Difference between revisions of "Eight step procedures"

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'''Step 1: Specify the stages of the problem''' <br />
 
'''Step 1: Specify the stages of the problem''' <br />
 
• The stages of a dynamic programming problem can be defined as points where decisions are made. These are often denoted with the variable n. <br />
 
• The stages of a dynamic programming problem can be defined as points where decisions are made. These are often denoted with the variable n. <br />
Step 2: Specify the states for each stage <br />
+
<br />
 +
'''Step 2: Specify the states for each stage''' <br />
 
• The states of a problem are defined as the knowledge necessary to make a decision, or s. We set C equal to the maximum value of s. <br />
 
• The states of a problem are defined as the knowledge necessary to make a decision, or s. We set C equal to the maximum value of s. <br />
Step 3: Specify the allowable actions for each state in each stage <br />
+
<br />
 +
'''Step 3: Specify the allowable actions for each state in each stage''' <br />
 
• This can be defined as: <br />
 
• This can be defined as: <br />
 
o Un(s) or j = 0,1,…,min{a[n], floor(s/w[n])} <br />
 
o Un(s) or j = 0,1,…,min{a[n], floor(s/w[n])} <br />
Step 4: Describe the optimization function using an English-language description. <br />
+
<br />
 +
'''Step 4: Describe the optimization function using an English-language description.''' <br />
 
• In this sentence, we describe the optimization function for each state, or s, and each stage, or n. This can also be called F*n(s) <br />
 
• In this sentence, we describe the optimization function for each state, or s, and each stage, or n. This can also be called F*n(s) <br />
Step 5: Define the boundary conditions <br />
+
<br />
 +
'''Step 5: Define the boundary conditions''' <br />
 
• This helps create a starting point to finding a solution to the problem. First, we set f*n+1(s) = 0 for all values of s. Here, we can note that s = 0,…,C <br />
 
• This helps create a starting point to finding a solution to the problem. First, we set f*n+1(s) = 0 for all values of s. Here, we can note that s = 0,…,C <br />
Step 6: Define the recurrence relation <br />
+
<br />
 +
'''Step 6: Define the recurrence relation''' <br />
 
• During this step, we make an allowable decision involving j items for the remaining capacity s for items n. We can write this statement as: <br />
 
• During this step, we make an allowable decision involving j items for the remaining capacity s for items n. We can write this statement as: <br />
 
o f*n(s) = max j=0,1,…,min{a[n], floor(s/w[n]0} {b[n,j]+f*n+a(s-j*w[n])} <br />
 
o f*n(s) = max j=0,1,…,min{a[n], floor(s/w[n]0} {b[n,j]+f*n+a(s-j*w[n])} <br />
Step 7: Compute the optimal value from the bottom-up <br />
+
<br />
 +
'''Step 7: Compute the optimal value from the bottom-up''' <br />
 
• In this step, a table is made <br />
 
• In this step, a table is made <br />
  

Revision as of 17:35, 20 November 2020

Author: Eljona Pushaj, Diana Bogdanowich, Stephanie Keomany
Steward: Fengqi You

Introduction

Theory, Methodology, and/or Algorithmic Discussion

Definition

To solve a problem using the 8-step procedure, one must follow the following steps:

Step 1: Specify the stages of the problem
• The stages of a dynamic programming problem can be defined as points where decisions are made. These are often denoted with the variable n.

Step 2: Specify the states for each stage
• The states of a problem are defined as the knowledge necessary to make a decision, or s. We set C equal to the maximum value of s.

Step 3: Specify the allowable actions for each state in each stage
• This can be defined as:
o Un(s) or j = 0,1,…,min{a[n], floor(s/w[n])}

Step 4: Describe the optimization function using an English-language description.
• In this sentence, we describe the optimization function for each state, or s, and each stage, or n. This can also be called F*n(s)

Step 5: Define the boundary conditions
• This helps create a starting point to finding a solution to the problem. First, we set f*n+1(s) = 0 for all values of s. Here, we can note that s = 0,…,C

Step 6: Define the recurrence relation
• During this step, we make an allowable decision involving j items for the remaining capacity s for items n. We can write this statement as:
o f*n(s) = max j=0,1,…,min{a[n], floor(s/w[n]0} {b[n,j]+f*n+a(s-j*w[n])}

Step 7: Compute the optimal value from the bottom-up
• In this step, a table is made

Numerical Example

Applications

Conclusion

References