Question

Apply the top-down (i.e., memory function) dynamic programming algorithm to the following instance of the knapsack problem. Input your results in the table shown below. For empty cells, input a single minus sign (-) into the cell.

item weight value 1 3 $25 2 2 $20 3 1 $15 14 4 $40 5 5 $50 and capacity W = 6.

Warning: When filling in the table below with your answers, be sure to type the number in each cell, with no decimal points or leading zeros or spaces. For example, if a cell should contain a value of 0, just type "0" and not "0.0" or "00" or " 0". Likewise, if a cell should contain a value of 5, just type "5" and not "5.0" or "05" or "05.0." For this problem, if you add a leading zero or a decimal, Canvas will mark answers as incorrect (even if they're numerically equal to the correct answer). Remember, to indicate a cell is empty, type one single minus sign: "-".

Answer 1

Sol)

If take V as a table, i and j as a row and column value then the value of

V[i, j] = max{ v[ i-1, j ] , v[ i-1 , j - w[ i ] + p[ i ] }

here, w[ i ] = weight of an object i

p[ i ] = profit/value of an object i

Initially we dont include any object, no profit will be gained so, we fill the first row with all zeros.

then calculate the value of each cell by using above formula.

Value(p)	weight(w)	i	capacity( j )
0	1	2	3	4	5	6
0	0	0	0	0	0	0	0
25	3	1	0	0	0	25	25	25	25
20	2	2	0	0	20	25	25	45	45
15	1	3	0	15	20	35	40	45	60
40	4	4	0	15	20	35	40	55	60
50	5	5	0	15	20	35	40	55	65

So, the answer table is :

i	capacity( j )
0	1	2	3	4	5	6
0	0	0	0	0	0	0	0
1	0	0	0	25	25	25	25
2	0	0	20	25	25	45	45
3	0	15	20	35	40	45	60
4	0	15	20	35	40	55	60
5	0	15	20	35	40	55	65

If we want to know which items are included in the bag

we start with maximum profit 65, 65 is generated in 5th row itself, so we include item 5

then we substract value of item5(50) from 65.

65 - 50 = 15.

15 is generated in 3rd row so we include item 3.

then we substract value of item3(15) from 15.

we get 0, no item is there, so we stop the process.

so, only item3 and item 5 (00101) will be included in the bag to get maximum profit value.