I would like to know how to classify the following optimization problem.
A lumber yard sells 2x4's in various stock lengths. For example, an 8ft could be $3 and a 10ft could be $4, while a 14ft might be $5.50. Importantly, the lengths are not linearly related to price and not all discrete lengths can be purchased as stock. It can be assumed that the available stock units are inexhaustible in these discrete lengths.
length cost
7.7ft $2.75
8ft $3.00
10ft $4.00
14ft $5.50
I need to create a set of 2x4's with given lengths by cutting them from the above stock (say I need lengths of 2ft, 2.5ft, 6ft once all is said and done). Also, each "cut" incurs a material cost of 1/8" (i.e. 0.0104ft). The solution of the problem is an assignment of each desired length to a piece of stock with the total cost of all stock minimized. In this example, the optimal solution minimizing cost is to buy a 14ft board at $5.50. (A runner-up solution is to buy two 8ft boards and allocate as {6ft} and {2ft, 0.0104ft, 2.5ft} for a cost of $6.)
It does not seem to be a Knapsack-class problem. It does not seem to be a cutting stock problem (because I would like to minimize cost rather than minimize waste). What sort of problem is this, and how can I go about efficiently solving it?
(As an after-note, this is a non-fictional problem I have solved in the obvious, inefficient way using multiset partitions and iteration in Haskell. The runtime is prohibitive to practical use with more than 23 desired lengths and 6 available stock sizes.)
I believe that this is a cutting stock problem, except that it's a multi-objective or multi-criteria cutting stock problem (where you want to minimize monetary cost as well as material cost), see for this example this article. Unfortunately almost all of the online resources I found for this breed of cutting stock problem were behind paywalls; in addition, I haven't done any integer-linear programming in several years, but if I remember correctly multi-objective problems are much more difficult than single-objective problems.
One option is to implement a two-pass algorithm. The first pass completely ignores the material cost of cutting the boards, and only uses the monetary cost (in place of the waste cost in a standard cutting stock problem) in a single-objective problem. This may leave you with an invalid solution, at which point you perform a local search to e.g. replace two 10-foot boards with a 14-foot board and an 8-foot bard until you reach a valid solution. Once you find a valid solution, you can continue the local search for several more iterations to see if you can improve on the solution. This algorithm will likely be sub-optimal when compared to a one-pass multi-objective solution, but it ought to be much easier to implement.
Related
I know that my question is general, but I'm new to AI area.
I have an experiment with some parameters (almost 6 parameters). Each one of them is independent one, and I want to find the optimal solution for maximum or minimum the output function. However, if I want to do it in traditional programming technique it will take much time since i will use six nested loops.
I just want to know which AI technique to use for this problem? Genetic Algorithm? Neural Network? Machine learning?
Update
Actually, the problem could have more than one evaluation function.
It will have one function that we should minimize it (Cost)
and another function the we want to maximize it (Capacity)
Maybe another functions can be added.
Example:
Construction a glass window can be done in a million ways. However, we want the strongest window with lowest cost. There are many parameters that affect the pressure capacity of the window such as the strength of the glass, Height and Width, slope of the window.
Obviously, if we go to extreme cases (Largest strength glass, with smallest width and height, and zero slope) the window will be extremely strong. However, the cost for that will be very high.
I want to study the interaction between the parameters in specific range.
Without knowing much about the specific problem it sounds like Genetic Algorithms would be ideal. They've been used a lot for parameter optimisation and have often given good results. Personally, I've used them to narrow parameter ranges for edge detection techniques with about 15 variables and they did a decent job.
Having multiple evaluation functions needn't be a problem if you code this into the Genetic Algorithm's fitness function. I'd look up multi objective optimisation with genetic algorithms.
I'd start here: Multi-Objective optimization using genetic algorithms: A tutorial
First of all if you have multiple competing targets the problem is confused.
You have to find a single value that you want to maximize... for example:
value = strength - k*cost
or
value = strength / (k1 + k2*cost)
In both for a fixed strength the lower cost wins and for a fixed cost the higher strength wins but you have a formula to be able to decide if a given solution is better or worse than another. If you don't do this how can you decide if a solution is better than another that is cheaper but weaker?
