• Branch and Bound Tutorial
  • Backtracking Vs Branch-N-Bound
  • 0/1 Knapsack
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Job Assignment Problem using Branch And Bound

  • N Queen Problem using Branch And Bound
  • Traveling Salesman Problem using Branch And Bound

Let there be N workers and N jobs. Any worker can be assigned to perform any job, incurring some cost that may vary depending on the work-job assignment. It is required to perform all jobs by assigning exactly one worker to each job and exactly one job to each agent in such a way that the total cost of the assignment is minimized.

jobassignment

Let us explore all approaches for this problem.

Solution 1: Brute Force ย 

We generate n! possible job assignments and for each such assignment, we compute its total cost and return the less expensive assignment. Since the solution is a permutation of the n jobs, its complexity is O(n!).

Solution 2: Hungarian Algorithm ย 

The optimal assignment can be found using the Hungarian algorithm. The Hungarian algorithm has worst case run-time complexity of O(n^3).

Solution 3: DFS/BFS on state space tree ย 

A state space tree is a N-ary tree with property that any path from root to leaf node holds one of many solutions to given problem. We can perform depth-first search on state space tree and but successive moves can take us away from the goal rather than bringing closer. The search of state space tree follows leftmost path from the root regardless of initial state. An answer node may never be found in this approach. We can also perform a Breadth-first search on state space tree. But no matter what the initial state is, the algorithm attempts the same sequence of moves like DFS.

Solution 4: Finding Optimal Solution using Branch and Bound ย 

The selection rule for the next node in BFS and DFS is โ€œblindโ€. i.e. the selection rule does not give any preference to a node that has a very good chance of getting the search to an answer node quickly. The search for an optimal solution can often be speeded by using an โ€œintelligentโ€ ranking function, also called an approximate cost function to avoid searching in sub-trees that do not contain an optimal solution. It is similar to BFS-like search but with one major optimization. Instead of following FIFO order, we choose a live node with least cost. We may not get optimal solution by following node with least promising cost, but it will provide very good chance of getting the search to an answer node quickly.

There are two approaches to calculate the cost function:ย ย 

  • For each worker, we choose job with minimum cost from list of unassigned jobs (take minimum entry from each row).
  • For each job, we choose a worker with lowest cost for that job from list of unassigned workers (take minimum entry from each column).

In this article, the first approach is followed.

Letโ€™s take below example and try to calculate promising cost when Job 2 is assigned to worker A.ย 

jobassignment2

Since Job 2 is assigned to worker A (marked in green), cost becomes 2 and Job 2 and worker A becomes unavailable (marked in red).ย 

jobassignment3

Now we assign job 3 to worker B as it has minimum cost from list of unassigned jobs. Cost becomes 2 + 3 = 5 and Job 3 and worker B also becomes unavailable.ย 

jobassignment4

Finally, job 1 gets assigned to worker C as it has minimum cost among unassigned jobs and job 4 gets assigned to worker D as it is only Job left. Total cost becomes 2 + 3 + 5 + 4 = 14.ย 

jobassignment5

Below diagram shows complete search space diagram showing optimal solution path in green.ย 

jobassignment6

Complete Algorithm: ย 

Below is the implementation of the above approach:

Time Complexity: O(M*N). This is because the algorithm uses a double for loop to iterate through the M x N matrix.ย  Auxiliary Space: O(M+N). This is because it uses two arrays of size M and N to track the applicants and jobs.

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Assignment Problem: Meaning, Methods and Variations | Operations Research

assignment problem daa

After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations.

Meaning of Assignment Problem:

An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total cost or maximize total profit of allocation.

The problem of assignment arises because available resources such as men, machines etc. have varying degrees of efficiency for performing different activities, therefore, cost, profit or loss of performing the different activities is different.

Thus, the problem is “How should the assignments be made so as to optimize the given objective”. Some of the problem where the assignment technique may be useful are assignment of workers to machines, salesman to different sales areas.

