Algorithm 1.

Find candidate solutions.

1:	procedure A. Find y minimal candidate solution
2:	for i ∈ I do
3:	$x_i^p\leftarrow 0$
4:	ℐ ← 0
5:	while $\sum _{i\in 1,\dots ℐ}{c}_i{x}_i^p\le b-p$ do
6:	$x_{ℐ}^p\leftarrow 1$
7:	ℐ ← ℐ + 1
8:	$x_{ℐ}^p\leftarrow \frac{b-p-\sum _{i\in 1,\dots ℐ-1}{c}_i{x}_i^p}{c_{ℐ}}$
9:	procedure B. Find y maximal candidate solution
10:	for i ∈ I do
11:	$x_i^P\leftarrow 0$
12:	ℐ ← 0
13:	while $\sum _{i\in 1,\dots ℐ}{c}_i{x}_i^P\le b-P$ do
14:	$x_{ℐ}^P\leftarrow 1$
15:	ℐ ← ℐ + 1
16:	$x_{ℐ}^P\leftarrow \frac{b-P-\sum _{i\in 1,\dots ℐ-1}{c}_i{x}_i^P}{c_{ℐ}}$
17:	procedure C. Find whole number candidate solutions
18:	for i″ ∈ I do
19:	for i′ ∈ I do
20:	if i′ ≤ i″ then
21:	$x_{i\prime }^w(i″)\leftarrow 1$
22:	else
23:	$x_{i\prime }^w(i″)\leftarrow 0$
	$y^w(i″)=b-\sum _{i\in \{1,‥,i″\}}{c}_i$
24:	procedure D. Find fractional candidate solutions
25:	for i″ ∈ I do
26:	for i′ ∈ I do
27:	if i′ ≤ i″ − 1 then
28:	$x_{i\prime }^f(i″)\leftarrow 1$
29:	if i′ ≥ i″ + 1 then
30:	$x_{i\prime }^f(i″)\leftarrow 0$
31:	$y^f(i″)\leftarrow \gamma [\frac{c_{i″}}{{\upsilon }_{i″}}\sum _{i\in \{1,‥,i″-1\}}{\upsilon }_i+b-\sum _{i\in \{1,‥,i″-1\}}{c}_i]/(\gamma +1)$
32:	$x_{i\prime }^f(i″)\leftarrow [b-\sum _{i\in \{1,‥,i″-1\}}{c}_i-y^f(i″)]/c_{i″}$