An exact approach for the bilevel knapsack problem with interdiction constraints and extensions

Abstract

We consider the bilevel knapsack problem with interdiction constraints, an extension of the classic 0–1 knapsack problem formulated as a Stackelberg game with two agents, a leader and a follower, that choose items from a common set and hold their own private knapsacks. First, the leader selects some items to be interdicted for the follower while satisfying a capacity constraint. Then the follower packs a set of the remaining items according to his knapsack constraint in order to maximize the profits. The goal of the leader is to minimize the follower’s total profit. We derive effective lower bounds for the bilevel knapsack problem and present an exact method that exploits the structure of the induced follower’s problem. The approach strongly outperforms the current state-of-the-art algorithms designed for the problem. We extend the same algorithmic framework to the interval min–max regret knapsack problem after providing a novel bilevel programming reformulation. Also for this problem, the proposed approach outperforms the exact algorithms available in the literature.

Appendix

1.1 Additional notation for MRKP

For further analysis, we define the set of items $l_i$ by L and the set of items $h_i$ by H. We consider the 2n items in the follower’s knapsack ordered by nonincreasing ratio profit over weight. For each item $j = 1, \dots , 2n$ in the ordering we denote by i(j) the corresponding index i. Similarly as in Sect. 2, we denote by KP(x) the follower’s knapsack problem induced by a leader’s solution x, here with item set

$$\begin{aligned} J := \left\{ \begin{array}{lr} j: x_{i(j)} = 1 \text { and } j \in L,\\ j: x_{i(j)} = 0 \text { and } j \in H,\\ x_i \in x \end{array}\right\} \end{aligned}$$

and denote by $KP^{LP}(x)$ the corresponding LP relaxation. We again denote as critical the last item with a strictly positive value in the optimal solution of $KP^{LP}(x)$. Notice that for each $x_i$ one item between $l_i$ and $h_i$ will be always available for insertion in the follower’s knapsack. Hence, under the standard assumption that $\sum \nolimits _{i = 1}^{n} w_i > C$ in the MRKP, we also have $\sum \nolimits _{j = 1}^{2n} w_{i(j)} > C$. This implies that there alway exists a critical item c in $KP^{LP}(x)$ and a split solution with value

$$\begin{aligned} \sum \limits _{\begin{array}{c} j \in J: \\ j< c, j \in L \end{array}} p_{i(j)}^{-} + \sum \limits _{\begin{array}{c} j \in J: \\ j< c, j \in H \end{array}} p_{i(j)}^{+} = \sum \limits _{j< c, j \in L} p_{i(j)}^{-}x_{i(j)} + \sum \limits _{j < c, j \in H} p_{i(j)}^{+}(1-x_{i(j)})\qquad \end{aligned}$$

(49)

1.2 Computing a lower bound for MRKP

As for BKP, we aim to guess the critical item of $KP^{LP}(x^*)$. To this purpose we modify model $CRIT_2(c)$ as follows. We replace term $\sum \nolimits _{h=1}^{w_c} hk_h$ with an integer variable $\theta$ associated with the weight contribution of the critical item c in the follower’s capacity constraint, with $1 \le \theta \le w_{i(c)}$. The objective function has term $\sum \nolimits _{i=1}^{n} p_{i}^{-}x_i$ as negative contribution plus the split solution value (49) and the additional profit contribution that can be gained by a feasible solution derived by the split solution. Such a profit is captured by variable $\pi$ through a set of constraints ${\mathcal {F}}(\pi ,x,\theta , v, \tau )$ involving additional binary variables $v_p$$(p = 1, \dots , \frac{|{\mathcal {F}}(\pi ,x,\theta , v,\tau )|}{2})$. We explain the construction of set ${\mathcal {F}}(\pi ,x,\theta , v,\tau )$ next. We obtain the following model, denoted as $CRIT_3(c)$.

