Big-M formulations are relatively straightforward, but the value of the M term This paper shows auxiliary variables are not strictly necessary, at least for binary variables: http://www.optimization-online.org/DB_FILE/2014/04/4309.pdf. endobj on the value of the integrality tolerance, trickle flow can result in generally subject to TAMSTAF1_TAF1TDMS_define {d in ND, t in F1[d], s in NS, c in NC}: I would like to ask you if this formulation is correct. Are indicator constraints immune to trickle flow or other numerics-induced logic 'errors'? << /S /GoTo /D (Outline0.6) >> Single reference for Mixed Integer Programming formulations to linearize, handle logical constraints and disjunctive constraints, do Big M, etc? In gurobipy this is written as model.addConstr ( (x == 1) >> (y + z <= 5)) where x is a binary variable, y and z are integer variables. If the big-M factor is eliminated in presolve. Given the trade-offs indicated by Ed Klotz, one might wonder: "Why not both"? What if due to That is, add the big-M for the strenghthened LP relaxation and the indicator for the "clean" branching effect. With Concert and CPLEX 10, the default formulation of linearized operators, Here some details in addition to the IBM Technote: Difference between using indicator constraints and a big-M formulation . endobj I read the article on indicator constraint but it was not lear to me 0 Please sign in Are indicator constraints immune to trickle flow or other For this reason, Gurobi might be able to produce a smaller or tighter representation of the general constraint than you would get from the most general formulation. = times [ 0 ]) introduce trickle-flow problems in node LP solutions. Why can we add/substract/cross out chemical equations for Hess law? Does the setting of can produce the wrong solution? Then, it introduces a new quadratic objective to balance the workload among the workers. Both of these constructions typically use a binary variable to I have placed the details, provided Ed Klotz of IBM, in a separate answer to this question. Use MathJax to format equations. 39 0 obj If neither CPLEX 10.0 nor 10.1 can efficiently solve the model formulated TAMS_TAF1[d,t,s,c]+TAF1_TDMS[d,t,s,c]=0 or 1; Consider using a <= condition for the "else" part. It must have unique values and cannot contain nulls. The price that you'll be paying is extra computing time. anystate-of-the-artcommercialMIQPsolverlikeIBM-Cplex,Gurobi andXpress. The goal is to have the constraint "always succeed" when I is false (essentially turning it off). If the model does not show any side effects from a big-M formulation. and big M formulation that is described in the above technote. I have no idea how they are encoded with SOS constraints. When the big-M factor remains very large, relative to other coefficients endobj Big M formulations are subject to logic "errors" due to "trickle flow". has been eliminated from your model by preprocessing. If the big-M factor is eliminated in presolve. See also the discussion here: Why is it important to choose big-M carefully and what are the consequences of doing it badly? << /S /GoTo /D (Outline0.2.1.10) >> For Gurobi, based on the material presented here, there are numerical issues associated with Big-M since the real numbers are not really real in computer. I have no idea how they are encoded with SOS constraints. I was formulating my answer when you posted this. 32 0 obj endobj which I just posted on the CPLEX forum. 2 ; You do not have permission to delete messages in this group, Either email addresses are anonymous for this group or you need the view member email addresses permission to view the original message, I have had this statement from AMPL:Gurobi 9.0.0: Gurobi can't handle nonquadratic nonlinear constraints. moment. The bigger the M, the more the solver will struggle. reference on how implementing it ? Hi, How do you encode an implication like the above in MIP? 3 0 obj If I figure it out I will inform you. similar to the content of the above technote, but it does provide a Sign in Are there any plans to expose the indicator constraint features of Gurobi? Only $z=0$ (zero) or $z=1$ (one) is allowed for the indicator variable because the indicator constraint implies that the indicator Would be nice to have an example code then. If the model does not show any side effects from a big-M formulation. 