Applying Dynamic Programming for Job Scheduling Problem Suppose you have one machine and a set of $n$ jobs $a_1, a_2, ..., a_n$ to process on that machine.
Each job $a_j$ has a processing time $t_j$ , a profit $p_j$ and a deadline $d_j$ . The machine can only do
one job at a time, and job $a_j$ must run uninterruptedly for $t_j$ consecutive time units. If
job $a_j$ is completed by its deadline $d_j$ , you receive a profit $p_j$ , but if itβs completed after
its deadline, you receive a profit of $0$ . Give a dynamic programming algorithm to find the
schedule that obtains the maximum amount of profit, assuming that all processing times are
integers between $1$ and $n$ .
Now we provide the solution for the problem, considering that we have the following data:
Job (j)
Processing time ($t_j$ )
Deadline ($d_j$ )
Profit ($p_j$ )
1
4
10
2
2
4
15
5
3
5
5
3
4
6
10
4
$π_1[π_1]= 2 \quad\quad\quad\quad π[1] = 1$ $π_1[π_2]= 5 \quad\quad\quad\quad π[2] = 2$ $π_1[π_3]= 3 \quad\quad\quad\quad π[3] = 3$ $π_1[π_4]= 4 \quad\quad\quad\quad π[4] = 4$
$π_2[π_1,a_2]= max \big[f_1[a_1]+h_2(t_1+t_2), f_1[a_2]+h_1(t_1+t_2)\big]$ $\quad\quad\quad\quad =max\big[2+5,5+2\big]$ $\quad\quad\quad\quad =max\big[7,7\big]$ $\quad\quad\quad\quad =7$ $\quad k[1,2] = 1,2$ $π_2[π_1,a_3]= max \big[f_1[a_1]+h_3(t_1+t_3), f_1[a_3]+h_1(t_1+t_3)\big]$ $\quad\quad\quad\quad =max\big[2+0,3+2\big]$ $\quad\quad\quad\quad =max\big[2,5\big]$ $\quad\quad\quad\quad =5$ $\quad k[1,3] = 1$ $π_2[π_1,a_4]= max \big[f_1[a_1]+h_4(t_1+t_4), f_1[a_4]+h_1(t_1+t_4)\big]$ $\quad\quad\quad\quad =max\big[2+4,4+2\big]$ $\quad\quad\quad\quad =max\big[6,6\big]$ $\quad\quad\quad\quad =6$ $\quad k[1,4] = 1,4$ $π_2[π_2,a_3]= max \big[f_1[a_2]+h_3(t_2+t_3), f_1[a_3]+h_2(t_2+t_3)\big]$ $\quad\quad\quad\quad =max\big[5+0,3+5\big]$ $\quad\quad\quad\quad =max\big[5,8\big]$ $\quad\quad\quad\quad =8$ $\quad k[2,3] = 2$ $π_2[π_2,a_4]= max \big[f_1[a_2]+h_4(t_2+t_4), f_1[a_4]+h_2(t_2+t_4)\big]$ $\quad\quad\quad\quad =max\big[5+4,4+5\big]$ $\quad\quad\quad\quad =max\big[9,9\big]$ $\quad\quad\quad\quad =9$ $\quad k[2,4] = 2,4 $ $π_2[π_3,a_4]= max \big[f_1[a_3]+h_4(t_3+t_4), f_1[a_4]+h_3(t_3+t_4)\big]$ $\quad\quad\quad\quad =max\big[3+0,4+0\big]$ $\quad\quad\quad\quad =max\big[3,4\big]$ $\quad\quad\quad\quad =4$ $\quad k[3,4] = 3 $
$π_3[π_1,a_2,a_3]= max \big[f_2[a_1,a_2]+h_3(t_1+t_2+t_3), f_2[a_1,a_3]+h_2(t_1+t_2+t_3), f_2[a_2,a_3]+h_1(t_1+t_2+t_3)\big]$ $\quad\quad\quad\quad\quad\quad =max\big[7+0,5+5,8+0\big]$ $\quad\quad\quad\quad\quad\quad =max\big[7,10,8\big]$ $\quad\quad\quad\quad\quad\quad =10$ $\quad\quad k[1,2,3] = 2$ $π_3[π_1,a_2,a_4]= max \big[f_2[a_1,a_2]+h_3(t_1+t_2+t_4), f_2[a_1,a_4]+h_2(t_1+t_2+t_4), f_2[a_2,a_4]+h_1(t_1+t_2+t_4)\big]$ $\quad\quad\quad\quad\quad\quad =max\big[7+0,6+5,9+0\big]$ $\quad\quad\quad\quad\quad\quad =max\big[7,11,9\big]$ $\quad\quad\quad\quad\quad\quad =11$ $\quad\quad k[1,2,4] = 2$ $π_3[π_1,a_3,a_4]= max \big[f_2[a_1,a_3]+h_3(t_1+t_3+t_4), f_2[a_1,a_4]+h_3(t_1+t_3+t_4), f_2[a_3,a_4]+h_1(t_1+t_3+t_4)\big]$ $\quad\quad\quad\quad\quad\quad =max\big[5+0,6+0,4+0\big]$ $\quad\quad\quad\quad\quad\quad =max\big[5,6,4\big]$ $\quad\quad\quad\quad\quad\quad =6$ $\quad\quad k[1,3,4] = 3$ $π_3[π_2,a_3,a_4]= max \big[f_2[a_2,a_3]+h_3(t_2+t_3+t_4), f_2[a_2,a_4]+h_3(t_2+t_3+t_4), f_2[a_3,a_4]+h_2(t_2+t_3+t_4)\big]$ $\quad\quad\quad\quad\quad\quad =max\big[8+0,9+0,4+5\big]$ $\quad\quad\quad\quad\quad\quad =max\big[5,9,9\big]$ $\quad\quad\quad\quad\quad\quad =9$ $\quad\quad k[2,3,4] = 2,3$
$π_4[π_1,a_2,a_3,a_4]= max \big[f_3[π_1,a_2,a_3]+h_4(t_1+t_2+t_3+t_4), f_3[π_1,a_2,a_4]+h_3(t_1+t_2+t_3+t_4), f_3[π_1,a_3,a_4]+$ $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad h_2(t_1+t_2+t_3+t_4), f_1[a_2,a_3,a_4]+h_4(t_1+t_2+t_3+t_4)\big]$ $\quad\quad\quad\quad\quad\quad =max\big[10+0,11+0,6+0,9+0\big]$ $\quad\quad\quad\quad\quad\quad =max\big[10,11,6,9\big]$ $\quad\quad\quad\quad\quad\quad =11$ $\quad\quad k[1,2,3,4] = 3$
1 --> 4 --> 2 --> 3
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