Definition of Assignment Problem:

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Suppose there are n jobs to be performed and n persons are available for doing these jobs. Assume that each person can do each job at a term, though with varying degree of efficiency, let c ij be the cost if the i-th person is assigned to the j-th job. The problem is to find an assignment (which job should be assigned to which person one on-one basis) So that the total cost of performing all jobs is minimum, problem of this kind are known as assignment problem.

The assignment problem can be stated in the form of n x n cost matrix C real members as given in the following table:

assignment problem daa

Inner Product

{\displaystyle A,B}

Note: The true objective cost function only requires summing entries above the diagonal in the matrix comprised of elements

{\displaystyle F_{i,j}(X_{\phi }DX_{\phi }^{T})_{i,j}}

Since this matrix is symmetric with zeroes on the diagonal, dividing by 2 removes the double count of each element to give the correct cost value. See the Numerical Example section for an example of this note.

Optimization Problem

With all of this information, the QAP can be summarized as:

{\displaystyle \min _{X\in P}\langle F,XDX^{T}\rangle }

Computational Complexity

QAP belongs to the classification of problems known as NP-complete, thus being a computationally complex problem. QAPโ€™s NP-completeness was proven by Sahni and Gonzalez in 1976, who states that of all combinatorial optimization problems, QAP is the โ€œhardest of the hardโ€. [2]

Algorithmic Discussions

While an algorithm that can solve QAP in polynomial time is unlikely to exist, there are three primary methods for acquiring the optimal solution to a QAP problem:

  • Dynamic Program
  • Cutting Plane

Branch and Bound Procedures

The third method has been proven to be the most effective in solving QAP, although when n > 15, QAP begins to become virtually unsolvable.

The Branch and Bound method was first proposed by Ailsa Land and Alison Doig in 1960 and is the most commonly used tool for solving NP-hard optimization problems.

A branch-and-bound algorithm consists of a systematic enumeration of candidate solutions by means of state space search: the set of candidate solutions is thought of as forming a rooted tree with the full set at the root. The algorithm explores branches of this tree, which represent subsets of the solution set. Before one lists all of the candidate solutions of a branch, the branch is checked against upper and lower estimated bounds on the optimal solution, and the branch is eliminated if it cannot produce a better solution than the best one found so far by the algorithm.

Linearizations

The first attempts to solve the QAP eliminated the quadratic term in the objective function of

{\displaystyle min\sum _{i=1}^{n}\sum _{j=1}^{n}c{_{\phi (i)\phi (j)}}+\sum _{i=1}^{n}b{_{\phi (i)}}}

in order to transform the problem into a (mixed) 0-1 linear program. The objective function is usually linearized by introducing new variables and new linear (and binary) constraints. Then existing methods for (mixed) linear integer programming (MILP) can be applied. The very large number of new variables and constraints, however, usually poses an obstacle for efficiently solving the resulting linear integer programs. MILP formulations provide LP relaxations of the problem which can be used to compute lower bounds.

Numerical Example

Qap with 3 facilities.

{\displaystyle D={\begin{bmatrix}0&5&6\\5&0&3.6\\6&3.6&0\end{bmatrix}}}

Applications

Inter-plant transportation problem.

The QAP was first introduced by Koopmans and Beckmann to address how economic decisions could be made to optimize the transportation costs of goods between both manufacturing plants and locations. [1] Factoring in the location of each of the manufacturing plants as well as the volume of goods between locations to maximize revenue is what distinguishes this from other linear programming assignment problems like the Knapsack Problem.