59 0 obj Demonstrates optimization with multiple objective functions. To the best of my knowledge the indicator constraints are just syntactic sugar for the user. Where M is some "very large number", a big M so to say, e.g. Now, the ROLL_NO field must have the value greater than 1000. See. We are currently running the latest tests with CBC to We are now working to improve the cbc binaries, but this will be my next task ! Sign up for a free GitHub account to open an issue and contact its maintainers and the community. Because I am having such a problem at the Then it very likely that your custom made big-M formulation yields a better bound compared to the automatically generated reformulation. << /pgfprgb [/Pattern /DeviceRGB] >> But for modest values of big M of say issue associated with big M formulations. xTMs0WB_ 0(5VS :&C,;N 0\{+D]KR! Thanks for contributing an answer to Operations Research Stack Exchange! endobj Since we are adding SOS, this relate a binary variable to a continuous variable or expression. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. 19 0 obj Modified date: And if so, would they be replicated in Python-MIP for CBC by using one of the encoding tricks above? python MIP version). By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. (With a little help of my friends) endobj (A class of surprising problems) If so, how should I write the code? If it is way too big, you'll get numerical errors. Could you edit to elaborate a bit? -- They make modelling complex boolean I write to you again because I have an another preoccupation. << /S /GoTo /D (Outline0.3.1.22) >> indicator constraints first appeared around CPLEX 10.0. The best answers are voted up and rise to the top, Not the answer you're looking for? 27 0 obj A constraint in Gurobi captures a restriction on the values that a set of variables may take. Use smallest possible value of big-M that doesnt violate intent of model, Bound strengthening in CPLEX presolve often does this automatically, Set simplex tolerances to minimum values, 1e-9, Ask for more accuracy on an ill-conditioned system. 66 0 obj << like IloIfThen, IloAbs, and IloOr, just slow solution, but wrong solution) if they are used in lieu of The Big-M formulation is the simplest but variables need tight bounds for it to work nicely. Is this constraint possible in Gurobi based on the Python language? Years ago, I attended an IBM workshop at an INFORMS conference. Stack Overflow for Teams is moving to its own domain! Connect and share knowledge within a single location that is structured and easy to search. turn on or turn off the enforcement of a constraint, or to https://www.scipopt.org/doc-7.0.0/html/cons__indicator_8h.php. Reply to this email directly, view it on GitHub, orunsubscribe. CPLEX 10 introduced indicator constraints as an alternative to big-M for example 1 or 2. in the blow is a semi code of my implementation: m = Model("mip1") subject to Contrainte_3 {d in ND, i in NS, j in NS: i <> j, c in NC, m in NC: m<>c}: sum {t in F1[d]}( X_F1[d,t,i,j]+TRUCK_CUST[d,t,i,c]+TRUCK_CUST[d,t,j,m]) <= Benefit number two - Gurobi may be able to leverage the implicit knowledge of what a constraint actually does in the solving process: However, general constraints have another potential advantage: Gurobi might be able to simplify parts of the MIP formulation if it can prove during presolve that the simplified version suffices for the correctness of the model. Already on GitHub? Indicator Constraints in Mixed-Integer Programming Andrea Lodi University of Bologna, Italy - andrea.lodi@unibo.it Amaya Nogales-Gmez, Universidad de Sevilla, Spain . This is the simplest possible way to get half-decent Ms. You really want to do some pre-solving where you minimize the Ms though, as Gurobi and CPLEX do, since that will give much better results by taking all constraints into account simultaneously (such that you have tighter bounds on every variable). in the model. Can indicator constraints ever be violated due to a similar Yes, I recently noticed after watching a workshop that the implementation is simple indeed. But how can I write TAMS_TAF1[d,t,s,c] + TAF1_TDMS[d,t,s,c] to tell that it must equal 0 or 1? In the below Operations Research Stack Exchange is a question and answer site for operations research and analytics professionals, educators, and students. 60 0 obj Please try again later or use one of the other support options on this page. this is not really a drawback. After that I can go back to my TODO Do not introduce indicator constraints if Big M can be eliminated from your model. There are different ways to implement this: (1) using binary variables, (2) using indicator constraints, and (3) using SOS1 sets. How many characters/pages could WordStar hold on a typical CP/M machine? << /S /GoTo /D (Outline0.4) >> To learn more, see our tips on writing great answers. Does it make sense to say that if someone was hired for an academic position, that means they were the "best"? indicator constraint, solving the associated relaxation, then INDICATOR CONSTRAINT Answered Parijat Mitra 1 year ago Can anybody explain what indicator constraint is in gurobi? I assume that if you're just going to plop down an arbitrarily large power of 10, indicators are preferable. Various optimization modeling languages and solvers allow for both indicator constraints (see for example here, here and here) and traditional binary variable and big-M approaches can be used to model whether a linear constraint such as $a'x \le b$ should be active in solving (mixed)-integer programs. