assignment problem in manufacturing

• Linear programming

Solving assignment problem with lpSolve in R

• Linnart Felkl

The assignment problem is a classic problem in linear program . If, for example, you have n  jobs that need to be manufactured during the upcoming shift (in a manufacturing plant) and you have  m machines to produce these tasks, then you want to assign the jobs to machines in an optimal way. In this say you might want to reduce the manufacturing costs incurred, hence you want to find the cost optimal production plan. The constraint, in this example, is that each machine can only fulfill one job during the upcoming shift. All jobs needs to be scheduled, i.e. assigned to a machine.

The problem can be stated in a mathematical model. Below the problem is stated for a case where there are 3 jobs and 3 machines. The cost of producing job 1 on machine 1 USD but 2 USD when producing it on machine 2. Job 2 costs 2 USD on machine 1 and 3 USD on machine 2. Job 3 costs 5 USD on machine 1 and 1 USD on machine 2. Machine 3 can perform job 1 for 2 USD and jobs 2 and 3 for 3 USD respectively.

The mathematical model for this looks as follows:

We can model and solve this problem with the lpSolve package in R, a package for linear programming (for continous and integer problems). The lp.assign function can do the job:

lp.assign is a function specifically meant for solving the assignment problem. The assignment problem by definition is a problem where all decision variables are integer variables. We therefore do not need to specifically tell lp.assign that the decision variables must be considered as integer variables.

Let us see the minimal costs that we must face during the upcoming shift:

Let us see the cost minimal production plan for the upcoming shift:

The transportation problem is another classical problem. You can see me solving it in R – here: Solving Bronson’s transport problem with lpSolve , using lp.transport .

Data scientist focusing on simulation, optimization and modeling in R, SQL, VBA and Python

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Assignment Problem: Meaning, Methods and Variations | Operations Research

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:

Nonlinear Assignment Problems in Manufacturing

Nonlinear Assignment Problems (NAPs) are combinatorial optimization problems for which no exact algorithm exists that can solve them in reasonable computational time. They enjoy applications in diverse areas, such as location theory, data association problems, physics, manufacturing and many others. This talk is divided into two parts. First an overview of NAPs will be presented, where di erent formulations, exact and heuristic solution methods, and other characteristics of NAPs will be discussed.

Related Papers

Gurdal Ertek

This paper discusses the design and application of local search methods to a real-life application at a steel cord manufacturing plant. The case study involves a layout problem that can be represented as a Quadratic Assignment Problem (QAP). Due to the nature of the manufacturing process, certain machinery need to be allocated in close proximity to each other. This issue is incorporated into the objective function through assigning high penalty costs to the unfavorable allocations. QAP belongs to one of the most difficult class of combinatorial optimization problems, and is not solvable to optimality as the number of facilities increases. We implement the well-known local search methods, 2-opt, 3-opt and tabu search. We compare the solution performances of the methods to the results obtained from the NEOS server, which provides free access to many optimization solvers on the internet.

AL-Rafidain Journal of Computer Sciences and Mathematics

Dr. Najla A Al-Saati

Computers & Operations Research

Ilhan Or , Ekrem Duman

Dennis Heffley

International Journal for Research in Applied Science & Engineering Technology (IJRASET)

IJRASET Publication

In this paper a new method is proposed for finding an optimal solution of a wide range of assignment problems, directly. A numerical illustration is established and the optimality of the result yielded by this method is also checked. The most attractive feature of this method is that it requires very simple arithmetical and logical calculations. The method is illustrated through an example.

Roseline Mostafa

Journal of Industrial Engineering International

Hossein Karimi

Mike Fontana

Discrete Applied Mathematics

catherine roucairol

Faicel HNAIEN

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A simulation-optimization based heuristic for the online assignment of multi-skilled workers subjected to fatigue in manufacturing systems, optimization of assignment problems in production lines with different skilled labor levels, a heuristic approach taking operators' fatigue into account for the dynamic assignment of workforce to reduce the mean flowtime, a new mathematical formulation for a potash-mine shift scheduling problem with a simultaneous assignment of machines and workers, multi-objective worker allocation optimisation in a multiple u-line system, serious game based on visual interactive simulation for dynamic workers assignment, workforce reconfiguration strategies in manufacturing systems: a state of the art, analysis of resource allocation problem with different techniques: a comparative study, embedding optimization with deterministic discrete event simulation for assignment of cross-trained operators: an assembly line case study, comparative study of whale optimization algorithm and flower pollination algorithm to solve workers assignment problem, 59 references, worker assignment in cellular manufacturing considering technical and human skills, dynamic cell formation and the worker assignment problem: a new model, heuristic for task-worker assignment with varying learning slopes, learning curve modelling of work assignment in mass customized assembly lines, simulation of the model of workers’ assignment in cellular manufacturing based on the multifunctional workers, recent developments in dual resource constrained (drc) system research.

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On solving the assembly line worker assignment and balancing problem via beam search

Development and evaluation of an assignment heuristic for allocating cross-trained workers, multi-level heterogeneous worker flexibility in a dual resource constrained (drc) job-shop, the impact of walking time on u-shaped assembly line worker allocation problems, related papers.

