Optimization Model with Days

I have the following situation: I have to find the most optimal days to palletize and to dispatch pallets to 22 different destination. The demand for each destination varies, ranging from 3 to 12 pallets. – Palletizing and dispatching can be done from Monday to Saturday, but not on the same day. – The maximum capacitiy for palletizing is 30 pallets/day, and for dispatching as well 30 pallets/day. – The goal is to minimize the total days between palletizing and dispatching. If I palletize for destination “A” on Monday, and dispatch it on Friday, there are 4 days. How could I find the best solution? What software should I use?

Optimization of Do loop with multiple variables

I am pretty new to Mathematica and I would like to optimize a Do loop with multiple variables that I have.

I have three lists of three elements each, with the condition that the first element must be bigger than the second and so on. From those, I construct matrices and make operations on them; since I want to scan on all the possible matrices, I have this big Do loop:

l1={a,b,c};l2={d,e,f};l3={g,h,i}; Do[ m1=Table[(l1[[i]]+l2[[j]]),{i,3},{j,3}]; (*just as an example, not the real operations*) m2=Table[(l1[[i]]+l3[[j]]),{i,3},{j,3}]; {vl,m1d,vr}=SingularValueDecomposition[m1]; {ul,m2d,ur}=SingularValueDecomposition[m2]; ... ,{a,2,5},{b,a-1},{c,0,b-1},{d,2,5},{e,d-1},{f,0,e-1},{g,2,5},{h,g-1},{i,0,h-1} ] 

The operations inside the loop are much more difficult, and this loop would take way too much time to complete. How can I improve my situation? I’ve read something about the use of Map but I didn’t quite understand how it should apply to my case.

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Optimization Problem

Let’s say I have 2 different types of machines responsible for 2 different duties.

I have the PRODUCTION_MACHINES that EACH are able to produce a single unit of goods within 2 hours.

Then I have TRANSPORTER_MACHINES which are EACH able to collect and package away a single unit of goods from the production line at 95 seconds per TRANSPORTER_MACHINE.

QUESTION:

Assuming there are 5 PRODUCTION_MACHINES working, and they are operated at different starting times, how many TRANSPORTER_MACHINES do you need in order to MINIMIZE the IDLE_TIME incurred in the production process ?

(Whereby “IDLE_TIME” is defined as the time expended by EITHER the PRODUCTION_MACHINE or a TRANSPORTER_MACHINE waiting to have its good collected or produced.

HINT: Too many TRANSPORTER_MACHINES will add to the idle time as there are prone to be extra TRANSPORTER_MACHINES idly waiting for struggling PRODUCTION_UNITS to produce quickly enough to keep up with DEMAND.

… and vice-versa, too few TRANSPORTER_UNITS will also increase the idle time, this time for the PRODUCTION_MACHINES as some of them will have to wait for the struggling TRANSPORTER_MACHINES to keep up with their SUPPLY.)

Thank you very much !

How to prove the optimization version problem (whose decision version is NP-complete) can be solved in poly-time iff P=NP?

I have proved the decision version of my problem be $ \mathcal{NP}$ -complete. And I know that if I can solve the optimization version in poly-time, then I can just to compare the obtained minimum (or maximum) with target value in decision version. Thus, the decision version can be solved in poly-time as well. Since the decision version is $ \mathcal{NP}$ -hard, so is the optimization version, i.e., the optimization version is $ \mathcal{NP}$ -hard.

My question is how to prove the converse direction: if the decision version can be solved in poly-time, can the optimization version be solved in poly-time as well?

I in advance thank you for any suggestions!

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How to program circle position optimization algorithm?

I am quite new to programming and I need to write a program to determine which are the optimal uniform distribution of circles. To make an example of what I want, let’s say that I have to inks, one blue and one red. The goal is to define the color that I want to achieve (in this example different types of violet) and that the program tells me where do I have to put the drops of each ink in order to achieve that.

The inputs would be the drop size (on the surface), drop spacing along X (on the surface), drop spacing along Y (on the surface) and final color. The outputs of the program should be X and Y positions, and which drop (red or blue) should be placed in each position.

Could you please give me some ideas or point me towards the right direction? I have tried looking for similar algorithms, but was unsuccessful, so I decided to ask here in hope that you can help.

Thank you!