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. Why is it important to choose big-M carefully and what are the consequences of doing it badly? When the big-M formulation is difficult to express, such as an if-then Bounds strengthen LP relaxations. IBM Technote: Why does a binary or integer variable take on a noninteger value in the solution?, depending endobj above? properties but is more complicated and needs more additional variables and Python code modeling a conditional statement in Gurobi might look similar to the following: import gurobipy as gp from gurobipy import GRB # Create a new model m = gp.Model ( "test") # Create variables I share your experience, using indicator constraints instead of "own big-M" can slow down the solving process. relaxation of an indicator constraint is obtained by removing the How do I simplify/combine these two methods for finding the smallest and largest int in an array? Haroldo.GambiniSantos@cs.kuleuven.be Hi, My intuition tells me that that should not really have a worse (maybe even better) performance than indicators. I wrote this equation and I got syntax error statement: Place all of the conditions at the end of the indexing expression, like this: On Fri, Apr 24, 2020 at 9:54 AM UTC, AMPL Modeling Language <, On Thu, Apr 23, 2020 at 4:00 AM UTC, AMPL Modeling Language <, On Thu, Apr 23, 2020 at 3:55 AM UTC, AMPL Modeling Language <, On Wed, Apr 22, 2020 at 7:13 PM UTC, AMPL Google Group <. %PDF-1.4 numerics-induced logic "errors"? A lazy constraint cannot have an indicator constraint. to consider supplying those bounds yourself in the formulation to From a talk given by IBM reps and other conversations, I think CPLEX handles them at least partially as big M constraints, where CPLEX deduces a suitable value of M. There may also be a tie-in to the branching logic, but I'm not sure how (and I hesitate to speculate). What can I do if my pomade tin is 0.1 oz over the TSA limit? 23 0 obj 44 0 obj phenomenon? tighten it. We do not plan to implement Gurobi only because I deleted all nonlinear constraints in myproblem. For example, a sample constraint is shown as follows: f = [1.0, 1.0, 1.0, 1.0] x = m.addVars (4, lb=0, ub=15, vtype=GRB.INTEGER) m.addConstr (x.prod (f) == 10 or 15, name="") This constraint can be equal to multiple values, such as 10 or 15. You can write out the You might have some logical expression in your constraints, which Gurobi would reject as "nonlinear" since it is not a linear expression. I'm not sure how much I trust the advice to use indicators in preference to big-M if M cannot be "reduced" (from what to what?). Example usage: # x7 = 1 -> x1 + 2 x3 + x4 = 1 model.addGenConstrIndicator (x7, True, x1 + 2*x2 + x4, GRB.EQUAL, 1.0) # alternative form model.addGenConstrIndicator (x7, True, x1 + 2*x2 + x4 == 1.0) # overloaded form model.addConstr ( (x7 == 1) >> (x1 + 2*x2 + x4 == 1.0)) become 36 0 obj endobj http://www.optimization-online.org/DB_FILE/2014/04/4309.pdf, https://link.springer.com/article/10.1007/s10589-016-9847-8, https://www.scipopt.org/doc-7.0.0/html/cons__indicator_8h.php. endobj constraints extremely easy, in fact Gurobi and CPLEX also include boolean AND and OR I rewrote the variable that I wrote incorrectly (var TAMSTAF1_TAF1TDMS {d in ND, t in F1[d], s in NS, c in NC} = if TAMS_TAF1[d,t,s,c]=TAF1_TDMS[d,t,s,c]=1 then 1 else 0 ) this way: On Sat, Apr 25, 2020 at 2:52 AM UTC, AMPL Modeling Language <, On Fri, Apr 24, 2020 at 5:07 PM UTC, AMPL Google Group <. (Bound tightening and bigM reduction) 40 0 obj In the above example we have set the check constraint on ROLL_NO column of STUDENT table. (Deactivating Linear Constraints) as low as possible). Namely, the endobj 43 0 obj Consider using the big-M form instead of indicators: Consider using indicator constraints instead of big-M: In all cases, defining upper bound information on the continuous I just asked a question on the CPLEX forum. "On handling indicator constraints in mixed integer programming" Many thanks in advance for your reply and your help. This obviously requires all variables to be bounded, the tighter the better. In SCIP, I think that the indicator constraint may be able to dynamically strenghten the linear relaxation during the solve process, whenever the bounds of the relevant variables are tightened (can not find a reference at the moment). For more information about indicator constraints, see the "Using If you choose to introduce Big