In some cases a correctly defined value requires a more complex function... for example for strength the value could increase up to a certain point (i.e. having a result stronger than a prescribed amount is just pointless) or a cost could have a cap (because higher than a certain amount a solution is not interesting because it would place the final price out of the market).
Once you find the criteria if the parameters are independent a very simple approach that in my experience is still decent is:
pick a random solution by choosing n random values, one for each parameter within the allowed boundaries
compute target value for this starting point
pick a random number 1 <= k <= n and for each of k parameters randomly chosen from the n compute a random signed increment and change the parameter by that amount.
compute the new target value from the translated solution
if the new value is better keep the new position, otherwise revert to the original one.
repeat from 3 until you run out of time.
Depending on the target function there are random distributions that work better than others, also may be that for different parameters the optimal choice is different.
Some time ago I wrote a C++ code for solving optimization problems using Genetic Algorithms. Here it is: http://create-technology.blogspot.ro/2015/03/a-genetic-algorithm-for-solving.html
It should be very easy to follow.
and thanks for reading my thread.
I have read some of the previous posts on formatting/normalising input data for a Neural Network, but cannot find something that addresses my queries specifically. I apologise for the long post.
I am attempting to build a radial basis function network for analysing horse racing data. I realise that this has been done before, but the data that I have is "special" and I have a keen interest in racing/sportsbetting/programming so would like to give it a shot!
Whilst I think I understand the principles for the RBFN itself, I am having some trouble understanding the normalisation/formatting/scaling of the input data so that it is presented in a "sensible manner" for the network, and I am not sure how I should formulate the output target values.
For example, in my data I look at the "Class change", which compares the class of race that the horse is running in now compared to the race before, and can have a value between -5 and +5. I expect that I need to rescale these to between -1 and +1 (right?!), but I have noticed that many more runners have a class change of 1, 0 or -1 than any other value, so I am worried about "over-representation". It is not possible to gather more data for the higher/lower class changes because thats just 'the way the data comes'. Would it be best to use the data as-is after scaling, or should I trim extreme values, or something else?
Similarly, there are "continuous" inputs - like the "Days Since Last Run". It can have a value between 1 and about 1000, but values in the range of 10-40 vastly dominate. I was going to scale these values to be between 0 and 1, but even if I trim the most extreme values before scaling, I am still going to have a huge representation of a certain range - is this going to cause me an issue? How are problems like this usually dealt with?
Finally, I am having trouble understanding how to present the "target" values for training to the network. My existing results data has the "win/lose" (0 or 1?) and the odds at which the runner won or lost. If I just use the "win/lose", it treats all wins and loses the same when really they're not - I would be quite happy with a network that ignored all the small winners but was highly profitable from picking 10-1 shots. Similarly, a network could be forgiven for "losing" on a 20-1 shot but losing a bet at 2/5 would be a bad loss. I considered making the results (+1 * odds) for a winner and (-1 / odds) for a loser to capture the issue above, but this will mean that my results are not a continuous function as there will be a "discontinuity" between short price winners and short price losers.
Should I have two outputs to cover this - one for bet/no bet, and another for "stake"?
I am sorry for the flood of questions and the long post, but this would really help me set off on the right track.
Thank you for any help anyone can offer me!
Kind regards,
Paul
The documentation that came with your RBFN is a good starting point to answer some of these questions.
Trimming data aka "clamping" or "winsorizing" is something I use for similar data. For example "days since last run" for a horse could be anything from just one day to several years but tends to centre in the region of 20 to 30 days. Some experts use a figure of say 63 days to indicate a "spell" so you could have an indicator variable like "> 63 =1 else 0" for example. One clue is to look at outliers say the upper or lower 5% of any variable and clamp these.
If you use odds/dividends anywhere make sure you use the probabilities ie 1/(odds+1) and a useful idea is to normalize these to 100%.
The odds or parimutual prices tend to swamp other predictors so one technique is to develop separate models, one for the market variables (the market model) and another for the non-market variables (often called the "fundamental" model).
