3-4 Practice Linear Programming Answer Key

[Finnis Springer]

10^13$ is beyond tractable on today's largest supercomputers due mostly to memory limits. The largest problems I've seen practically solved have had on the order of $10^5$ rows and columns, but the most important factor tends to be the number of nonzeros, where we are just crossing into solving problems with $10^6$ nonzeros. See Mittelman's parallel benchmarks page to get a feel for what the best freely available and commercial solvers can do on a range of problems at this size.

A huge number of inequality constraints is generally tractable by constraint generation techniques, processing at each time only a limited number of constraints, and adding constraints violated at the current solution to the constraint set (while deleting strongly satisfied ones). But in the cases I have seen, this requires that the number of non-slack variables is limited, too.

When I was hunting around for data to answer this question, I found papers 10+ years ago that were solving problems of roughly the same size, which suggests to me that we might be able to do better now with the infrastructure we have. Certainly not $10^13$ variables, but based on the $n^3.5$ scaling of interior point methods, if the linear algebra and parallelism were implemented well, and you had the time and a modestly sized cluster, I don't see why you couldn't attempt to solve a problem with $10^7$ or maybe $10^8$ variables (only if you had special structure you could exploit with decomposition methods like Benders' decomposition or Dantzig-Wolfe, plus cutting-plane generation algorithms). (I will add that I am ignoring the effect of constraints, which complicate matters depending on how many bits are stored; this effect only makes the estimates below more pessimistic.)

I believe GAMS has a parallel implementation, and since it uses solvers like CPLEX, Gurobi, MOSEK, and Xpress (i.e., the four solvers in the benchmark Aron cites), I don't see why one couldn't run parallel instances of those solvers (in fact, I know this is possible for CPLEX and Gurobi). The limiting factors will be memory (because memory is expensive) and communication more than processing power, since a linear program reduces, ultimately, to constructing and solving a system of linear equations repeatedly (a massive oversimplification, yes, but linear algebra is something we know how to parallelize).

But $10^13$ variables is too much. Assuming that memory and communication were no object, you'd need to take the largest problem in the benchmark, and scale the run time on that machine by a factor of roughly $10^24$ before possibly exploiting special structure that your particular problem may have. That's not to say that you couldn't try to solve it approximately using the methods that Professor Neumaier has suggested, but a solution to optimality is likely impossible without waiting a really long time, using a huge computer, and having a scalable implementation of an LP solver tuned to that huge computer.

As a very rough order of magnitude estimate, the Intel Core 2 Quad used in the benchmark Aron cites can operate at a peak speed of 40 gigaflops. Assuming you were to get on Jaguar, Oak Ridge National Lab's supercomputer, and you could use the whole machine (extremely unlikely, but let's dream big), you'd have roughly 2 petaflops at your fingertips (based on the TOP 500 numbers here, or roughly 50000 times the computing power, not counting penalties due to communication, memory limitations, or the fact that no one ever runs at peak speed (not even the LINPACK benchmark).

Going from $10^6$ to $10^7$ variables means roughly a factor of 10,000 increase, which you could conceivably split among a cluster of 50-100 machines, and waiting a month (assuming you're willing to wait, you have the machines, and again, memory and communication aren't limiting, all of which are big "ifs"). Going from $10^6$ to $10^13$ variables means going from your desktop to using all of Jaguar, and waiting roughly $10^17$ to $10^18$ years. (And again, these effects also ignore the fact that you're going to have more constraints!)

I would like to see the generating equations, perhaps it's smart to do some serious decomposition before you throw this problem to the algorithms, and I would like to say as a practitioner that maybe it is smart to chew on it before feeding the hardware. Also sounds like the size that would induce to numerical errors in the formulation given limited computer memory and precision.

Optimization of resources (cost and time) is required in every aspect of our lives. We need the optimization because we have limited time and cost resources, and we need to take the maximum out of them. Every aspect of the business world today requires optimization, from manufacturing to resolving supply chain issues to stay competitive.