$\quad CRIT_3(c)$:

$$\begin{aligned}&\text {min}\quad \sum \limits _{j< c, j \in L} p_{i(j)}^{-}x_{i(j)} + \sum \limits _{j < c, j \in H} p_{i(j)}^{+}(1-x_{i(j)}) + \pi - \sum \limits _{i=1}^{n} p_{i}^{-}x_i \end{aligned}$$

(50)

$$\begin{aligned}&\text {subject to}\quad \sum \limits _{i=1}^{n} w_{i}x_{i} \le C \end{aligned}$$

(51)

$$\begin{aligned}&\sum \limits _{j< c, j \in L} w_{i(j)}x_{i(j)} + \sum \limits _{j < c, j \in H} w_{i(j)}(1-x_{i(j)}) + \theta = C \end{aligned}$$

(52)

$$\begin{aligned}&1 \le \theta \le w_{i(c)} \end{aligned}$$

(53)

$$\begin{aligned}&{\mathcal {F}}(\pi ,x,\theta ,v,\tau ) \end{aligned}$$

(54)

$$\begin{aligned}&x_{i(c)} = 1 \text { if } c\in L \;, x_{i(c)} = 0 \text { if } c\in H \end{aligned}$$

(55)

$$\begin{aligned}&x_{i} \in \{0,1\} \quad i= 1,\dots ,n \end{aligned}$$

(56)

$$\begin{aligned}&v_p \in \{0,1\} \quad p= 1,\dots ,\frac{|{\mathcal {F}}(\pi ,x,\theta , v,\tau )|}{2} \end{aligned}$$

(57)

$$\begin{aligned}&\theta \in N \end{aligned}$$

(58)

$$\begin{aligned}&\pi \ge 0 \end{aligned}$$

(59)

The objective function (50) minimizes the difference between the value of a follower’s knapsack solution and sum $\sum \nolimits _{i=1}^{n} p_{i}^{-}x_i$. Constraint (51) represents the leader’s capacity constraint. Constraints (52) and (53) guarantee that item c is critical as it is the last item packed, with a weight in the interval $[1, w_{i(c)}]$. Constraint (55) enforces the availability of item c in the follower’s knapsack: if $c\in L$ we must have $x_{i(c)} = 1$ or else $x_{i(c)} = 0$. Constraints (56)–(59) define the domain of the variables.

We build a proper set ${\mathcal {F}}(\pi ,x,\theta ,v,\tau )$, namely a set of constraints that considers only feasible solutions in the follower’s knapsack, as follows. For a given subset $\tau$ of items around the critical item c, the corresponding profit $p^{\tau }$ and weight $w^{\tau }$ now are:

$$\begin{aligned} p^{\tau }&= - \sum \limits _{ \begin{array}{c} j \in \tau : \\ j< c, j \in L \end{array}} p_{i(j)}^{-} - \sum \limits _{ \begin{array}{c} j \in \tau : \\ j < c, j \in H \end{array}} p_{i(j)}^{+} + \sum \limits _{ \begin{array}{c} j \in \tau : \\ j \ge c, j \in L \end{array}} p_{i(j)}^{-} + \sum \limits _{ \begin{array}{c} j \in \tau : \\ j \ge c, j \in H \end{array}} p_{i(j)}^{+} \end{aligned}$$

(60)

$$\begin{aligned} w^{\tau }&= -\sum \limits _{j \in \tau : i < c} w_{i(j)} + \sum \limits _{j \in \tau : j \ge c} w_{i(j)}. \end{aligned}$$

(61)