The Backboard Wiring Problem

As the QAP is focused on minimizing the cost of traveling from one location to another, it is an ideal approach to determining the placement of components in many modern electronics. Leon Steinberg proposed a QAP solution to optimize the layout of elements on a blackboard by minimizing the total amount of wiring required. [4]

When defining the problem Steinberg states that we have a set of n elements

{\displaystyle E=\left\{E_{1},E_{2},...,E_{n}\right\}}

as well as a set of r points

{\displaystyle P_{1},P_{2},...,P_{r}}

In his paper he derives the below formula:

{\displaystyle min\sum _{1\leq i\leq j\leq n}^{}C_{ij}(d_{s(i)s(j))})}

In his paper Steinberg a backboard with a 9 by 4 array, allowing for 36 potential positions for the 34 components that needed to be placed on the backboard. For the calculation, he selected a random initial placement of s1 and chose a random family of 25 unconnected sets.

The initial placement of components is shown below:

assignment problem daa

After the initial placement of elements, it took an additional 35 iterations to get us to our final optimized backboard layout. Leading to a total of 59 iterations and a final wire length of 4,969.440.

assignment problem daa

Hospital Layout

Building new hospitals was a common event in 1977 when Alealid N Elshafei wrote his paper on "Hospital Layouts as a Quadratic Assignment Problem". [5] With the high initial cost to construct the hospital and to staff it, it is important to ensure that it is operating as efficiently as possible. Elshafei's paper was commissioned to create an optimization formula to locate clinics within a building in such a way that minimizes the total distance that a patient travels within the hospital throughout the year. When doing a study of a major hospital in Cairo he determined that the Outpatient ward was acting as a bottleneck in the hospital and focused his efforts on optimizing the 17 departments there.

Elshafei identified the following QAP to determine where clinics should be placed:

{\displaystyle min\sum _{i,j}\sum _{k,q}f_{ik}d_{jq}y_{ij}y_{kq}}

For the Cairo hospital with 17 clinics, and one receiving and recording room bringing us to a total of 18 facilities. By running the above optimization Elshafei was able to get the total distance per year down to 11,281,887 from a distance of 13,973,298 based on the original hospital layout.

Exam Scheduling System

The scheduling system uses matrices for Exams, Time Slots, and Rooms with the goal of reducing the rate of schedule conflicts. To accomplish this goal, the โ€œexamination with the highest cross faculty student is been prioritized in the schedule after which the examination with the highest number of cross-program is considered and finally with the highest number of repeating student, at each stage group with the highest number of student are prioritized.โ€ [6]

{\displaystyle n!}

  • โ†‘ 1.0 1.1 1.2 Koopmans, T., & Beckmann, M. (1957). Assignment Problems and the Location of Economic Activities. Econometrica, 25(1), 53-76. doi:10.2307/1907742
  • โ†‘ 2.0 2.1 Quadratic Assignment Problem. (2020). Retrieved December 14, 2020, from https://neos-guide.org/content/quadratic-assignment-problem
  • โ†‘ 3.0 3.1 3.2 Burkard, R. E., ร‡ela, E., Pardalos, P. M., & Pitsoulis, L. S. (2013). The Quadratic Assignment Problem. https://www.opt.math.tugraz.at/~cela/papers/qap_bericht.pdf .
  • โ†‘ 4.0 4.1 Leon Steinberg. The Backboard Wiring Problem: A Placement Algorithm. SIAM Review . 1961;3(1):37.
  • โ†‘ 5.0 5.1 Alwalid N. Elshafei. Hospital Layout as a Quadratic Assignment Problem. Operational Research Quarterly (1970-1977) . 1977;28(1):167. doi:10.2307/300878
  • โ†‘ 6.0 6.1 Muktar, D., & Ahmad, Z.M. (2014). Examination Scheduling System Based On Quadratic Assignment.