I am optimizing algorithmic strategies. In the process of choosing from a pool of many optimized strategies, I am in the phase of searching (evaluating) for robustness of the strategy.
Following the guidelines of Dr. Pardo's book "The Evaluation of Trading Strategies" in page 231 Dr. Pardo recomends, in the Numeral 3 to apply the following ratio to the optimized data:
" 3. The ratio of the total profit of all profitable simulations divided by the
total profit of all simulationsis significantly positive"
The Question: from the optimization results, I am not being able to properly understand what does Mr. Pardo means by stating "...all simulationsis significantly positive"; what does Mr. Pardo means by 'significantly positive?
a.) with 95% confidence level?
b.) with a certain p value?
c.) the relation of the average net profit of each simulation minus it' standard deviation
Even though the sentence might seem 'simple' I would REALLY like to understand what Mr. Pardo means by the statement and HOW to calculate it, in order to filter the most robust algorithmic strategies.
The aim of analyzing the optimization profile of an algorithmic simulation is to be able to filter robust strategies.
Therefore the ratio should help us to uncover if the simulation results are on the right track or not.
So, we would like to impose some 'penalties' to our results, so we can select the robust cases from those of doubtful (not robust) result.
I came to the following penalizing measures (found in the book of Mr. Pardo and other sources).
a.) we can use a market return (yearly value) as a benchmark, so all the simulations whose result are below such level, can be excluded from our analysis,
b.) some other benchmark to divide those 'robust' results from those more 'doubtful' (for example, deducing to each result one standard deviation)
From (a) and (b), we can create the ratio:
the total sum of all profitable simulations divided by the profitable results considered robust
The ratio should be greater or equal than 1.
If the ratio is equal to 1 then it means that our simulation result has given interesting results (we are analyzing the positive values in this ratio, but profitable results should always be compared to the negative results also).
If the ratio is greater from 1, then we have not reach the possible scenario, and the result should be compared with the other tests for optimizations.
While simulating trading algorithms, no result is absolute but partial and it's value is taken in relationship to what we expect from the algorithm.
If someone has a better explanation or idea or concept you might find interesting please share, I would gladly read it.
Best regards to all.
Remark on the subject
With all due respect to the subject ( published in 2008 ) the term robustness has its own meaning if-and-only-if the statement also clarifies in which particular respect is the robustness measured and against what phenomena is it to be exposed & tested the Model-under-review's response ( against what perturbances -- type and scale -- shall the Model-under-test hold its robust behaviour, measures of which were both defined and quantified a-priori the test ).
In any case, where such context of the robustness is not defined, the material, be it printed by any bold name, sounds -- and forgive me to speak in plain English -- just like a PR-story, an over-hyped e-zine headline or like a paid advertorial.
Serious quantitative model evaluations, the more if one strives to perform an optimisation ( with respect to some defined quantitative goal ), requires a more thorough insight into the subject than to axiomatically post a trivial "must-have" imperative of
large-average && small-HiLo-range && small StDev.
Any serious Quant-Modelling effort, if it were not to just spoil the consumed hundreds-of-thousands CPU core hours of deep parametric-spaces' scans, shall incorporate a serious parametrisation decision in either dimension of the main TruTrading Strategy sub-spaces --
{ aSelectPOLICY, aDetectPOLICY, anActPOLICY, anAllocatePOLICY, aTerminatePOLICY }
A failure to do so, either cripples the model or leads to a blind-belief, where it is hard to guess, whether the former or the latter is a greater of the both Quant-sins.
Remark on the cited hypothesis
The book states, without any effort to proof the construction, that:
The more robust trading strategywill have an optimization proļ¬le with a: 1. Largeaverageproļ¬t 2. Small maximum-minimumrange3. Small standarddeviation
Is it correct?
Now kindly spend a few moments and review this 4D-animated view of a Model-under-test ( visualisation of which is reduced into just four dimensions for easier visual perception ), where none of the above stands true.
<aMouseRightCLICK>.openPictureOnAnotherTab to see full HiRes picture details
Based on contemporary state-of-art adaptive money-management practice, that fails to be correct, be it due to a poor parametrisation ( thus artificially leading the model into a rather "flat-profits" sub-space of aParamSetVectorSPACE )
or due to a principal mis-concept or a poor practice ( including the lack thereof ) of the implementation of the most powerful profit-booster ever -- the very money-management model sub-space.