Linear programming offers the easiest way to do optimization as it simplifies the constraints and helps to reach a viable solution to a complex problem. In this article, we will solve some of the linear programming problems through the graphing method.

A transport company has two types of trucks, Type A and Type B. Type A has a refrigerated capacity of and a non-refrigerated capacity of . In contrast, Type B has the same overall volume with equal refrigerated and non-refrigerated stock sections. A grocer must hire trucks to transport of refrigerated stock and of non-refrigerated stock. The cost per kilometre of Type A is , and for Type B. How many trucks of each type should the grocer rent to achieve the minimum total cost?

A school is preparing a trip for 400 students. The transportation company has 10 buses of 50 seats each and 8 buses of 40 seats but only has 9 drivers available. The rental cost for a large bus is and for a small bus. Calculate how many buses of each type should be used for the trip for the least possible cost.

A store wants to liquidate 200 shirts and 100 pairs of pants from last season. They have decided to put together two offers, A and B. Offer A is a package of one shirt and a couple of pants which will sell for . Offer B is a package of three shirts and a pair of pants, which will sell for . The store does not want to sell less than 20 packages of Offer A and less than 10 of Offer B. How many boxes do they have to deal with to maximize the money generated from the promotion?

A school is preparing a trip for 400 students. The company who is providing the transportation has 10 buses of 50 seats each and 8 buses of 40 seats, but only has 9 drivers available. The rental cost for a large bus is and for a small bus. Calculate how many buses of each type should be used for the trip for the least possible cost.

A store wants to liquidate 200 shirts and 100 pairs of pants from last season. They have decided to put together two offers, A and B. Offer A is a package of one shirt and a pair of pants which will sell for . Offer B is a package of three shirts and a pair of pants, which will sell for . The store does not want to sell less than 20 packages of Offer A and less than 10 of Offer B. How many packages of each do they have to deal to maximize the money generated from the promotion?

I very much appreciate the exercises and solutions however, it seems that the answer to the third exercise has an issue. A couple of jeans refers to two pairs not one so the equation x + y = 100 is supposed to be 2x+ y = 100. Perhaps there was an issue with the wording because I just realized it is a pair in the solution but a couple in the question

An agricultural Research institute suggested to a farmer to spread out at least 4800kg of a special phosphate fertilizer and not less than 7200kg of a special nitrogen fertilizer to raise productivity of crops in his fields. There are two sources for obtaining these: Mixture A and B, both of these are available in bags weighting 100 kg each and they cost sh 40 and sh24 respectively. Mixture A contains phosphate and nitrogen equivalent of 20 kg and 80 kg respectively, while mixture B contains these ingredients equivalent of 50 kg each.

Required: Write this as a linear programming problem and determine how many bags of each type the farmer should buy in order to obtain the required fertilizer at a minimum cost

Please help to do under this question

A company produces two types of TVs, one of which is black and white, the

other colour. The company has the resources to make at most 200 sets a week. Creating a black

and white set includes Birr 2700 and Birr 3600 to create a colored set. The business should

spend no more than Birr 648,000 a week producing TV sets. If it benefits from Birr 525 per set

of black and white and Birr 675 per set of colours.

Construct the linear programing model.

How many sets of black/white and colored sets it should produce in order to get

maximum profit using

Graphical Method and Simplex Method

A company manufacture two types of product A1 and A2. Each product using milling and drilling machine. The process time per unit of A1 on the milling is 10 hours and the drilling is 8 hours, the process time per unit of A2 on the milling is 15 minute and on the drilling is 10 hours, the maximum number of hours available per week on the drilling and milling machine are 80 hours and 60 hours respectively also the profit per selling of A1 and A2 are 25 naira and 35 naira respectively. Formulate a LP model to determine the production volume of each of the product such that the total profit is maximized

Linear Programming is widely used in Mathematics and some other fields such as economics, business, telecommunication, and manufacturing fields. In this article, let us discuss the definition of linear programming, its components, and different methods to solve linear programming problems.

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