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<div class="section" id="Multidimensional-Knapsack">
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<h1><span class="section-number">3. </span>Multidimensional Knapsack<a class="headerlink" href="#Multidimensional-Knapsack" title="Permalink to this headline">¶</a></h1>
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<div class="section" id="Problem-definition">
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<h2><span class="section-number">3.1. </span>Problem definition<a class="headerlink" href="#Problem-definition" title="Permalink to this headline">¶</a></h2>
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<p>Given a set of <span class="math notranslate nohighlight">\(n\)</span> items and <span class="math notranslate nohighlight">\(m\)</span> types of resources (also called <em>knapsacks</em>), the problem is to find a subset of items that maximizes profit without consuming more resources than it is available. More precisely, the problem is:</p>
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<div class="math notranslate nohighlight">
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\[\begin{split}\begin{align*}
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\text{maximize}
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& \sum_{j=1}^n p_j x_j
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\\
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\text{subject to}
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& \sum_{j=1}^n w_{ij} x_j \leq b_i
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& \forall i=1,\ldots,m \\
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& x_j \in \{0,1\}
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& \forall j=1,\ldots,n
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\end{align*}\end{split}\]</div>
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</div>
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<div class="section" id="Random-instance-generator">
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<h2><span class="section-number">3.2. </span>Random instance generator<a class="headerlink" href="#Random-instance-generator" title="Permalink to this headline">¶</a></h2>
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<p>The class <code class="docutils literal notranslate"><span class="pre">MultiKnapsackGenerator</span></code> can be used to generate random instances of this problem. The number of items <span class="math notranslate nohighlight">\(n\)</span> and knapsacks <span class="math notranslate nohighlight">\(m\)</span> are sampled from the user-provided probability distributions <code class="docutils literal notranslate"><span class="pre">n</span></code> and <code class="docutils literal notranslate"><span class="pre">m</span></code>. The weights <span class="math notranslate nohighlight">\(w_{ij}\)</span> are sampled independently from the provided distribution <code class="docutils literal notranslate"><span class="pre">w</span></code>. The capacity of knapsack <span class="math notranslate nohighlight">\(i\)</span> is set to</p>
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<div class="math notranslate nohighlight">
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\[b_i = \alpha_i \sum_{j=1}^n w_{ij}\]</div>
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<p>where <span class="math notranslate nohighlight">\(\alpha_i\)</span>, the tightness ratio, is sampled from the provided probability distribution <code class="docutils literal notranslate"><span class="pre">alpha</span></code>. To make the instances more challenging, the costs of the items are linearly correlated to their average weights. More specifically, the price of each item <span class="math notranslate nohighlight">\(j\)</span> is set to:</p>
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<div class="math notranslate nohighlight">
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\[p_j = \sum_{i=1}^m \frac{w_{ij}}{m} + K u_j,\]</div>
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<p>where <span class="math notranslate nohighlight">\(K\)</span>, the correlation coefficient, and <span class="math notranslate nohighlight">\(u_j\)</span>, the correlation multiplier, are sampled from the provided probability distributions <code class="docutils literal notranslate"><span class="pre">K</span></code> and <code class="docutils literal notranslate"><span class="pre">u</span></code>.</p>
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<p>If <code class="docutils literal notranslate"><span class="pre">fix_w=True</span></code> is provided, then <span class="math notranslate nohighlight">\(w_{ij}\)</span> are kept the same in all generated instances. This also implies that <span class="math notranslate nohighlight">\(n\)</span> and <span class="math notranslate nohighlight">\(m\)</span> are kept fixed. Although the prices and capacities are derived from <span class="math notranslate nohighlight">\(w_{ij}\)</span>, as long as <code class="docutils literal notranslate"><span class="pre">u</span></code> and <code class="docutils literal notranslate"><span class="pre">K</span></code> are not constants, the generated instances will still not be completely identical.</p>
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<p>If a probability distribution <code class="docutils literal notranslate"><span class="pre">w_jitter</span></code> is provided, then item weights will be set to <span class="math notranslate nohighlight">\(w_{ij} \gamma_{ij}\)</span> where <span class="math notranslate nohighlight">\(\gamma_{ij}\)</span> is sampled from <code class="docutils literal notranslate"><span class="pre">w_jitter</span></code>. When combined with <code class="docutils literal notranslate"><span class="pre">fix_w=True</span></code>, this argument may be used to generate instances where the weight of each item is roughly the same, but not exactly identical, across all instances. The prices of the items and the capacities of the knapsacks will be calculated as above, but using these perturbed weights instead.</p>