Item 1 becomes insignificant at all.
Item 2 works right on the contrary to the stated postulate.
Item 3 cannot yield anything but the opposite due to 1 & 2 above.
For a number of financial instruments, Bloomberg scales the prices that are shown in the Terminal - for example:
FX Futures at CME: e.g. ADZ3 Curncy (Dec-2013 AUD Futures at CME) show as 93.88 (close on 04-Oct-2013), whereas the actual (CME) market/settlement price was 0.9388
FX Rates: sometimes FX rates are scaled - this may vary by which way round the FX rate is asked for, so EURJPY Curncy (i.e. JPY per EUR) has a BGN close of 132.14 on 04-Oct-2013. The inverse (EUR per JPY) would be 0.007567. However, for JPYEUR Curncy (i.e. EUR per JPY), BGN has a close of 0.75672 for 04-Oct-2013.
FX Forwards: Depending on whether you are asking for rates or forward points (which can be set by overrides)... if you ask for rates, you might get these in terms of the original rate, so for EURJPY1M Curncy, BGN has a close of 132.1174 on 04-Oct-2013. But if you ask for forward points, you would get these scaled by some factor - i.e. -1.28 for EURJPY1M Curncy.
Now, I am not trying to criticise Bloomberg for the way that they represent this data in the Terminal. Goodness only knows when they first wrote these systems, and they have to maintain the functionality that market practitioners have come to know and perhaps love... In that context, scaling to the significant figures might make sense.
However, when I am using the API, I want to get real-world, actual prices. Like... the actual price at the exchange or the actual price that you can trade EUR for JPY.
So... how can I do that?
Well... the approach that I have come to use is to find the FLDS that communicate this scaling information, and then I fetch that value to reverse the scale that they have applied to the values. For futures, that's PX_SCALING_FACTOR. For FX, I've found PX_POS_MULT_FACTOR most reliable. For FX forward points, it's FWD_SCALE.
(It's also worth mentioning that how these are applied vaires - so PX_SCALING_FACTOR is what futures prices should be divided by, PX_POS_MULT_FACTOR is what FX rates should be multipled by, and FWD_SCALE is how many decimal places to divide the forward points by to get to a value that can be added to the actual FX rate.)
The problem with that is that it doubles the number of fetchs I have to make, which adds a significant overhead to my use of the API (reference data fetches also seem to take longer than historical data fetches.) (FWIW, I'm using the API in Java, but the question should be equally applicable to using the API in Excel or any of the other supported languages.)
I've thought about finding out this information and storing it somewhere... but I'd really like to not have to hard code that. Also, that would require to spend a very long time finding out the right scaling factors for all the different instruments I'm interested in. Even then, I would have no guarantee that they wouldn't change their scale on me at some point!
What I would really like to be able to do is apply an override in my fetch that would allow me specify what scale should be used. (And no, the fields above do not seem to be override-able.) I've asked the "helpdesk" about this on lots and lots of occasions - I've been badgering them about it for about 12 months, but as ever with Bloomberg, nothing seems to have happened.
So...
has anyone else in the SO community faced this problem?
has anyone else found a way of setting this as an override?
has anyone else worked out a better solution?
Short answer: you seem to have all the available information at hand and there is not much more you can do. But these conventions are stable over time so it is fine to store the scales/factors instead of fetching the data everytime (the scale of EURGBP points will always be 4).
For FX, I have a file with:
number of decimal (for spot, points and all-in forward rate)
points scale
spot date
To answer you specific questions:
FX Futures at CME: on ADZ3 Curncy > DES > 3:
For this specific contract, the price is quoted in cents/AUD instead of exchange convention USD/AUD in order to show greater precision for both the futures and options. Calendar spreads are also adjusted accordingly. Please note that the tick size has been adjusted by 0.01 to ensure the tick value and contract value are consistent with the exchange.
Not sure there is much you can do about this, apart from manually checking the factor...