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An optimization method for task assignment for industrial manufacturing organizations

• Published: 17 May 2017
• Volume 47 , pages 1144–1156, ( 2017 )

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• Yuhong Li 1 ,
• Mengyuan Sun 1 ,
• Haipeng Kong 1 &
• Guanghong Gong 1  

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An industrial manufacturing organization is an aggregation of collaborative units and employees during the process of product development and production. The rapid growing degree of product complexity has resulted in a rising scale of corresponding manufacturing organizations. An effective and optimal schema is essential for assigning human resources to tasks to save costs. This paper proposes an optimization method for this task assignment issue based on a dynamic industrial manufacturing process model and an improved quantum genetic algorithm (QGA) with heuristic information: the heuristic QGA (HQGA). The dynamic process model adopts a hierarchical network to illustrate task composition in a complex industrial manufacturing process and dynamically describes the task completing process on the basis of individual performance and cooperative performance. To reduce the complexity of the fitness evaluation and assignment optimization in the model, the HQGA is presented for three types of optimization objective functions. The HQGA introduces a heuristic principle to accelerate convergence toward an optimal solution, where quantum bitbased encoding design can reflect the degree of participation of different individuals for different tasks. In four case studies, the HQGA successfully completed task assignment based on our dynamic process model and showed better optimization performance compared with conventional QGA and GA.

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Li, N., Li, Y., Sun, M. et al. An optimization method for task assignment for industrial manufacturing organizations. Appl Intell 47 , 1144–1156 (2017). https://doi.org/10.1007/s10489-017-0940-1

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DOI : https://doi.org/10.1007/s10489-017-0940-1

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Optimizing Storage Allocation and Supply for Manufacturing with Time Constraints

I am currently working on a storage location assignment problem (SLAP), which involves optimizing the allocation of various items with different dimensions to different storage areas. My model comprises two phases: warehousing, where there is one storage area with a specific number of racks, and manufacturing, which has its storage area and a specific number of racks as well.

In this problem, I aim to determine both the number of supplies for each SKU item and the allocation of storage spaces for each item. These allocations need to adhere to the capacity constraints of each storage area while maximizing the production of final products. Each final product or order is a combination of several items.

While addressing this problem, I've noticed that there are processes in manufacturing and final products that have sequence and timing considerations (there is a timing/sequence of operations to produce one product). This makes the problem somewhat resemble a flexible flow shop scheduling problem. However, unlike traditional flow shop scheduling, I am not looking to minimize makespan. Instead, I want to focus on the allocation of items to space and the supply of each SKU item based on the entire process of warehousing and manufacturing.

Could you please help me identify the specific type of problem or hybrid problem that my model falls under? Additionally, any insights or guidance on how to approach this complex optimization challenge would be very helpful.

• assignment-problem
• job-shop-scheduling

I guess, as you correctly mentioned, this problem is too complex as actually many of the SCM problems are complex and I doubt one math formulation can handle all of its aspects. Instead, the problem can be decomposed into at least three separated models with some interchanges between them. (please, be aware it is an initial thought and may extend to more than these three models).

• First, a lot-sizing or MRP model to capture the optimum number of orders based on the production needs and supplier capacity.
• Second, a packing and storing model to determine the best package form and the amount of racks needed to store these packages. In both warehouses and also production lines.
• Third, a sequencing model to represent how the products should be allocated to resources based on the production routes. (specifically, in the mentioned hybrid flow shop).

Based on the above three mentioned models, you will then decide how to develop either a math model, or heuristics, or hybrid solutions.

• $\begingroup$ For the first part of the model, could you provide further clarification? The number of orders is inherently stochastic, and we can only estimate a range based on market demand. What exactly do we mean by 'optimum' in this context? As for the second point, the composition of packages (i.e., how many of each item type is included in an order) is a known variable. However, determining the optimal storage allocation within racks and spaces remains a challenge. To formulate this problem effectively, are there any resources or communities where I can seek guidance? $\endgroup$ –  Maryam Commented Sep 24, 2023 at 3:38
• $\begingroup$ I'm reading books, articles, and trying to work on a simple version of the problem myself, but I was wondering if there's a community or experts I can turn to for guidance? $\endgroup$ –  Maryam Commented Sep 24, 2023 at 3:38
• $\begingroup$ @Maryam, 1) for the first part pleaes, search lot-sizing or MRP keywords. 2) I meant by optimum was actually based on solving a math model either in deterministic or stochastic form. 3) If you can determine how many of each item type are included in an order and also you know how you can package them, e.g. in a specific carton and then laying them down in the pallet, you only need to determine the number of racks needed based on the simple calculation without having any math model. $\endgroup$ –  A.Omidi Commented Sep 24, 2023 at 15:44
• $\begingroup$ 4) pleaes, see these links, 1 - 2 - 3 . And also by searching the community you absolutely find many related resources. $\endgroup$ –  A.Omidi Commented Sep 24, 2023 at 15:44

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Computer Science > Data Structures and Algorithms

Title: multi-slot tag assignment problem in billboard advertisement.

Abstract: Nowadays, billboard advertising has emerged as an effective advertising technique due to higher returns on investment. Given a set of selected slots and tags, how to effectively assign the tags to the slots remains an important question. In this paper, we study the problem of assigning tags to the slots such that the number of tags for which influence demand of each zone is satisfied gets maximized. Formally, we call this problem the Multi-Slot Tag Assignment Problem. The input to the problem is a geographical region partitioned into several zones, a set of selected tags and slots, a trajectory, a billboard database, and the influence demand for every tag for each zone. The task here is to find out the assignment of tags to the slots, such the number of tags for which the zonal influence demand is satisfied is maximized. We show that the problem is NP-hard, and we propose an efficient approximation algorithm to solve this problem. A time and space complexity analysis of the proposed methodology has been done. The proposed methodology has been implemented with real-life datasets, and a number of experiments have been carried out to show the effectiveness and efficiency of the proposed approach. The obtained results have been compared with the baseline methods, and we observe that the proposed approach leads to a number of tags whose zonal influence demand is satisfied.

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