A column and constraint generation algorithm for the dynamic knapsack problem with stochastic item sizes

Abstract

We consider a version of the knapsack problem in which an item size is random and revealed only when the decision maker attempts to insert it. After every successful insertion the decision maker can choose the next item dynamically based on the remaining capacity and available items, while an unsuccessful insertion terminates the process. We propose an exact algorithm based on a reformulation of the value function linear program, which dynamically prices variables to refine a value function approximation and generates cutting planes to maintain a dual bound. We provide a detailed analysis of the zero-capacity case, in which the knapsack capacity is zero, and all item sizes have positive probability of equaling zero. We also provide theoretical properties of the general algorithm and an extensive computational study. Our main empirical conclusion is that the algorithm is able to significantly reduce the gap when initial bounds and/or heuristic policies perform poorly.

Appendix

1.1 Proof of Proposition 2

Proof

For simplicity in notation, let us denote $v^*_N(0)$ as $v_N$ and $w^*_N(0)$ as $w_N$. We proceed by induction on the size of N, with the trivial base case that $w_{\{i\}} = q_i c_i$. We now observe

$$\begin{aligned} w_N&= v_{ \{ 1, \dots , n \}} - \sum _{\begin{array}{c} U \subsetneq N \\ n \not \in U \end{array}} w_U - \sum _{\begin{array}{c} U \subsetneq N \\ n \in U \end{array}} w_U = v_{ \{ 1, \dots , n \}} - v_{ \{ 1, \dots , n-1 \}} - \sum _{\begin{array}{c} U \subsetneq N \\ n \in U \end{array}} w_U \nonumber \\&= \left( v_{ \{ 1, \dots , n-1\}} + q_n c_n \, \prod _{i \le n-1} q_i \right) - v_{ \{ 1, \dots , n-1 \}} - \sum _{\begin{array}{c} U \subsetneq N \\ n \in U \end{array}} w_U \nonumber \\&= q_n c_n r_{N \backslash n} - \sum _{\begin{array}{c} U \subsetneq N \\ n \in U \end{array}} w_U = q_n c_n r_{N \backslash n} - \sum _{\begin{array}{c} U \subsetneq N \\ n \in U \end{array}} \Bigl [ (-1)^{|U| + 1} \, q_n c_n \, \prod _{i \in U \backslash n} p_i \Bigr ] \nonumber \\&= q_n c_n r_{N \backslash n} - q_n c_n \left( \sum _{U \subsetneq N \backslash n} \Bigl [ (-1)^{|U| + 2} \prod _{i \in U} p_i \Bigr ] \right) , \end{aligned}$$

(16)

where $r_U := \prod _{i \in U} q_i$, and the second to last equality follows from the induction hypothesis. Considering the following identity,

$$\begin{aligned} \prod _{i \in N} p_i = \sum _{U \subseteq N} (-1)^{|U|} r_U, \end{aligned}$$

we can thus simplify (16) above to yield

$$\begin{aligned} w_N&= q_n c_n r_{N \backslash n} - q_n c_n \sum _{U \subsetneq N \backslash n} (-1)^{|U|} \sum _{V \subseteq U} (-1)^{|V|} r_V \nonumber \\&= q_n c_n \left[ r_{N \backslash n} - \sum _{U \subsetneq N \backslash n} \sum _{V \subseteq U} (-1)^{|U| + |V|} \, r_V \right] \nonumber \\&= q_n c_n \left[ r_{N \backslash n} + \sum _{U \subset N \backslash n} \sum _{V \subseteq U} (-1)^{|U| + |V| + 1} \, r_V \right] . \end{aligned}$$

(17)

Examining the double sum in (17), for a set X, $r_X$ appears once for each (strict) subset of $(N \backslash n) \backslash X$. Thus we have

$$\begin{aligned} \sum _{U \subsetneq N \backslash n} \sum _{V \subseteq U} (-1)^{|U| + |V| + 1} \, r_V = \sum _{X \subsetneq N \backslash n} r_X \sum _{Y \subsetneq (N \backslash n) \backslash X} (-1)^{|Y|+ 1}, \end{aligned}$$

(18)

where the exponent for $-1$ is taken from the substitution of $V = X$ and $U = X \cup Y$.

On the other hand, the right hand side of (9) can be rewritten as

$$\begin{aligned}&q_n c_n \, (-1)^{|N| + 1} \prod _{i < n} p_i = q_n c_n \, (-1)^{|N| + 1} \sum _{X \subseteq N \backslash n} (-1)^{|X|} \, r_X \nonumber \\&\quad = q_n c_n \, \sum _{X \subseteq N \backslash n} (-1)^{|N| + |X| + 1} \, r_X \nonumber \\&\quad = q_n c_n \Bigl [ r_{N \backslash n} + \sum _{X \subset N \backslash n} (-1)^{|N| + |X| + 1} \, r_X \Bigr ] \end{aligned}$$

(19)

