Eight step procedures

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

Introduction

The eight-step procedures are a simplified, multi-stage approach for determining optimal solutions in mathematical optimization. Dynamic programming, developed by Richard Bellman in the 1950s, is used to solve for the maximization or minimization of the objective function by transforming the problem into smaller steps and enumerating all the different possible solutions and finding the optimal solution.

In the eight-step procedure, a problem can be broken down into subproblems to solve. Using the solutions from the subproblems in a recursive manner, the solution can be determined after all the solutions of the subproblems are calculated to find the best solution, which demonstrates the principle of optimality: Any optimal policy has the property that, whatever the current state and current decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the current decision. Such a standard framework is used so that dynamic programming store the values of the subproblems to avoid recomputing, and thus, reduce time to solve the problem.

Theory, Methodology, and/or Algorithmic Discussion

Methodology

To solve a problem using the 8-step procedure, one must use 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. Specifying the stages also divides the problem into smaller pieces.

Step 2: Specify the states for each stage
The states of a problem are defined as the knowledge necessary to make a decision. There are multiple states for each stage. In general, the states consists of the information that is needed to solve the smaller problem within each stage.

Step 3: Specify the allowable actions for each state in each stage
This helps create for a decision that must be made at each stage.

Step 4: Describe the optimization function using an English-language description.

Step 5: Define the boundary conditions
This can help create a starting point to finding a solution to the problem.

Step 6: Define the recurrence relation
This is often denoted with a function, and shows the relationsip between the value of a decision at a particular stage and the value of optimal decision made at the previous stages.

Step 7: Compute the optimal value from the bottom-up
This step can be done manually or by using programming. Note that for each state, an optimal decision made at the remaining stages of the problem is independent from the decisions of the previous states.

Step 8: Arrive at the optimal solution
This is the final step for solving a problem using the eight step procedure.

Numerical Example

Suppose we have a knapsack with a weight capacity of C=5 and N=2 types of items. An item of type n weighs W [n] and generates a benefit of b [n,j] when packing j items of type n to the knapsack however only a[n] units of this item are available.

To solve a Knapsack problem we use the following steps:

Step 1: Specify the stages of the problem

Weight capacity of C=5 and N=2

Step 2: Specify the states for each stage

Item types are stages: n=1,2

Step 3: Specify the allowable actions for each state in each stage

$U_{2}(5)\,=\,0,1,...,min\left\{a[2],\left\lfloor {\frac {5}{w[2]}}\right\rfloor \right\}$= {0,1,2}

Step 4: Describe the optimization function using an English-language description.

$f_{n}^{*}(s)$

Remaining capacity s= 1,2,3,4,5

Step 5: Define the boundary conditions

Boundary Conditions:

$f_{n+1}^{*}(s)=0$, s=0,1,2,3,4,5 C=5

Step 6: Define the recurrence relation

$f_{2}^{*}(5)=max\left\{b[2,j]+f_{3}^{*}(5-j*w[2])\right\}$

Step 7: Compute the optimal value from the bottom-up

$f_{2}^{*}(5)=max\left\{b[2,j]+f_{3}^{*}(5-j*w[2])\right\}$

$f_{n+1}^{*}(s)=0$, s=0,1,2,3,4,5 C=5

Unused Capacity s	$f_{1}^{*}(s)$	Type 1 opt $U_{1}^{*}(s)$	$f_{2}^{*}(s)$	Type 2 opt $U_{2}^{*}(s)$	$f_{3}^{*}(s)$
5	9	0	9	2	0
4	9	0	9	2	0
3	4	0	4	1	0
2	4	0	4	1	0
1	0	0	0	0	0
0	0	0	0	0	0

Step 8: Arrive at the optimal solution

$f_{1}^{*}(5)=26$

Applications

The following are some applications where dynamic programming is used. The criteria for applying dynamic programming to an optimization problem are if the objective function involves maximization, minimization, or counting and if the problem is determined by finding all the solutions to find the optimal solution.

Shortest/ Longest Path Problem

In the shortest path problem, the path with the least amount of cost or value must be determined in a problem with multiple nodes in between the beginning node s to the final node e. Travelling from one node to another incurs a value or cost c(p, q), and the objective is to reach t with the smallest cost possible. The eight-step procedure can be used to determine the possible solutions which the optimal solution can be determined from.

Likewise, but in a maximization function, the longest path can be determined in a problem by determining the solution with the highest cost involved to travel from node s to node e.

Knapsack Problem

The knapsack problem is an example of determining the distribution of effort or when there are limited resources to be shared with competing entities, and the goal is to maximize the benefit of the distribution. Dynamic programming is used when the increase in benefit in regard to increasing the quantity of resources is not linearly proportional. The volume may also be considered in addition to the weight of the resources. A volume constraint is added to the problem and represented in the state by stage n by an ordered pair (s, v) for remaining weight and volume. By considering d constraints, the number of states can grow exponentially with a d -dimensional state space even if the value of d is small. The problem becomes infeasible to solve and is referred to as the curse of dimensionality. However, the curse has faded due to advances in computational power.

Inventory Planning Problem

In inventory management, dynamic programming is used to determine how to meet anticipated and unexpected demand in order to minimize overall costs. Tracking an inventory system involves establishing a set of policies that monitor and control the levels of inventory, determining when a stock must be replenished, and the quantity of parts to order. For example, a production schedule can be computationally solved by knowing the demand, unit production costs, and inventory supply limits in order to keep the production costs below a certain rate.

Conclusion

The eight-step procedure is an approach used in dynamic programming to transform a problem into simpler problems to yield an optimal solution. The recursive nature of the procedure allows for the optimization problems to be solved using computational models that reduce time and effort and can be used in many applications across many industries.