From 8a55ce784328f7161482dc385d81bf44543bd58f Mon Sep 17 00:00:00 2001
From: Alessandro Zocca <a.zocca@vu.nl>
Date: Fri, 20 Oct 2023 14:00:22 +0200
Subject: [PATCH] Update dinner-seat-allocation.ipynb

---
 notebooks/04/dinner-seat-allocation.ipynb | 31 +++++++++++++----------
 1 file changed, 17 insertions(+), 14 deletions(-)

diff --git a/notebooks/04/dinner-seat-allocation.ipynb b/notebooks/04/dinner-seat-allocation.ipynb
index ec7e5980..b9660104 100644
--- a/notebooks/04/dinner-seat-allocation.ipynb
+++ b/notebooks/04/dinner-seat-allocation.ipynb
@@ -904,9 +904,14 @@
     "To illustrate this, we first visualize the seating problem using a graph where:\n",
     "* the nodes on the left-hand side stand for the families and the numbers next to them provide the family size;\n",
     "* the nodes on the left-hand side stand for the tables and the numbers next to them provide the table size;\n",
-    "* each left-to-right arrow stand comes with a number denoting the capacity of arc $(f, t)$ -- how many people of family $f$ can be assigned to table $t$.\n",
-    "\n",
-    "<img src=\"seating_model_basic.png\" width=\"800\" height=\"600\">"
+    "* each left-to-right arrow stand comes with a number denoting the capacity of arc $(f, t)$ -- how many people of family $f$ can be assigned to table $t$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "![](seating_model_basic.png)"
    ]
   },
   {
@@ -1088,9 +1093,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "By adding two more nodes to the graph above, we can formulate the problem as a slightly different flow problem where all the data is formulated as the arc capacity, see figure below. In a network like this, we can imagine a problem of sending resources from the _root node_ \"door\" to the _sink node_ \"seat\", subject to the restriction that for any node that is not the start nor the target, the sum of incoming and outgoing flows are equal (_balance constraint_). If there exists a flow in this new graph that respects the arc capacities and the sum of outgoing flows at the source is equal to the total number of individuals, that is $\\sum_{f \\in F} m_f = 39$, it means that there exists a family-to-table assignment that meets our requirements.\n",
-    "\n",
-    "<img src=\"seating_flow_model.png\" width=\"800\" height=\"600\">"
+    "By adding two more nodes to the graph above, we can formulate the problem as a slightly different flow problem where all the data is formulated as the arc capacity, see figure below. In a network like this, we can imagine a problem of sending resources from the _source node_ \"door\" to the _target node_ \"seat\", subject to the restriction that for any node that is neither the source nor the target, the sum of incoming and outgoing flows are equal (_balance constraint_). If there exists a flow in this new graph that respects the arc capacities and the sum of outgoing flows at the source is equal to the total number of individuals, that is $\\sum_{f \\in F} m_f$, it means that there exists a family-to-table assignment that meets our requirements."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "![](seating_flow_model.png)"
    ]
   },
   {
@@ -1234,7 +1244,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Even for this very small example, we see that network algorithms generate a solution significantly faster than using the MILO formulation above."
+    "Even for this very small example, we see that network algorithms generate a solution significantly faster than using the MILO formulation above. Realizing that the optimization problem we are tackling is a particular problem class for which a tailored algorithm is available can result in solving times orders of magnitude faster, particularly for large instances."
    ]
   },
   {
@@ -1377,13 +1387,6 @@
     "plt.tight_layout()\n",
     "plt.show()"
    ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
   }
  ],
  "metadata": {