M values in your model anyway, use the smallest possible value of << /S /GoTo /D (Outline0.1.1.2) >> ibm.com/developerworks/community/forums/html/. % It might not properly disable the constraint then. too large, the model may become numerically difficult or exhibit G33NNVSW45C7NFSM4HJ7HKQA.gif]. How to add Binary Variable with condition in LP, How can I solve a linear optimization problem with bounds that are a function of the decision, Is a convex or MILP (without big-M) formulation possible for this problem, Heuristics for mixed integer linear and nonlinear programs. >> endobj 56 0 obj << /S /GoTo /D (Outline0.2.2.15) >> Well, if b is 0, then the constraint becomes irrelevant, i.e, we don't want it to actually constraint anything. If [the solver] can not efficiently solve the model formulated with indicator constraints. For example, for a linear program in canonical form: max c t x Ax = b x 0 The interface takes the matrix A and vectors b and c, and returns the optimal solution. IBM Technote: Difference between using indicator constraints and a big-M formulation, Klotz and Wunderling:Tools for Adapting Math Programming Solutions in the Real World, Mobile app infrastructure being decommissioned. dummy 311 is integer and dummy312 is binary. How can we build a space probe's computer to survive centuries of interstellar travel? For Gurobi, define a binary variable and a separate constraint: On Tue, Apr 21, 2020 at 8:38 PM UTC, AMPL Modeling Language <, On Tue, Apr 21, 2020 at 4:28 PM UTC, AMPL Google Group <. }cpzPHYVS`Bn iu gh2S+. g&g? << /S /GoTo /D (Outline0.2) >> You can write out the presolved model to check this condition. It can be calculated by getting the upper bound of (a * x) + (b * y) and subtracting c from it. fixing and so forth. strengthening and other MIP preprocessing methods, then you might want a deadly grudge lifetime cast; pop os emacs hornby r8213 polymer 80 rails not lining up; virgin vs chad female version middle child and only child compatibility how to find arlo wifi password; narcissist and flying monkeys youtube mint mobile mnc 260 not working mint mobile mnc 260 not working Universidade Federal de Ouro Preto - UFOP [AB4VZOQ5GS55JKWFAUUBYT3RHAFSPA5CNFSM4IURLVU2YY3PNVWWK3TUL52HS4DFVREXG43VMVBW63LNMVXHJKTD Here is the advice in the IBM CPLEX documentation. Big-M formulations are relatively straightforward, but the value of Best Practices with Indicator Constraints. Making statements based on opinion; back them up with references or personal experience. (Interpreting the numbers) I will have to defer to someone else as to how indicator constraints are handled internally in the solver, for instance in CPLEX, and to what extent SOS may or may not be involved. Slides 19-23 of Klotz and Wunderling:Tools for Adapting Math Programming Solutions in the Real World may help a bit; the info is x y - OR x y + . These interfaces use a matrix representation. Full support for indicator constraints, if it happens, should be at the CBC level and not Python-MIP. Unfortunately, indicator constraints are not supported by the Gurobi MATLAB and R interfaces. endobj The following model has been developed to deter-mine how much of each type should be produced to maximize profit subject to a . # Indicator constraints z = [] # Initialization of an input for nonlinear-type formulation t = -1 for i in range ( 10 ): m.addConstr ( (b [i] == 1) >> (t == i)) z.append (t) # Objective function obj = gp.quicksum (x [i] + y [i] for i in range ( 10 )) m.setObjective (obj + tmp_f (z), GRB.MAXIMIZE) m.optimize () 0 For huge big M values, constraint on complex expressions. integrality tolerance have any effect on whether indicator constraints Computing Department As discussed at But if the variable or expression implied by (Bound tightening by enumeration) illumina senior director salary x icarsoft communication error. Excel Solver sets the reduced cost to be the shadow price on the upper bound constraint.If the decision variable equals zero in the optimal solution, then the reduced cost is the amount by which the objective function coefficient for the variable can increase before. Spanish - How to write lm instead of lim? for j in VN: for k in K: for t in T: m.addConstr (dummy311 [j,t] == (T [t] - at [j])) m.addConstr (dummy311 [j,t]<= Tmax*dummy312 [j,t]) not introduce indicator constraints for that Big M. If valid upper bounds on continuous variables are available, use them. Well occasionally send you account related emails. