multiple job assignment problem

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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.

Let us explore all approaches for this problem.

Solution 1: Brute Force  

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  

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  

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  

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 “intelligent” 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:  

• 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.

Let’s take below example and try to calculate promising cost when Job 2 is assigned to worker A. 

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). 

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. 

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. 

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

Complete Algorithm:  

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.  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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Job Assignment Problem and Traveling Salesman Problem: A Linked Optimisation Problem

• Conference paper
• First Online: 05 December 2022
• Cite this conference paper

• Akinola Ogunsemi 9 ,
• John McCall 9 ,
• Mathias Kern 11 ,
• Benjamin Lacroix 9 ,
• David Corsar 10 &
• Gilbert Owusu 11  

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13652))

Included in the following conference series:

• International Conference on Innovative Techniques and Applications of Artificial Intelligence

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Linked decision-making in service management systems has attracted strong adoption of optimisation algorithms. However, most of these algorithms do not incorporate the complexity associated with interacting decision-making systems. This paper, therefore, investigates the linkages between two classical problems: job assignment problem and travelling salesman problem (JAPTSP) of a service chain system where service personnel perform tasks at different locations. We formulate a novel mathematical model from a linked optimisation perspective with objectives to minimise job cost and total travel distance simultaneously. We present three algorithmic approaches to tackling the JAPTSP: Nondominated Sorting Genetic Algorithm for Linked Problem (NSGALP), Multi-Criteria Ranking Genetic Algorithm for Linked Problem (MCRGALP), and Sequential approach. We evaluate the performance of the three algorithmic approaches on a combination of JAP and TSP benchmark instances. Results show that selecting an appropriate algorithmic approach is highly driven by specific considerations, including multi-objective base performance metrics, computation time, problem correlation and qualitative analysis from a service chain perspective.

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Akinola Ogunsemi, John McCall & Benjamin Lacroix

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Mathias Kern & Gilbert Owusu

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Ogunsemi, A., McCall, J., Kern, M., Lacroix, B., Corsar, D., Owusu, G. (2022). Job Assignment Problem and Traveling Salesman Problem: A Linked Optimisation Problem. In: Bramer, M., Stahl, F. (eds) Artificial Intelligence XXXIX. SGAI-AI 2022. Lecture Notes in Computer Science(), vol 13652. Springer, Cham. https://doi.org/10.1007/978-3-031-21441-7_2

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MLB Fans Call for Angel Hernandez’s Job After Another Truly Horrific Game

• Author: Tim Capurso

In this story:

MLB umpire Angel Hernandez, notorious for his wishy-washy strike zone, was in rare form during Friday night's game between the Texas Rangers and Houston Astros, a 12-8 victory for the visiting Rangers.

Hernandez's performance definitely didn't pass the eye test, as he left the Rangers broadcasters practically gobsmacked after calling three consecutive strikes on pitches that appeared to be outside the zone.

The veteran umpire didn't fare much better on a deeper dive into the numbers behind his performance Friday, as Umpire Scorecards, an account on X (formerly Twitter) that grades the efforts of the game's officials, rated his performance quite harshly.

Umpire: Angel Hernandez Final: Rangers 12, Astros 8 #StraightUpTX // #Relentless #TEXvsHOU // #HOUvsTEX More stats for this game 👇 https://t.co/RJKSQogI6y pic.twitter.com/uRM1ykGr3R — Umpire Scorecards (@UmpScorecards) April 13, 2024

MLB fans proceeded to react to Hernandez's less-than-stellar performance, which included 12 called strikes that Umpire Scorecard regarded as true balls, exactly as one would expect.

WHY IS HE NOT FIRED ALREADY — . (@HalToHell) April 13, 2024

Yet another flawless performance by Angel Hernandez — Sill E. Gewce (@ParahoyitsAJ) April 13, 2024

I’m really not sure why or how he’s still allowed to umpire MLB games. — Texas Rangers Insider (@RangersInsiders) April 13, 2024

78% accuracy — Space City Sports (@LiveSCS) April 13, 2024

Hernandez has been an MLB umpire since 1991. It's safe to say that the majority of baseball fans are ready for his tenure to end.

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