FX Rates: PX_POS_MULT_FACTOR is your best bet indeed - note that the value of that field for a given pair is extremely unlikely to change. Alternatively, you could follow market conventions for pairs and AFAIK the rates will always be the actual rate. So use EURJPY instead of JPYEUR. The major currencies, in order, are: EUR, GBP, AUD, NZD, USD, CAD, CHF, JPY. For pairs that don't involve any of those you will have to fetch the info.
FX Forwards: the points follow the market conventions, but the scale can vary (it is 4 most of the time, but it is 3 for GBPCZK for example). However it should not change over time for a given pair.
This is my problem set for one of my CS class and I am kind of stuck. Here is the summary of the problem.
Create a program that will:
1) take a list of grocery stores and its available items and prices
2) take a list of required items that you need to buy
3) output a supermarket where you can get all your items with the cheapest price
input: supermarkets.list, [tomato, orange, turnip]
output: supermarket_1
The list looks something like
supermarket_1
$2.00 tomato
$3.00 orange
$4.00 tomato, orange, turnip
supermarket_2
$3.00 tomato
$2.00 orange
$3.00 turnip
$15.00 tomato, orange, turnip
If we want to buy tomato and orange, then the optimal solution would be buying
from supermarket_1 $4.00. Note that it is possible for an item to be bough
twice. So if wanted to buy 2 tomatoes, buying from supermarket_1 would be the
optimal solution.
So far, I have been able to put the dataset into a data structure that I hope will allow me to easily do operations on it. I basically have a dictionary of supermarkets and the value would point to a another dictionary containing the mapping from each entry to its price.
supermarket_1 --> [turnip --> $2.00]
[orange --> $1.50]
One way is to use brute force, to get all combinations and find whichever satisfies the solution and find the one with the minimum. So far, this is what I can come up with. There is no assumption that the price of a combination of two items would be less than buying each separately.
Any suggestions hints are welcome
Finding the optimal solution for a specific supermarket is a generalization of the set cover problem, which is NP-complete. The reduction goes as follows:
Given an instance of the set cover problem, just define a cost function assigning 1 to each combination, apply an algorithm that solves your problem, and you obtain an optimal solution of the set cover instance. (Finding the minimal price hence corresponds to finding the minimum number of covering sets.) Thus, your Problem is NP-hard, and you cannot expect to finde a solution that runs in polynomial time.
You really should implement the brute-force method you mentioned. I too recommand you to do this as a first step. If the performance is not sufficient, you can try a
using a MIP-formulation and a solver like CPLEX, or you have to devolop a heuristic approach.
For a single supermarket, it is rather trivial to find a mixed integer program (MIP). Let x_i be the integer number how often product combination i is contained in a solution, c_i its cost and w_ij the number how often product j is contained in product combination i. Then, you are minimizing
sum_i x_i * c_i
subject to conditions like
sum_i x_i * w_ij >= r_j,
where r_j is the number how often product j is required.
Well, you have one method, so implement it now so you have something that works to submit. A brute-force solution should not take long to code up, then you can get some performance data and you can think about the problem more deeply. Guesstimate the number of supermarkets in a reasonable shopping range in a large city. Create that many supermarket records and link them to product tables with random-ish prices, (this is more work than the solution).
Run your brute-force solution. Does it work? If it outputs a solution, 'manually' add up the prices and list them against three other 'supermarket' records taken at random, (just pick a number), showing that the total is less or equal. Modify the price of an item on your list so that the solution is no longer cheap and re-run, so that you get a different solution.
Is it so fast that no further work is justified? If so, say so in the conclusions section of your report and post the lot to your prof/TA. You understood the task, thought about it, came up with a solution, implemented it, tested it with a representative dataset and so shown that functionality is demonstrable and performance is adequate - your assignment is over, go to the bar, think about the next one over a beer.
I am not sure what you mean by "brute force" solution.
Why don't you just calculate the cost of your list of items in each of the supermarkets, and then select the minimum? Complexity would be in O(#items x #supermarkets) which is good.
Regarding your data structure you can also simply have a matrix (2 dimension array) price[super][item], and use ids for your supermarkets/items.