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<p>By default, all generated prices, weights and capacities are rounded to the nearest integer number. If <code class="docutils literal notranslate"><span class="pre">round=False</span></code> is provided, this rounding will be disabled.</p>
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<div class="admonition note">
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<p class="admonition-title">References</p>
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<ul class="simple">
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<li><p><strong>Freville, Arnaud, and Gérard Plateau.</strong> <em>An efficient preprocessing procedure for the multidimensional 0–1 knapsack problem.</em> Discrete applied mathematics 49.1-3 (1994): 189-212.</p></li>
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<li><p><strong>Fréville, Arnaud.</strong> <em>The multidimensional 0–1 knapsack problem: An overview.</em> European Journal of Operational Research 155.1 (2004): 1-21.</p></li>
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</ul>
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</div>
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</div>
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<div class="section" id="Challenge-A">
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<h2><span class="section-number">3.3. </span>Challenge A<a class="headerlink" href="#Challenge-A" title="Permalink to this headline">¶</a></h2>
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<ul class="simple">
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<li><p>250 variables, 10 constraints, fixed weights</p></li>
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<li><p><span class="math notranslate nohighlight">\(w \sim U(0, 1000), \gamma \sim U(0.95, 1.05)\)</span></p></li>
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<li><p><span class="math notranslate nohighlight">\(K = 500, u \sim U(0, 1), \alpha = 0.25\)</span></p></li>
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<li><p>500 training instances, 50 test instances</p></li>
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</ul>
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<div class="nbinput nblast docutils container">
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<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[ ]:
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</pre></div>
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</div>
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<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
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<span></span><span class="n">MultiKnapsackGenerator</span><span class="p">(</span>
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<span class="n">n</span><span class="o">=</span><span class="n">randint</span><span class="p">(</span><span class="n">low</span><span class="o">=</span><span class="mi">250</span><span class="p">,</span> <span class="n">high</span><span class="o">=</span><span class="mi">251</span><span class="p">),</span>
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<span class="n">m</span><span class="o">=</span><span class="n">randint</span><span class="p">(</span><span class="n">low</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">high</span><span class="o">=</span><span class="mi">11</span><span class="p">),</span>
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<span class="n">w</span><span class="o">=</span><span class="n">uniform</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mf">1000.0</span><span class="p">),</span>
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<span class="n">K</span><span class="o">=</span><span class="n">uniform</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mf">500.0</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mf">0.0</span><span class="p">),</span>
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<span class="n">u</span><span class="o">=</span><span class="n">uniform</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mf">1.0</span><span class="p">),</span>
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<span class="n">alpha</span><span class="o">=</span><span class="n">uniform</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mf">0.0</span><span class="p">),</span>
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<span class="n">fix_w</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
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|
<span class="n">w_jitter</span><span class="o">=</span><span class="n">uniform</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mf">0.95</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mf">0.1</span><span class="p">),</span>
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<span class="p">)</span>
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</pre></div>
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</div>
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</div>
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</div>
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</div>
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</div>
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<div class='prev-next-bottom'>
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<a class='left-prev' id="prev-link" href="../stab/" title="previous page"><span class="section-number">2. </span>Maximum Weight Stable Set</a>
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<a class='right-next' id="next-link" href="../tsp/" title="next page"><span class="section-number">4. </span>Traveling Salesman</a>
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