Comparing the double sum in (17) with the sum in (19), identity (18) implies that it thus suffices to show

$$\begin{aligned} S_{N,X} := \sum _{Y \subset (N \backslash n) \backslash X} (-1)^{|Y|+ 1} = (-1)^{|N + |X| + 1}. \end{aligned}$$

But viewing the sum $S_{N,X}$ combinatorially, we can rewrite

$$\begin{aligned} S_{N,X}= & {} \sum _{i = 0}^{|N| - |X| - 2} \left( {\begin{array}{c}|N| - |X| - 1\\ i\end{array}}\right) (-1)^{i + 1} \\= & {} \sum _{i = 0}^{|N| - |X| - 1} \left( {\begin{array}{c}|N| - |X| - 1\\ i\end{array}}\right) (-1)^{i + 1} - \, (-1)^{|N| - |X| } \\= & {} - \sum _{i = 0}^{k} \left( {\begin{array}{c}k\\ i\end{array}}\right) (-1)^{i} + \, (-1)^{k} = (-1)^{k}, \end{aligned}$$

where the third equality substitutes $k = |N| - |X| - 1$, and the last equality follows from the identity that the alternating sum of binomial coefficients is 0. Hence,

$$\begin{aligned} S_{N,X} ={\left\{ \begin{array}{ll} -1 = (-1)^{|N| + |X| + 1} &{} \text {if}\, |N|\,\text {and}\, |X|\,\text {have the same parity} \\ 1 = (-1)^{|N| + |X| + 1} &{} \text {if}\, |N|\,\text {and}\,|X|\,\text {have different parity.} \end{array}\right. } \end{aligned}$$

The entire argument above is identical for any subset $M \subseteq N$. $\square$

1.2 Computational experiments—10-item instances discussion

Table 3 provides summary results for the computational experiments described in Sect. 5. The success rate is defined as the percentage of instances that reached the optimality gap within the time limit, while run time is the average run time in hours for the successful instances. The incomplete remaining gap refers to the geometric mean of the remaining gap of any instance that did not reach the target optimality gap within the time limit. We observe from Table 3 that distributions D6 and D7, the two distributions that do not have support for 0, have the highest success rate. Other distributions seem to have a lower success rate as the variance of the distribution decreases; these results suggest that both including 0 in the support and smoothing the distribution can make closing the final gap of less than 0.5% difficult. We also observed that uncorrelated instances tend to take less time to complete than correlated instances, which makes sense as correlated item sizes and profits tend to make more difficult deterministic knapsack problems.

Additionally, recalling that the 10-item instances actually solve to (numerical) optimality, Figs. 6 and 7 provide insight into which set cardinalities are more prevalent in the optimal solution. In particular, we observe a noticeable difference in the set size distributions between the Bernoulli instances (D1, D3, D4, D5) and non-Bernoulli instances (D2, D6, D7). The plot for Bernoulli instances is more skewed to the right and has a mode of two items, which further explains why the Quadratic relaxation (4) (which includes all singleton and pair item sets) was the best-performing approximation in earlier experiments for these instance types [6]. However, the plot for non-Bernoulli instances seems to become more normally distributed as the instance’s variance decreases: D2 is centered around set sizes of 3 and 4, D6 centered around set sizes 4 and 5, and D7 centered around set size 5. This suggests that, at optimality, less extreme instances favor variables that correspond to larger item set sizes, because individual items have a relatively smaller impact, as opposed to in the more extreme Bernoulli instances.

Table 3 10-Item instances summary

Fig. 6

10 Items: $w_{U}(\sigma )$ set size frequencies, Bernoulli

Fig. 7

10 Items: $w_{U}(\sigma )$ set size frequencies, non-Bernoulli

Fig. 8

30 Items-distribution variance versus relative remaining gap

Fig. 9

30 Items-fill rate versus relative remaining gap

Fig. 10

30 Items-initial gap versus relative remaining gap

Fig. 11

20 Items-distribution variance versus relative gap closed per loop

1.3 Additional plots—20 and 30-item instances

Figures 8, 9, 10 display various parameters of the 30 item instances against the relative remaining gap, while Figs. 11, 12, 13, 14, 15, 16 display various parameters of the 20 and 30-item instances against the relative gap closed per loop (RGPL), an alternative metric for the general algorithm’s progress.

Fig. 12

20 Items-fill rate versus relative gap closed per loop

Fig. 13

20 Items-initial gap versus relative gap closed per loop

Fig. 14

30 Items-distribution variance versus relative gap closed per loop

Fig. 15

30 Items-fill rate versus relative gap closed per loop

Fig. 16

30 Items-initial gap versus relative gap closed per loop

1.4 Raw data

Tables 4, 5, 6 present the raw data used to calculate the summaries in Tables 1, 2, 3 of the general algorithm’s performance in Sect. 5. They are separated by the number of items in each instance, recording results for 10, 20, and 30-items. For fairness in comparisons, the Avg. Inner Loops Metric for 20 and 30-item instances is only recorded for instances that completed 16 algorithm loops.

Table 4 General algorithm performance data, 10 Items

Table 5 General Algorithm Performance Data, 20 Items

Table 6 General Algorithm Performance Data, 30 Items