I'm new to linear programming and trying to develop an ILP model around a problem I'm trying to solve.
My problem is analogous to a machine resource scheduling problem. I have a set of binary variables to represent paired-combinations of machines with a discrete-time grid. Job A takes 1 hour, Job B takes 1 hr and 15 minutes, so the time grid should be in 15 minute intervals. Therefore Job A would use 4 time units, and Job B would use 5 time units.
I'm having difficulty figuring out how to express a constraint such that when a job is assigned to a machine, the units it occupies are sequential in the time variable. Is there an example of how to model this constraint? I'm using PuLP if it helps.
Thanks!
You want to implement the constraint:
x(t-1) = 0 and x(t) = 1 ==> x(t)+...+x(t+n-1) = n
One way is:
x(t)+...+x(t+n-1) >= n*(x(t)-x(t-1))
Notes:
you need to repeat this constraint for each t.
A slightly better version is:
x(t+1)+...+x(t+n-1) >= (n-1)*(x(t)-x(t-1))
There is also a disaggregated version of this constraint that may help performance (depending on the solver: some solvers can do this disaggregation automatically).
Things can become interesting near the beginning and end of the planning period. E.g. machine started at t=-1.
Update:
A different approach is just to limit the "start" of a job to 1. I.e. allow only the combination
x(j,t-1) = 0 and x(j,t) = 1
for a given job j. This can be handled in a similar way:
start(j,t) >= x(j,t)-x(j,t-1)
sum(t, start(j,t)) <= 1
0 <= start(j,t) <= 1
Related
Detailed business problem:
I'm trying to solve a production scheduling business problem as below:
I have two plants producing FG A and B respectively.
Both the products consume the same Raw Material x
I need to create a 30 day production schedule looking at the Raw Material availability.
FG A and B can be produced if there is sufficient raw material available on the day.
After every 6 days of production the plant has to undergo maintenance and the production on that day will be zero.
Objective is to maximize the margin looking at the day level Raw material available and adhere to the production constraint (i.e. shutdown after every 6th day)
I need to build a linear programming to address the below problem:
Variable y: (binary)
variable z: cumulative of y
When z > 6 then y = 0. I also need to reset the cumulation of z after this point.
Desired output:
How can I build the statement to MILP constraint. Are there any techniques for solving this problem. Thank you.
I think you can model your maintenance differently. Just forbid any sequences of 7 ones for y. I.e.
y[t-6]+y[t-5]+y[t-4]+y[t-3]+y[t-2]+y[t-1]+y[t] <= 6 for t=1,..,T
This is easier than using your accumulator. Note that the beginning needs some attention: you can use historic data for this. I.e., at t=1, the values for t=0,-1,-2,.. are known.
Your accumulator approach is not inherently wrong. We often use it to model inventory. An inventory capacity is a restriction on how large the accumulated inventory can be.
A construction company has 6 projects, for each they need $d_i$ workers. The company has no workers at the beginning of project 1.
Each new worker must take a safety course that costs 300, and 50 more for each worker.
If there is no new worker there is no course.
Firing a worker does not cost any money, and a workers can't be rehired.
Given that the salary of a worker is 100 per project, formulate a linear programming problem that minimizes the workers costs.
What I tried:
Let $x_i$ be the number of new workers for project $i$.
Let $y_i$ be the number of old workers remaining from previous projects until project $i$ (all the workers hired - all the workers that were fired)
Let $z_i$ be an indicator such that $z_i =0 \iff x_i>0$
The function I'm trying to solve is:
$\min(\sum_{i=1}^6 150x_i + 300(1-z_i) + 100y_i)$
s.t:
\begin{align}
x_i,y_i,z_i &\ge 0 \\
z_i &\ge 1-x_i \\
y_i + x_i &\ge d_i \\
y_i &\ge y_{i-1} + x_i
\end{align}
Something feels not right to me. The main reason is that I tried to use matlab to solve this and it failed.
What did I do wrong? How can I solve this question?
When I see this correctly you have two small mistakes in your constraints.
The first appears when you use z_i >= 1-x_i. This allows z_i to take the value 1 all the time, which will never give you the extra cost of 300. You need to upper bound z_i such that z_i will not be 1 when you have x_i>0. For this constraint you need something called big M. For sufficiently large M you would then use z_i <= 1-x_i/M. This way when x_i=0 you can have z_i=1, otherwise the right hand side is smaller than 1 and due to integrality z_i has to be zero. Note that you usually want to choose M as tight as possible. So in your case d_i might be a good choice.
The second small mistake lays in y_i >= y_{i-1} + x_i. This way you can increase y_i over y_{i-1} without having to set any x_i. To force x_i to increase you need to flip the inequality. Additionally by the way you defined y_i this inequality should refer to x_{i-1}. Thus you should end up with y_i <= y_{i-1} + x_{i-1}. Additionally you need to take care of corner cases (i.e. y_1 = 0)
I think with these two changes it should work. Let me know whether it helped you. And if it still doesn't work I might have missed something.
I am trying to overcome a machine allocation problem with time horizon of 5 day. Production plan is hard to catch up, so my objective is to minimize total machines working time spent. Machines uses molds to produce and there are molds for each type of product. If a product is produced at the end of the day and if there will be production later day, total setup needed for that machine should be decreased by one. For this reason,
sets
i: mold type
j:jobs
k: days
parameters
x(i,k) ith mold production needed at day k
y(i,j) 1 if ith mold is compatible with jth machine
Decision variable
m(i,j,k) : 1 if ith mold processed in jth machine in day k 0 o/w
b(j,k) setup number of jth machine in day k
While computing the setup number for day 1, b(j,’1’), is simply equal to the sum of m(i,j,k).
