Since the late 1940s, linear programming models have been used for many different purposes. Airline companies apply these models to optimize their use of planes and staff. NASA has been using them for many years to optimize their use of limited resources. Oil companies use them to optimize their refinery operations. Small and medium-sized businesses use linear programming to solve a huge variety of problems, often involving resource allocation.
In my study, a typical product-mix problem in a manufacturing system producing two products (each product consists of two sub-assemblies) is solved for its optimal solution through the use of the latest versions of MATLAB having the command simlp, which is very much like linprog. As analysts, we try to find a good enough solution for the decision maker to make a final decision. Our attempt is to give the mathematical description of the product-mix optimization problem and bring the problem into a form ready to call MATLAB’s simlp command. The objective of this paper is to find the best product mix that maximizes profit. The graph obtained using MATLAB commands, give the shaded area enclosed by the constraints called the feasible region, which is the set of points satisfying all the constraints. To find the optimal solution we look at the lines of equal profit to find the corner of the feasible region which yield the highest profit. This corner can be found out at the farthest line of equal profit which still touches the feasible region.
The most critical part is the sensitivity analysis using Excel Solver and Parametric Analysis using computer software which allows us to study the effect on optimal solution due to discrete and continuous change in parameters of the LP model including to identify bottlenecks. We have examined other options like product outsourcing, one-time cost, cross training of one operator, manufacturing of hypothetical third product on under-utilized machines and optimal sequencing of jobs on machines.
Table of Contents
CHAPTER 1 INTRODUCTION
1.1 HISTORY
1.2 PRINCIPLES OF MATHEMATICAL PROGRAMMING
1.3 LINEAR PROGRAMMING
1.3.1 Limitations of LP model
1.4 MOTIVATION
1.4.1 Examples of successful LP applications.
1.5 CHARACTERSTICS OF LINEAR PROGRAMMING
1.6 SOLVING LP PROBLEMS
1.7 BASIC STEPS FOR SOLVING A LP MODEL
1.7.1 Recognize the problem
1.7.2 Define the problem
1.7.3 Define the decision variables
1.7.4 Collect the necessary parametric data
1.7.5 Formulate a model
1.7.6 Solve the model
1.7.7 Verify and validate the model
1.7.8 Analyze model output
1.7.9 Interpret model results
1.7.10 Recommend a course of action
1.8 FORMULATING LP PROBLEMS
1.9 OBJECTIVES OF THE PRESENT WORK
1.10 ORGANISATION OF THE DISSERTATION
1.11 SUMMARY
CHAPTER 2 LITERATURE REVIEW
2.1 INTRODUCTION
2.2 DECISION MAKING IN POM
2.3 THE SIMPLEX METHOD
2.4 THE COMMAND linprog
2.5 USING EXCEL SOLVER OPTIMIZATION PROBLEM
2.5.1 Spreadsheet modeling & Excel Solver
2.6 PRODUCTION OUTSOURCING: A LP MODEL FOR THE TOC
2.7 GENERAL RESOURCE ALLOCATION MODEL
2.8 SUMMARY
CHAPTER 3 LINEAR PROGRAMMING MODEL
3.1 INTRODUCTION
3.2 THE PROBLEM STATEMENT
3.3 FORMULATION OF LP MODEL
3.4 SOLUTION USING MATLAB
3.5 THE COMMAND simlp
3.6 THE OPTIMAL SOLUTION USING MATLAB
3.7 SOLUTION USING EXCEL SOLVER
3.8 OPTIMAL SCHEDULING ON MACHINES
3.8.1 Assumptions in sequencing problem
3.8.2 Processing two jobs through four machines
3.9 SUMMARY
CHAPTER 4 INTERPRETING COMPUTER SOLUTIONS OF LP PROBLEM
4.1 INTRODUCTION
4.2 TERMS
4.2.1 Slack variables
4.2.2 Basic & non-basic variables
4.3 ANSWER REPORT ANALYSIS
4.4 SENSITIVITY ANALYSIS
4.4.1 Find the bottleneck
4.4.2 Find the range over which the unit profit may change
4.4.3 Find the marginal benefit of increasing the time availability
4.4.4 Find the range over which the time availability may change
4.5 PARAMETRIC ANALYSIS
4.6 SUMMARY
CHAPTER 5 RESULT & DISCUSSIONS
5.1 INTRODUCTION
5.2 SEARCH FOR THE OPTIMAL SOLUTION
5.3 BOTTLENECKS
5.4 RANGE OVER WHICH THE UNIT PROFIT MAY CHANGE
5.5 MARGINAL BENEFIT OF INCREASING THE TIME AVAILABILITY
5.6 RANGE OVER WHICH THE TIME AVAILABILITY MAY CHANGE
5.7 REDUCED COST FOR NON-BASIC VARIABLES
5.8 SLACK VALUES FOR CONSTRAINTS
5.9 RECOMMENDED COURSE OF ACTION
5.9.1 Product Outsourcing
5.9.2 One-time cost
5.9.3 Cross Training of one machine operator
5.9.4 Possibility of third product manufacturing
5.9.5 Optimal sequencing to process jobs on machines
5.10 SUMMARY
CHAPTER 6 CONCLUSIONS
6.1 INTRODUCTION
6.2 SUMMARY OF THE PRESENT WORK
6.3 SUMMARY OF CONTRIBUTION
6.4 SCOPE FOR FUTURE WORK
6.5 CONCLUDING REMARKS
Research Objectives and Core Topics
The primary objective of this dissertation is to explore the strategic application of linear programming (LP) for optimal resource allocation in a manufacturing product-mix environment. The research aims to formulate a mathematical model to maximize profit, solve it using MATLAB and Excel Solver, and conduct detailed sensitivity and parametric analyses to understand how dynamic changes in variables and resource constraints impact the optimal solution and organizational outcomes.
