A method for finding an optimal solution of mixed integer programming problems with one constraint is proposed. Initially, this method lessens the number of variables and the interval of their change; then, for the resulting problem one derives recurrent relations of dynamic programming that are used for computing. dynamic programming is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. Using a matrix for information storage, we can solve problems of a sufficiently large dimension. The computational experiments demonstrate that the method in question is highly efficient. In this paper shows study about Knapsack problem.

1. International Journal of Current Trends in Engineering & Research
e-ISSN 2455–1392 Volume 1 Issue 2, December 2015 pp. 1-3
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A Survey- Knapsack Problem Using Dynamic Programming
Vijay Tiwari1
, Ashok Gupta2
1,2
Allahabad University
Abstract— A method for finding an optimal solution of mixed integer programming problems with
one constraint is proposed. Initially, this method lessens the number of variables and the interval of
their change; then, for the resulting problem one derives recurrent relations of dynamic programming
that are used for computing. dynamic programming is a method for solving a complex problem by
breaking it down into a collection of simpler subproblems, solving each of those subproblems just
once, and storing their solutions. Using a matrix for information storage, we can solve problems of a
sufficiently large dimension. The computational experiments demonstrate that the method in
question is highly efficient. In this paper shows study about Knapsack problem.
Keyword— Dynamic Programming, Knapsack
I. INTRODUCTION
The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a
set of items, each with a weight and a value, determine the number of each item to include in a
collection so that the total weight is less than or equal to a given limit and the total value is as large
as possible. It derives its name from the problem faced by someone who is constrained by a fixed-
size knapsack and must fill it with the most valuable items.
The problem often arises in resource allocation where there are financial constraints and is studied in
fields such as combinatorics, computer science, complexity theory, cryptography and applied
mathematics.
The knapsack problem has been studied for more than a century, with early works dating as far back
as 1897.It is not known how the name "knapsack problem" originated, though the problem was
referred to as such in the early works of mathematician Tobias Dantzig (1884–1956), suggesting that
the name could have existed in folklore before a mathematical problem had been fully defined
Optimization is one of the mathematical disciplines that aim to find the maximum or minimum value
of a function with certain limitations (constraints). One example is optimization knapsack problem
that defined as follows: If there’s several item, each with their own weight and value, the knapsack
problem is to determine the item that will be incorporated into the knapsack such that the total
weight is less than or equal to the limit and have maximum total value. There are two categories
known in knapsack problem, the fractional knapsack and integer knapsack. In the fractional
knapsack objects can be inserted into the container in fraction, while the integer knapsack items
should be included as a whole. One form of the integer knapsack that is often discussed is the 0/1
knapsack [2], which is mathematically formulated in equation (1).
Maximize
pj xj (1)
Requirement wj xj ≤ M
xj ∊ {0,1}, j=1,..,n.
In this paper, we propose an extension of the 0/1 knapsack problem, called MinMax 0/1 knapsack
problem, which is formulated in equation (2).

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Minimize pj xj (2)
Requirement M1 ≤ wj xj ≤ M2
xj ∊ {0,1}, j=1,..,n
where M1 is the minimum limit of the knapsack and M2 is the maximum limit.
One example of the MinMax 0/1 knapsack problem is the loading of goods into a container which
aims to minimize the use of container space available on the delivery of goods between islands or
countries. In this problem the items are inserted into the container must meet the minimum limit and
not exceed the maximum capacity.
In the 0-1 knapsack problem given some items say n and a knapsack, the aim is to pack the
knapsack to get the maximum total value. Each item has some weight and some value or profit. Total
weight that we can carry is no more than some fixed number W that is the maximum weight
knapsack can carry. So we must consider weights of items as well as their values. The aim is to fill
the knapsack using various items so that the total weight of the items does not exceed the capacity of
the knapsack i.e. W simultaneously maximizing the total profit of the included objects. The problem
is called a 0-1 problem, because each item must be entirely accepted or rejected. Every object has a
weight wi and profit pi. The goal is to maximize the value/profit of the included objects in the
knapsack. The value of xi will be 0 if object is not included else xi will be 1.
Figure 1 Knapsack problem
II. PROPOSED METHOD
This section describes our proposed method of solving MinMax 0/1 knapsack problem using
dynamic programming. To describe the workings of the dynamic programming in finding the
optimal solution of MinMax 0/1 knapsack, we use Example 1 below.
Example 1.
Consider 4 items (x1, x2, x3, x4), each of which has a weight and value (w1, w2, w3, w4) =
(3,4,2,2), (p1, p2, p3, p4) = (12, 14, 7, 6). If minimum capacity M1 = 5 and the maximum capacity
M2 = 6, then some alternative solutions to these problems is shown in Table 1. In the table it can be
seen that under the MinMax 0/1 knapsack problem it is alternative number 5, 6, 7, and 8 that meet
the total weight of 5. Of the four alternatives, the optimal solution is simply an alternative number 6
with a total value of the item for 18 (minimum).

3. International Journal of Current Trends in Engineering & Research (IJCTER)
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Finding optimal solution using Dynamic Programming contains several steps, namely 0:
1. Determine the optimal solution’s structure.
2. Recursively define the optimal solution.
3. Determine the optimal solution in forward or reverse.
4. Construct optimal solution.
III. CONCLUSION
Based on the discussion in this topic, we conclude that:
1. MinMax knapsack can be solved using dynamic programming so that the total value of items
is optimal (in this case minimal) while a minimum limit requirement is met without
exceeding the maximum capacity limit.
2. MinMax knapsack problem can be applied to the problem of loading of goods into the
container so that the total weight is minimum and at the same time the minimum capacity
requirement of container is met without exceeding the maximum capacity of the containers.
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