We consider in set ${\mathcal {F}}(\pi ,x,\theta ,v,\tau )$ only subsets of items improving upon the split solution, i.e. with $p^\tau > 0$, that could be feasible, i.e. with $w^{\tau } \le w_{i(c)}$. Also we do not consider subsets with two items j and $j^{\prime }$ corresponding to the same $x_i$, namely with $i(j) = i(j')$, as these items cannot be both available for packing. For the $p-th$ subset $\tau$ considered for selection in ${\mathcal {F}}(\pi ,x,\theta ,v,\tau )$, we will have binary variable $v_p$ equal to one only if the follower’s capacity constraint is not violated, i.e. $w^{\tau } \le \theta$. This condition will be enforced by adding to ${\mathcal {F}}(\pi ,x,\theta ,v,\tau )$ the constraint $(w_{i(c)} - w^{\tau }) v_p \ge \theta - w^{\tau },$ that we translate as follows to be added as a linear constraint

$$\begin{aligned} \left( w_{i(c)} - w^{\tau } + \frac{1}{2}\right) v_p \ge \theta - w^{\tau } + \frac{1}{2}, \end{aligned}$$

(62)

where constant term $\frac{1}{2}$ is introduced to have $v_p = 1$ when $w^{\tau } = \theta$. Then an improvement $\pi$ can be determined by adding to ${\mathcal {F}}(\pi ,x,\theta ,v,\tau )$ the following constraint

$$\begin{aligned} \pi \ge p^\tau \left( v_p - \sum \limits _{j \in \tau , j \in L} (1-x_{i(j)}) - \sum \limits _{j \in \tau , j \in H} x_{i(j)}\right) . \end{aligned}$$

(63)

Notice that constraints (62) and (63) ensure that only items available for insertion in the follower’s knapsack are considered and the follower’s capacity constraint is not violated, validating tuple $\tau$. The following proposition holds.

Proposition 3

If $KP^{LP}(x^*)$ has critical item c and model $CRIT_3(c)$ has a proper set ${\mathcal {F}}(\pi ,x,\theta ,v,\tau )$, then $z(CRIT_3(c)) \le z^*$.

Proof

We apply the same argument of Propositions 1 and 2 and observe that model $CRIT_3(c)$ considers feasible solutions for $KP(x^*)$. Let $z_1$ denote the solution value of the best of these KP solutions. We have

$$\begin{aligned}&z(CRIT_3(c)) \le z_1 - \sum \limits _{i=1}^{n} p_{i}^{-}x_i^* \le z(KP(x^*)) - \sum \limits _{i=1}^{n} p_{i}^{-}x_i^* = z^*, \end{aligned}$$

where the first inequality is implied by noticing that $x^*$ is feasible but not necessarily optimal for model $CRIT_3(c)$. $\square$

We finally notice that the introduction of variable $\theta$ is motivated by the presence of large follower’s weights (with values up to 10,000) in the benchmark MRKP literature instances we considered in our computational tests. As remarked in Sect. 3, large weights compromise the use of variables $k_h$ and the solution of the corresponding ILP models.

1.3 A new exact approach for MRKP

We employ the same algorithmic framework of BKP with the following few modifications. We construct set ${\mathcal {F}}(\pi ,x,\theta ,v,\tau )$ in model $CRIT_3(c)$ by a procedure denoted as MRKPTuples. The procedure is very similar to DefineTuples, so we do not report the corresponding pseudo code. The only differences are the presence of 2n items in the follower’s knapsack and the insertion of $\mu$ pairs of constraints (62)–(63) to set ${\mathcal {F}}(\pi ,x,\theta ,v,\tau )$ and constraints related to the adding of the critical item (Lines 6–7 in the pseudo code of DefineTuples).

Similarly to (28), we then define the first possible critical item q as

$$\begin{aligned} q := \min \left\{ j: \sum \limits _{k=1}^{j} w_{i(k)} \ge C\right\} . \end{aligned}$$

(64)

We also test variables $x_i$ and $v_p$ for fixing by reduced costs and consider the following constraint instead of constraint (35) during the exploration of a subproblem $CRIT_3(c)$

$$\begin{aligned} \sum \limits _{i : \bar{x_i} = 0}^{n} x_i + \sum \limits _{i : \bar{x_i} = 1}^{n} (1-x_i) \ge 1. \end{aligned}$$

(65)

Finally notice that for MRKP we do not have to handle the absence of the critical item in the follower’s knapsack problem (see Sect. 8.1). Below is reported the pseudo code of the derived approach.