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9.11 An ILP model for the Multiple-Choice Knapsack Problem

Part 2 · Formulations and their properties · Chapter 9 · Chapter 9: Knapsack problems · Lecture notes, volume 2 (PDF) · Slides · Exercises: Knapsack problems · App: LP relaxation and upper bounds for the 0/1 KP

The Multiple-Choice Knapsack Problem and its reference instance (Example 2.7) are defined in Chapter 2 «Knapsack problems», Section 2.8.

  • We introduce the following \(n\) binary variables:

    \[\begin{equation*} \label{BIN_VAR_MCKP} x_j = \left\{ \begin{array}{ll} 1 & \textrm{if item $j$ is selected in the knapsack}\\[1.5ex] 0 & \textrm{otherwise} \end{array} \qquad \forall j \in \mathscr{I}. \right. \tag{9.30} \end{equation*}\]

Using these variables, an Integer Linear Programming (ILP) model for the Multiple-Choice Knapsack Problem (MCKP) reads as follows:

\[\begin{align*} && \max \;\;{\sum_{j \in \mathscr{I}} p_j \, x_j} \label{obj_MCKP} \tag{9.31a}\\[1ex] && \text{subject to} \quad {\sum_{j \in \mathscr{I}} w_j \, x_j} &\leq c, \label{cons1_MCKP} \tag{9.31b}\\[1ex] && {\sum_{j \in \mathscr{J}_h} x_j} &= 1, & \forall h \in \{1,2,\dots,q\}, \label{cons2_MCKP} \tag{9.31c}\\[1ex] &&x_j &\in \{0,1\}, & \forall j \in \mathscr{I}. \label{cons3_MCKP} \tag{9.31d} \end{align*}\]

This ILP model is referred to as the \(\mathrm{ILP}_{\mathrm{MCKP}}\) model.

  • Description of the objective function and constraints of the \(\mathrm{ILP}_{\mathrm{MCKP}}\) model:

    • the linear objective function \(\eqref{obj_MCKP}\) maximizes the total profit of the items selected in the knapsack;

    • constraint \(\eqref{cons1_MCKP}\) ensures that the total weight of the selected items does not exceed the knapsack capacity (one linear constraint);

    • constraints \(\eqref{cons2_MCKP}\) ensure that exactly one item is selected from each class (\(q\) linear constraints, one for each class);

    • finally, constraints \(\eqref{cons3_MCKP}\) define the variables of the ILP model.

Example 9.15: \(\mathrm{ILP}_{\mathrm{MCKP}}\) model

We consider a knapsack of capacity \(c = 15\) and a set \(\mathscr{I} = \{1,2,3,4,5,6\}\) of \(6\) items (\(n = 6\)), partitioned into \(q = 3\) classes \(\mathscr{J}_1 = \{1,2\}\), \(\mathscr{J}_2 = \{3,4\}\) and \(\mathscr{J}_3 = \{5,6\}\), with the following profits and weights:

\[ \begin{array}{l|rrrrrrr} \hline & {j = 1} & {j = 2} & {j = 3} & {j = 4} & {j = 5} & {j = 6} \\ \hline {p_j} & {6} & {8} & {5} & {7} & {4} & {9} \\ {w_j} & {4} & {6} & {5} & {8} & {3} & {7} \\ \hline \end{array} \]

This is the MCKP instance of Example 2.7. The \(\mathrm{ILP}_{\mathrm{MCKP}}\) model for this MCKP instance is:

\[ \begin{array}{rrrrrrrrrrrrr} {\max} & 6 \, {x_1} & + 8 \, {x_2} & + 5 \, {x_3} & + 7 \, {x_4} & + 4 \, {x_5} & + 9 \, {x_6} \\[0.5ex] \text{subject to} & 4 \, {x_1} & + 6 \, {x_2} & + 5 \, {x_3} & + 8 \, {x_4} & + 3 \, {x_5} & + 7 \, {x_6} & {\le} & 15 \\[0.5ex] & {x_1} & + {x_2} & & & & & {=} & 1 \\[0.5ex] & & & {x_3} & + {x_4} & & & {=} & 1 \\[0.5ex] & & & & & {x_5} & + {x_6} & {=} & 1 \\[0.5ex] & {x_1}, & {x_2}, & {x_3}, & {x_4}, & {x_5}, & {x_6} & { \in} & { \{0,1\}} \end{array} \]

An optimal solution \(~{\tilde{\vet x}}(\mathrm{ILP}_{\mathrm{MCKP}})\) is:

\[ \tilde x_1=1,~~\tilde x_2=0,~~\tilde x_3=0,~~\tilde x_4=1,~~\tilde x_5=1,~~\tilde x_6=0 \]

The optimal value \(z(\mathrm{ILP}_{\mathrm{MCKP}})\) is \(17\). The associated optimal solution of the MCKP is \(\tilde{S} =\{1,4,5\}\). The optimal solution is not unique: the subset \(\{2,3,5\}\) has the same value.

Next: 9.12 An ILP model for the Knapsack Problem with Setups