Problem 45415. Find an optimal placement of coolers on a grid

Created by Karl Ezra Pilario

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In a certain chemical plant, 6 new pieces of cooling equipment (coolers) are to be installed in a vacant space. This vacant space was divided into a grid of 6 cells by 6 cells. Your task is to assign the 6 coolers to 6 cells in this grid with the following requirements:<ol><li>There must be one cooler for every column in the grid.</li><li>If you decide to install a cooler at row R, column C, then the cooler at column C+1 must be installed either on row R-1, R, or R+1 only. This is done to ease the connection of coolers by piping.</li><li>The total installation cost for the 6 coolers must be minimum.</li></ol>For this problem, you are given a cost matrix (COST) in relation to the third requirement above. COST is a 6 x 6 matrix where the r,c-th element is the cost of assigning any cooler to row r, column c (All the coolers are identical). Write a function that accepts the matrix COST, and output the value of the minimum cost of installation. You are ensured that all elements of COST are integers in the range [1,1000].In the sample test case below, the optimal placement is at the following rows: 4,3,3,2,2,2.<pre class="language-matlab">&gt;&gt; COST = [695 766 710 119 752 548;
 318 796 755 499 256 139;
 951 187 277 960 506 150;
 35 490 680 341 700 258;
 439 446 656 586 891 841;
 382 647 163 224 960 255];
&gt;&gt; coolers(COST)
ans = 
 1393
</pre>Meanwhile, the optimal placement for the case below is at rows: 5,6,5,6,5,6<pre class="language-matlab">&gt;&gt; COST = [815 617 918 76 569 312;
 244 474 286 54 470 529;
 930 352 758 531 455 988;
 350 831 754 780 12 602;
 197 586 381 935 163 263;
 252 550 568 130 795 100];
&gt;&gt; coolers(COST)
ans = 
 1521
</pre>

In a certain chemical plant, 6 new pieces of cooling equipment (coolers) are to be installed in a vacant space. This vacant space was divided into a grid of 6 cells by 6 cells. Your task is to assign the 6 coolers to 6 cells in this grid with the following requirements:

# There must be one cooler for every column in the grid.
# If you decide to install a cooler at row R, column C, then the cooler at column C+1 must be installed either on row R-1, R, or R+1 only. This is done to ease the connection of coolers by piping.
# The total installation cost for the 6 coolers must be minimum.

For this problem, you are given a cost matrix (COST) in relation to the third requirement above. COST is a 6 x 6 matrix where the _r,c_-th element is the cost of assigning any cooler to row _r_, column _c_ (All the coolers are identical). Write a function that accepts the matrix COST, and output the value of the minimum cost of installation. You are ensured that all elements of COST are integers in the range [1,1000].

In the sample test case below, the optimal placement is at the following rows: 4,3,3,2,2,2.

>> COST = [695   766   710   119   752   548;
             318   796   755   499   256   139;
             951   187   277   960   506   150;
              35   490   680   341   700   258;
             439   446   656   586   891   841;
             382   647   163   224   960   255];
  >> coolers(COST)
  ans = 
        1393
    
Meanwhile, the optimal placement for the case below is at rows: 5,6,5,6,5,6

>> COST = [815   617   918    76   569   312;
             244   474   286    54   470   529;
             930   352   758   531   455   988;
             350   831   754   780    12   602;
             197   586   381   935   163   263;
             252   550   568   130   795   100];
  >> coolers(COST)
  ans = 
        1521

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Last Solution submitted on May 05, 2023

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Alfonso Nieto-Castanon on 19 Aug 2020

nice problem (hint: viterbi) and overall an excellent series / problem-group!

Jimmy Sølvsteen Nielsen on 10 Sep 2020

I wish I had done a clever solution but I resorted to brute forcing it!

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Problem 45415. Find an optimal placement of coolers on a grid

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