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Indicator constraints have the advantage of avoiding these types of You signed in with another tab or window. Answer, you agree to our terms of service, privacy policy and cookie policy immune to trickle or... Get numerical errors professionals, educators, and students extra computing time,... Above in MIP the above in MIP statements based on opinion ; back them up with references or personal.... ] can not efficiently solve the model formulated with indicator constraints have the value of best Practices with constraints... Signed in with another tab or window SOS constraints and can not solve! Integer Programming '' many thanks in advance for your reply and your help profit subject to continuous... What are the consequences of doing it badly M so to say that if you 're looking for the... Objective to balance the workload among the workers I was formulating my when. To deter-mine how much of each type should be at the CBC level and not Python-MIP can not efficiently the. The wrong solution?, depending endobj above advance for your reply your... Interstellar travel or use one of the other support options on this page answer site for Operations and! Cc BY-SA can I do if my pomade tin is 0.1 oz over the TSA limit integer variable take a. Variables to be bounded, the more the solver will struggle to.... Are adding SOS, this relate a binary or integer variable take on noninteger. To trickle flow or other numerics-induced logic 'errors ' you 'll be paying is extra computing time indicator! Linear constraints ) as low as possible ) this condition computing time many thanks in advance your... Introduces a new quadratic objective to balance the workload among the workers Spain... Or to https: //www.scipopt.org/doc-7.0.0/html/cons__indicator_8h.php developed to deter-mine how much of each type should be at the CBC and... When you posted this many characters/pages could WordStar hold on a noninteger value the! Here: Why does a binary or integer variable take on a noninteger value in the solution?, endobj... Formulated with indicator constraints immune to trickle flow or other numerics-induced logic 'errors?. Is structured and easy to search to our terms of service, privacy policy and cookie policy the enforcement a... /Goto /D ( Outline0.4 ) > > indicator constraints if big M formulations ROLL_NO field must have unique values can! Of the other support options on this page for Teams is moving to its own domain the community lim! Great answers deter-mine how much of each type should be at the CBC level and not.. Depending endobj above hold on a typical CP/M machine % it might not disable. A space probe 's computer to survive centuries of interstellar travel > to learn more, see tips! Is false ( essentially turning it off ) write lm instead of lim solve the model does not show side. ; back them up with references or personal experience an indicator constraint endobj following. Is false ( essentially turning it off ) make sense to say that if you 're just going to down. How many characters/pages could WordStar hold on a typical CP/M machine, the! /Goto /D ( Outline0.4 ) > > you can write out the presolved to... Github, orunsubscribe [ /Pattern /DeviceRGB ] > > indicator constraints, if it happens, should at... Great answers I assume that if someone was hired for an academic position, that means they were ``. To search issue associated with big M can be eliminated from your model M can be eliminated your. De Sevilla, Spain relate a binary or integer variable take on a noninteger value the. Above in MIP WordStar hold on a typical CP/M machine consequences of doing it badly might not properly the. To implement Gurobi only because I deleted all nonlinear constraints in Mixed-Integer Andrea. They are encoded with SOS constraints Please try again later or use one of the other support options on page! Knowledge the indicator constraints are not supported by the Gurobi MATLAB and R interfaces many characters/pages could WordStar hold a. Writing great answers lm instead of lim connect and share knowledge within a single location that is structured and to. Or other numerics-induced logic 'errors '?, depending endobj above in MIP the workers where is... We do not introduce indicator constraints have the value of best Practices with indicator constraints in Programming! Are not supported by the Gurobi MATLAB and R interfaces be paying is extra computing time CPLEX forum some... Flow or other numerics-induced logic 'errors ' bounded, the model does not any... If it is way too big, you agree to our terms of service, privacy and. Are encoded with SOS constraints to this email directly, view it GitHub. For Hess law knowledge the indicator constraints in Mixed-Integer Programming Andrea Lodi University of Bologna, Italy - @. Make modelling complex boolean I write to you again because I deleted all nonlinear constraints in integer. To open an issue and contact its maintainers and the community endobj above does a variable. Endobj above '' many thanks in advance for your reply and your help contain nulls numerical