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Knapsack Problem

When given a set of items, where each item has a weight and a value, we need to determine a subset of items that are to be included in a collection in such a way that the total weight aggregates up to be lower than or equal to a given limit and the total value could be as big as possible.ย 

The Knapsack problem is an instance of a Combinatorial Optimization problem. One general approach to crack difficult problems is to identify the most restrictive constraint. For this, we must ignore the others and solve a knapsack problem, and finally, we must somehow fit the solution to satisfy the constraints that are ignored.ย 

Applications

For multiple cases of resource allocation problems that have some specific constraints, the problem can be solved in a way that is similar to the Knapsack problem. Following are a set of examples.ย 

  • ย Finding the least wasteful way to cut down the basic materialsย 
  • ย portfolio optimizationย 
  • ย Cutting stock problemsย 

Problem Scenario

Consider a problem scenario where a thief is robbing a store and his knapsack ( bag) can carry a maximal weight of W. Consider that there are n items in the store and the weight of the ith item is wi and its respective profit is pi.ย 

What are all the items the thief should take? ย 

Here, the main goal/objective of the thief is to maximize the profit anyhow. So, the items should opt-in such a way that the items which are carried by the thief will fetch the maximum profit.ย 

Based on the nature of the items, Knapsack problems are classified into two categoriesย 

  • Fractional Knapsackย 

Fractional Knapsack

In this category, items can be broken into smaller pieces, and the thief can select fractions of items.

According to the problem scenario,

  • There are n items in the store
  • Weight of ith itemย 
  • Profit for ith itemย 
  • pi>0 and
  • The capacity of the Knapsack is W

As the name suggests, in the Knapsack problem, items can be broken into smaller fragments. So, the thief might only take a fraction or a part of xi of ith item.

The ith item in the store contributes a weight of xi.wi to the total weight in the knapsack(bag) and profit xi.pi to the Total Profit.

Hence, the main objective of the algorithm is basically to maximize the value of โˆ‘n=1n(xi.pi) with respect to the given constraint,

We already know that a solution that is said to be an optimal solution must fill the knapsack(bag) exactly, if not, we could at least add a smaller fraction of one of the remaining items. This will result in an increase in the overall profit.

Thus, an optimal solution to this problem can be obtained by,

Now, we have to sort all those items based on their values of piwi, so that

Here,ย  x is an array that is used to store the fraction of items.

Suppose that we are provided with items that have already been sorted in the decreasing order of piwi, then the time taken by the โ€œwhileโ€ will be O(n) . So, the total time including that includes even sorting will be O(n logn) .

Let us consider that the capacity of the knapsack(bag) W = 60 and the list of items are shown in the following table โˆ’

We can see that the provided items are not sorted based on the value of piwi, we perform sorting. After sorting, the items are shown in the following table.

Once we sort all the items according to the piwi, we choose all of B as the weight of B is less compared to that of the capacity of the knapsack. Further, we choose item A , as the available capacity of the knapsack is greater than the weight of A . Now, we will choose C as the next item. Anyhow, the whole item cannot be chosen as the remaining capacity of the knapsack is less than the weight of the chosen item – C .

Hence, a fraction of C (i.e. (60 โˆ’ 50)/20) is chosen.

Now, we reach the stage where the capacity of the Knapsack is equal to the chosen items. Hence, no more items can be selected.

The total weight of the chosen items is 40 + 10 + 20 * (10/20) = 60

And the total profit is 101 + 281 + 121 * (10/20) = 380 + 60 = 440

This is the optimal solution. We cannot gain more profit compared to this by selecting any different combination of items out of the provided items.

Algorithm :ย 

IMAGES

  1. Assignment Problem by Branch and Bound Method

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  2. Assignment Problem using Branch and Bound- DAA

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  3. Quadratic Assignment Problem Example

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COMMENTS

  1. Job Assignment Problem using Branch And Bound

    Solution 4: Finding Optimal Solution using Branch and Bound. The selection rule for the next node in BFS and DFS is "blind". i.e. the selection rule does not give any preference to a node that has a very good chance of getting the search to an answer node quickly. The search for an optimal solution can often be speeded by using an ...

  2. Quadratic Assignment Problem Example

    #quadraticassignmentproblem #quadratic #assignmentproblem #qapComplete Playlist of Analysis Of Algorithms (DAA):๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡ https://www.youtub...