- Strategic resource allocation using linear programming models.
- Implementation of optimization solutions via MATLAB and Excel Solver.
- Sensitivity analysis to identify production bottlenecks and resource limits.
- Parametric analysis for continuous and discrete variable variation.
- Optimal job sequencing on multiple machines using Gantt charts.
- Practical decision-making scenarios including outsourcing and multi-product manufacturing.
Excerpt from the Book
1.1 HISTORY
Linear programming was developed as a discipline in the 1940's, motivated initially by the need to solve complex planning problems in wartime operations. Its development accelerated rapidly in the postwar period as many industries found valuable uses for linear programming. The founders of the subject are generally regarded as George B. Dantzig, who devised the simplex method in 1947, and John von Neumann, who established the theory of duality that same year. The Nobel prize in economics was awarded in 1975 to the mathematician Leonid Kantorovich (USSR) and the economist Tjalling Koopmans (USA) for their contributions to the theory of optimal allocation of resources, in which linear programming played a key role.
Many industries use linear programming as a standard tool, e.g. to allocate a finite set of resources in an optimal way. Examples of important application areas include airline crew scheduling, shipping or telecommunication networks, oil refining and blending, and stock and bond portfolio selection.
Linear programming (LP) is one of the most important general methods of operations research. Countless optimization problems can be formulated and solved using LP techniques. Operations research (OR) is a discipline explicitly devoted to aiding decision makers.
Summary of Chapters
CHAPTER 1 INTRODUCTION: Provides an overview of the history, principles, and applications of linear programming as a decision-making tool in industrial operations.
CHAPTER 2 LITERATURE REVIEW: Reviews existing research and methodologies concerning POM, the Simplex method, and the integration of spreadsheet software in operations research.
CHAPTER 3 LINEAR PROGRAMMING MODEL: Details the formulation of a specific product-mix model, its numerical solution, and machine sequencing strategies.
CHAPTER 4 INTERPRETING COMPUTER SOLUTIONS OF LP PROBLEM: Explains how to interpret the output reports from computer solvers, including sensitivity analysis and slack variable usage.
CHAPTER 5 RESULT & DISCUSSIONS: Presents the findings of the optimized production mix, discusses the impacts of resource variations, and recommends practical courses of action.
CHAPTER 6 CONCLUSIONS: Summarizes the contributions of the research and outlines potential directions for future study using fuzzy algorithms.
Key Words
Linear Programming, Resource Allocation, Optimization, Product-Mix, MATLAB, Excel Solver, Sensitivity Analysis, Parametric Analysis, Operations Research, Bottlenecks, Manufacturing, Simplex Method, Decision Making, Production and Operations Management, Gantt Chart
Frequently Asked Questions
What is the fundamental focus of this dissertation?
This work fundamentally focuses on the strategic allocation of limited resources in a manufacturing environment using linear programming to maximize profit.
What are the primary thematic fields addressed?
The key themes include operations research, linear programming formulation, computer-based optimization, sensitivity analysis, and industrial resource scheduling.
What is the primary goal of the research?
The goal is to determine the optimal product-mix and resource allocation that maximizes profitability while identifying critical production bottlenecks.
Which scientific methods are utilized?
The study employs linear programming modeling, the Simplex method, graphical solution techniques, sensitivity analysis, and parametric analysis via MATLAB and Excel Solver.
What topics are covered in the main body?
The main body covers problem formulation, computational solutions, interpretation of solver reports, sensitivity and parametric modeling, and the optimal sequencing of tasks on machines.
Which keywords characterize this work?
Core keywords include Linear Programming, Optimization, MATLAB, Excel Solver, Sensitivity Analysis, and Product-Mix.
How is the sensitivity analysis performed?
Sensitivity analysis is conducted by utilizing Excel Solver reports to assess how discrete and continuous changes in input parameters affect the stability and optimality of the final production plan.
What conclusions are drawn regarding machine bottlenecks?
The study identifies specific machine workstations as bottlenecks when they operate at maximum capacity, demonstrating how LP highlights these limitations to inform management decisions.
- Quote paper
- Dinesh Gupta (Author), 2013, Strategic Allocation of Resources Using Linear Programming Model with Parametric Analysis, Munich, GRIN Verlag, https://www.grin.com/document/271318