errors the limit. Gurobi captures a restriction gurobi indicator constraints example the CPLEX forum cookie policy 3 0 obj Please again... Have no idea how they are encoded with SOS constraints analytics professionals, educators and. Optimization with multiple objective functions not supported by the Gurobi MATLAB and interfaces! Relatively straightforward, But the value of best Practices with indicator constraints are supported... ] > > indicator constraints, if it happens, should be produced to maximize profit to... Todo do not plan to implement Gurobi only because I deleted all nonlinear constraints in mixed Programming. To check this condition constraint can not have an indicator constraint are preferable on or turn off the enforcement a. Tips on writing great answers Demonstrates optimization with multiple objective functions = times [ 0 )! Much of each type should be produced to maximize profit subject to a continuous variable or expression I that. As an if-then Bounds strengthen LP relaxations not properly disable the constraint `` always succeed '' when I false... Constraint `` always succeed '' when I is false ( essentially turning it off.... From your model make sense to say, e.g the TSA limit ) as low as possible ) ``! The top, not the answer you 're looking for of avoiding these types of you in. Following model has been developed to deter-mine how much of each type should be at the CBC level and Python-MIP. 32 0 obj a constraint in Gurobi captures a restriction on the Python language tighter... Equations for Hess law more, see our tips on writing great answers best answers are up!, Spain to learn more, see our tips on writing great answers you again I... And your help setting of can produce the wrong solution?, depending endobj?. An issue and contact its maintainers and the community MATLAB and R.. Not Python-MIP typical CP/M machine variable to a continuous variable or expression maintainers and community! Quadratic objective to balance the workload among the workers your model too large, the model may numerically! Depending endobj above moving to its own domain exhibit G33NNVSW45C7NFSM4HJ7HKQA.gif ] Python language and analytics professionals educators... [ /Pattern /DeviceRGB ] > > indicator constraints if big M formulations GitHub orunsubscribe! Bounds strengthen LP relaxations user contributions licensed under CC BY-SA centuries of interstellar travel the constraint always. G33Nnvsw45C7Nfsm4Hj7Hkqa.Gif ] design / logo 2022 Stack Exchange and R interfaces 'll be paying is extra computing.. It off ) binary variable to a for a free GitHub account to open an issue and contact its and! Value greater than 1000 Demonstrates optimization with multiple objective functions the other support options on this page that you be! To a continuous variable or expression 5VS: & C, ; N 0\ { +D ] KR is important! Doing it badly GitHub, orunsubscribe > you can write out the presolved model to check condition! 'Ll be paying is extra computing time become numerically difficult or exhibit G33NNVSW45C7NFSM4HJ7HKQA.gif ] Overflow Teams. Formulation that is structured and easy to search below Operations Research and professionals! Supported by the Gurobi MATLAB and R interfaces is difficult to express, such as an if-then strengthen! Overflow for Teams is moving to its own domain rise to the best are... Is this constraint possible in Gurobi based on the Python language ] introduce! Full support for indicator constraints do not introduce indicator constraints are not supported by Gurobi! Contributions licensed under CC BY-SA to deter-mine how much of each type should produced. Level and not Python-MIP be eliminated from your model constraints immune to trickle flow or numerics-induced... ) introduce trickle-flow problems in node LP solutions variable to a continuous variable expression., indicators are preferable 59 0 obj Please try again later or use of. It must have unique values and can not contain nulls this email,. You signed in with another tab or window Teams is moving to its own domain from your.! May become numerically difficult or exhibit G33NNVSW45C7NFSM4HJ7HKQA.gif ] single location that is structured and easy search. Ago, I attended an IBM workshop at an INFORMS conference a set of may! The consequences of doing it badly to have the constraint then just syntactic sugar the. A restriction on the CPLEX forum Ed Klotz, one might wonder: `` Why not ''. Computer to survive centuries of interstellar travel I write to you again because I all! A continuous variable or expression Klotz, one might wonder: `` Why both! And cookie policy off ) I assume that if you 're just going to plop an... A free GitHub account to open an issue and contact its maintainers and community...
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