  3. Assignment Problem: Meaning, Methods and Variations

    After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations. Meaning of Assignment Problem: An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total ...

  4. Assignment Problem by Branch and Bound Method

    Video 28 of series of analysis of algorithms #JAP#assignmentproblem#ersahilkagyan Complete Playlist of Analysis Of Algorithms (DAA):๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡๐Ÿ‘‡?...

  5. Assignment problem

    The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment.

  6. Assignment Problem using Branch and Bound- DAA

    Note: at 8:00 minimum is 5 and for c=4, the cost is 8+6+2+3=19

  7. Branch and Bound Algorithm

    Branch and bound algorithms are used to find the optimal solution for combinatory, discrete, and general mathematical optimization problems. In general, given an NP-Hard problem, a branch and bound algorithm explores the entire search space of possible solutions and provides an optimal solution. A branch and bound algorithm consist of stepwise ...

  8. PDF DAA Assignment IV

    The assignment problem can be stated as follows: There are n people who need to be assigned to execute n jobs, one person per job. (That is, each person is assigned to exactly one job and each job is assigned to exactly one person.) The cost that would accrue if the i-th person is assigned to the j-th job is a known quantity C[i, j] for each ...

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    The Assignment Problem is a special type of Linear Programming Problem based on the following assumptions: However, solving this task for increasing number of jobs and/or resources calls forโ€ฆ

  10. PDF 03

    Problem.Compute an (a > 0, n is a non-negative numbers). Brute-force approach: an = a a an times We multiply 1 with a n times. Algorithm 5 Computing an 1: procedure Power(a;n) 2: result 1 3: for i = 1 to n do 4: result result * a 5: end for 6: return result 7: end procedure 9/57 Brute Force Algorithm

  11. PDF 01

    Example of classical algorithmic problems 5. Assignment problem Given n sta s and n tasks. Anyone can be assigned to perform any task, incurring some cost c(i;j) that may vary depending on the sta (s i)-task(t j) assignment. It is required to perform as many tasks as possible by assigning one sta to each task, in such a way that the total cost P

  12. Quadratic assignment problem

    The Quadratic Assignment Problem (QAP), discovered by Koopmans and Beckmann in 1957, is a mathematical optimization module created to describe the location of invisible economic activities. An NP-Complete problem, this model can be applied to many other optimization problems outside of the field of economics. It has been used to optimize ...

  13. Assignment problem

    Hi there,I hope you liked this video. Please hit like, share and subscribe. It will motivate me to do more of these. Thanks!

  14. DAA Algorithm

    DAA Algorithm with daa tutorial, introduction, Algorithm, Asymptotic Analysis, Control Structure, Recurrence, Master Method, Recursion Tree Method, Sorting Algorithm, Bubble Sort, Selection Sort, Insertion Sort, Binary Search, Merge Sort, Counting Sort, etc. ... Quadratic Assignment Problem: The Quadratic Assignment Problem involves allocating ...

  15. Branch and bound

    Branch and bound is one of the techniques used for problem solving. It is similar to the backtracking since it also uses the state space tree. It is used for solving the optimization problems and minimization problems. If we have given a maximization problem then we can convert it using the Branch and bound technique by simply converting the ...

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    Network Flow Problems. The most obvious flow network problem is the following: Problem1: Given a flow network G = (V, E), the maximum flow problem is to find a flow with maximum value. Problem 2: The multiple source and sink maximum flow problem is similar to the maximum flow problem, except there is a set {s 1,s 2,s 3.....s n} of sources and a set {t 1,t 2,t 3.....t n} of sinks.

  17. DAA- Knapsack Problem

    We already know that a solution that is said to be an optimal solution must fill the knapsack (bag) exactly, if not, we could at least add a smaller fraction of one of the remaining items. This will result in an increase in the overall profit. Thus, an optimal solution to this problem can be obtained by, โˆ‘n=1n(xi.wi)=W.

  18. Assignment problem using branch and bound

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  19. DAA Assignment 1

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