Updated the scheduler tutorial to explain more about the dependency offset constraints

- explained differences between 'minimum' and 'exact' in how the constraints work

Author: shousner

famodel/irma/scheduler_tutorial.ipynb

@@ -1225,9 +1225,15 @@
    "id": "38f7e788",
    "metadata": {},
    "source": [
+    "In this example, we have additional inputs that specify that Task 0 (Install Mooring) must be done AT LEAST 1 period (offset) after Task 1 (Install Anchor) starts. The constraints that it makes are like the following:\n",
+    "\n",
+    "$ X_{t,s}[0,0] + X_{t,s}[1,0] \\leq 1 \\quad X_{t,s}[0,0] + X_{t,s}[1,1] \\leq 1 \\quad X_{t,s}[0,1] + X_{t,s}[1,1] \\leq 1 $\n",
+    "\n",
+    "which says that if Task 1 starts in period 0, then Task 0 cannot start in period 0 (but it could start anywhere else). Similarly, if Task 1 starts in period 1, then Task 0 cannot start in period 0 nor 1.\n",
+    "\n",
     "**Notes**:\n",
-    "- This minimum example does coincidentally allows Task 0 to start 1 period after the start of Task 1, but it wasn't required too.\n",
-    "- Using the dependency type of 'finish_start' creates 1's in Xta variables in the constraints. Using 'start_start' does not include 1's in the constraint row"
+    "- Task 0 does not have to start exactly 1 period after Task 1 starts (for this case), but it will try too since there are penalty objective values on later start times\n",
+    "- Using the dependency type of 'finish_start' creates 1's in Xta variables in the constraints because the duration of the 'finish' depends on the duration of the specific task-asset group combination. Using 'start_start' does not include 1's in the constraint row because it does not require information about the duration."
    ]
   },
   {
@@ -1271,19 +1277,29 @@
     "else:\n",
     "    print(\"No task dependencies defined - no constraints needed\")\n",
     "\n",
-    "result_exact = scheduler_exact.optimize()\n"
+    "result_exact = scheduler_exact.optimize()"
    ]
   },
   {
    "cell_type": "markdown",
    "id": "2969cb3f",
    "metadata": {},
    "source": [
-    "Can add more explanation for how the constraints look for different dependency offsets.\n",
+    "In this example, we run the same scenario but specify that Task 0 MUST start EXACTLY 1 period after the start of Task 1. The constraints are made like the following\n",
     "\n",
-    "They mainly update the $X_{t,s}$ variables to ensure each task does not start in a period that would violate the dependency.\n",
+    "$$ -X_{t,s}[0,1] + X_{t,s}[1,0] \\leq 0 $$\n",
     "\n",
-    "There are some -1's in the 'exact' method to ensure that a certain start period does exactly happen for a task"
+    "$$ X_{t,s}[0,0] + X_{t,s}[1,0] \\leq 1 \\quad X_{t,s}[0,2] + X_{t,s}[1,0] \\leq 1 \\quad X_{t,s}[0,3] + X_{t,s}[1,0] \\leq 1 $$\n",
+    "\n",
+    "which says in the first equation first, if Task 1 starts in period 0, then Task 0 must start in period 1. And in the second set of equations, if Task 1 starts in period 0, then Task 0 cannot start in any other period (0, 2, 3, 4)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "677af285",
+   "metadata": {},
+   "source": [
+    "Can add more explanations/examples for different dependency types and offsets"
    ]
   },
   {