Enhance Knapsack example

docs/src/examples/knapsack.md

@@ -10,8 +10,56 @@ We start with some instances of the discrete [Knapsack Problem](https://en.wikip
 \end{array}
 ```
 
-Now, lets fill a few knapsacks using [JuMP](https://github.com/jump-dev/JuMP.jl).
-We will generate uniform random costs ``\mathbf{c}`` and weights ``\mathbf{w}`` then set the knapsack's capacity ``C`` to be a fraction of the total available weight i.e. ``80\%``.
+First, consider the following items
+
+| Item (``i``) | Value (``c_{i}``) | Weight (``w_{i}``) |
+|:------------:|:-----------------:|:------------------:|
+|       1      |         1         |       0.3          |
+|       2      |         2         |       0.5          |
+|       3      |         3         |       1.0          |
+
+to be carried in a knapsack with capacity ``C = 1.6``.
+
+Writing down the data above as a linear program, we have
+
+```math
+\begin{array}{r l}
+    \max        & x_{1} + 2 x_{2} + 3 x_{3} \\
+    \text{s.t.} & 0.3 x_{1} + 0.5 x_{2} + x_{3} \le 1.6 \\
+    ~           & \mathbf{x} \in \mathbb{B}^{3}
+\end{array}
+```
+
+## Simple JuMP Model
+
+Writing this in [JuMP](https://github.com/jump-dev/JuMP.jl) we end up with
+
+```@example dwave-knapsack
+using JuMP
+using ToQUBO
+using DWave
+
+model = Model(() -> ToQUBO.Optimizer(DWave.Neal.Optimizer))
+
+@variable(model, x[1:3], Bin)
+@objective(model, Max, x[1] + 2 * x[2] + 3 * x[3])
+@constraint(model, 0.3 * x[1] + 0.5 * x[2] + x[3] ≤ 1.6)
+
+optimize!(model)
+
+solution_summary(model)
+```
+
+The final decision is to take items ``2`` and ``3``, i.e., ``x_{1} = 0, x_{2} = 1, x_{3} = 1``.
+
+```@example dwave-knapsack
+value.(x)
+```
+
+## Using DataFrames
+
+Now, lets fill a few more knapsacks.
+First, we generate uniform random costs ``\mathbf{c}`` and weights ``\mathbf{w}`` then set the knapsack's capacity ``C`` to be a fraction of the total available weight i.e. ``80\%``.
 
 This example was inspired by [D-Wave's knapsack example repository](https://github.com/dwave-examples/knapsack).
 
@@ -24,8 +72,8 @@ using Random
 rng = MersenneTwister(1)
 
 df = DataFrame(
-   :cost   => rand(rng, 1:100, 16),
-   :weight => rand(rng, 1:100, 16),
+   :cost   => rand(rng, 1:100, 8),
+   :weight => rand(rng, 1:100, 8),
 )
 
 CSV.write("knapsack.csv", df)
@@ -57,7 +105,7 @@ C = round(0.8 * sum(w))
 optimize!(model)
 
 # Add Results as a new column
-df[:,:select] = map((xi) -> (xi > 0.0) ? "✅" : "❌", value.(x))
+df[:,:select] = map(xi -> ifelse(xi > 0, "✅", "❌"), value.(x))
 
 df
 ```