Eight step procedures

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

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} equal to the maximum value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} .

Step 3: Specify the allowable actions for each state in each stage
• This can be defined as:
o Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{n}(s)\, or\, j\, =\, 0,1,...,min\left \{ a[n], \left \lfloor \frac{s}{w[n]} \right \rfloor \right \} }

Step 4: Describe the optimization function using an English-language description.
• In this sentence, we describe the optimization function for each state, or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} , and each stage, or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} . This can also be called Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{*}_{n+1}(s) = 0} for all values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} . Here, we can note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=0,...,C }

Step 6: Define the recurrence relation
• During this step, we make an allowable decision involving Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} items for the remaining capacity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} for items Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} . We can write this statement as:
o Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{*}_{n}(s)= \overset{max}{j=0,1,...,min\left \{ a[n],\left \lfloor \frac{s}{w[n]} \right \rfloor \right \}} \left \{ b[n,j]+ f^{*}_{n+1}(s-j*w[n]) \right \} }

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