, cvx_base, mono_base, bounds_base, expo, self, cvx_f, cvx_g, mono_f, mono_g, bounds_f, bounds_g, self, visitor, base, expo, cvx_expo, mono_expo, bounds_expo, cog-imperial / suspect / tests / monotonicity / test_rules.py, pyomo.core.kernel.component_map.ComponentMap, pyomo.core.kernel.component_set.ComponentSet, pyomo.pysp.util.config.safe_register_common_option. Why is proving something is NP-complete useful, and where can I use it? Formulate a MILP to maximize Dorians profit. \left[\text{start}_{j,m}+\text{dur}_{j,m} + t_{clean} \leq \text{start}_{k,m}\right] \vee \left[\text{start}_{k,m}+\text{dur}_{k,m} + t_{clean} \leq \text{start}_{j,m}\right] To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Production Models with Linear Constraints, 2.2. PS - I am also struggling to model the binary variables used to define the precedence of a job. Why are only 2 out of the 3 boosters on Falcon Heavy reused? 2 (1960): 219-223. The idea is that there are M machines and also i jobs. Our selection contains examples from a wide variety of industries including manufacturing , financial services , energy and utilities , telecommunications, and oil & gas as well as examples that have relevance across various industries such as marketing campaign optimization and supply network design.

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