For computing other days setup number I tried these but these made problem nonlinear and it takes months to solve.
b(j,'2')=e=sum(i,m(i,j,'2')) - sum(i,m(i,j,'2')*m(i,j,'1'))
By this way, if mold i is produced in both days, there will not be any setup made at second day. In order to restrain multiple setup reduction I put: sum(i,m(i,j,'2')*m(i,j,'1')) =l= 1
So, how can I decrease the setup number for a machine if it has used a mold a day before without making the problem nonlinear.
It is possible to linearize m(i,j,'2')*m(i,j,'1'):
Both(i,j) <= m(i,j,'2')
Both(i,j) <= m(i,j,'1')
Both(i,j) >= m(i,j,'2')+m(i,j,'1')-1
Both(i,j) is a binary variable
This transformation is done automatically by some solvers.
Note that there are alternative ways to model the start of a run, and often there are things to exploit (depending on the details).
I have difficulty formulating the constraints correctly. Dumbed down version of the problem:
There are 12 time units, 3 products, demand d_{i,t} for the product i at time $t$ is known in advance and the resources r_{i,t} (all 8, product i uses the non-i resources) necessary for product i at time t are known as well. We need to minimize holding costs h_i by deciding how much of product i we need to produce at time t, this is called x_{i,t}. Starting inventory for every product is 6. To help I introduce stock level s_{i,t}. This equates to the following formulation:
I got this working using the Excel solver but I need to do this in AIMMS. The Stock variable s is giving me issues, I can't get it to work using an if statement to condition on if t=1 nor would I know how to split it in two constraints, given the first iteration of the second constraint would need to reference the first constraint.
You can specify index domain in your constraint attributes as follows:
(t,i) | t > 1
if t > 1 statement should work if the set of time instances is a subset of integers. If not - you should use ord(t) > 1, i.e.
if ord(t) > 1
then
Your_Constraint
endif
I have a relatively large project, which up to now had almost no automatic tests.
Now the management has given us several weeks of time to stabilize the product, including test automation.
In order to get the most value of these X weeks we can spend on automatic testing, I need to know what classes/methods to test first.
How can I prioritize testing efforts (decide, which class/method to test now, which later) apart from the approaches listed below?
Calculate the dependents for each class (how many other classes use class, including transitive dependencies). Classes with the greatest number of dependent classes should be tested first.
Find out, which classes change most frequently (according to the version control system). Frequent changes may be a symptom of either lots of bugs or active development in these classes. In both cases, it makes sense to write unit tests for them.
Find out, which classes are involved in bug reports from testers and/or customers.
All of your ideas seems good. This article can help you with prioritizing and automating.
This is formula of how to do Estimating Testing effort:
Method for Testing Process (based on use case driven approach).
Step 1 : count number of use cases (NUC) of system
step 2 : Set Avg Time Test Cases(ATTC) as per test plan
step 3 : Estimate total number of test cases (NTC)
Total number of test cases = Number of usecases X Avg testcases per a use case
Step 4 : Set Avg Execution Time (AET) per a test case (idelly 15 min depends on your system)
Step 5 : Calculate Total Execution Time (TET)
TET = Total number of test cases * AET
Step 6 : Calculate Test Case Creation Time (TCCT)
usually we will take 1.5 times of TET as TCCT
TCCT = 1.5 * TET
Step 7 : Time for ReTest Case Execution (RTCE) this is for retesting
usually we take 0.5 times of TET
RTCE = 0.5 * TET
Step 8 : Set Report generation Time (RGT
usually we take 0.2 times of TET
RGT = 0.2 * TET
Step 9 : Set Test Environment Setup Time (TEST)
it also depends on test plan
Step 10 : Total Estimation time = TET + TCCT+ RTCE + RGT + TEST + some buffer...;)
Here is example how it works:
Total No of use cases (NUC) : 227
Average test cases per Use cases(AET) : 10
Estimated Test cases(NTC) : 227 * 10 = 2270
Time estimation execution (TET) : 2270/4 = 567.5 hr
Time for creating testcases (TCCT) : 567.5*4/3 = 756.6 hr
Time for retesting (RTCE) : 567.5/2 = 283.75 hr
Report Generation(RGT) = 100 hr
Test Environment Setup Time(TEST) = 20 hr.
Total Hrs 1727.85 + buffer
4 means Number of test cases executed per hour
i.e 15 min will take for execution of each test case
And since you are going to automate almost from scratch
up to now had almost no automatic tests
I think you may consider not only benefits, but Myths about Automated testing too:
Automation cannot replace the human
Once automated, cost savings is a given
Every test case can be automated
Testing can be fully automated
One test tool is suitable for all tasks
Automated Testing doesn’t mean Automatic
After all Testing ... It means Computer aided testing.
Other than the above answer, I would like to add few more points regrading priority and coverage for automation:
1. Before you start any coding, understand what all test cases can give you maximum coverage like a full flow or end to end test scenario.
2. Always maintain the usability of automation test suite - like code those test cases first which can be utilised all the time and can cover most of regression part rather than concentrating a specific feature of application.
3. When you've a limited time, try to avoid implementing fancy things like logger, email summary report, html report etc...
4. Better to have data driven framework rather than having a keyword or hybrid framework because you can cover many test cases by just varying your test data in a limited time.
5. Maintain an excel sheet or csv for test data, test results, you can use JXL Lib to handle excel sheet in java.
6. Read excel sheet, write excel sheet are the most common method which you may want to use in your automation code. You can get reference about this from blog: http://testingmindzz.blogspot.in/