How can calculus be used to optimize manufacturing processes

Numerous companies in industry require Operations Research professionals to apply mathematical techniques to a wide range of challenging questions.

Operations Research can be defined as the science of decision-making. It has been successful in providing a systematic and scientific approach to all kinds of government, military, manufacturing, and service operations.

Operations Research is a splendid area for graduates of mathematics to use their knowledge and skills in creative ways to solve complex problems and have an impact on critical decisions. The term? Operations Research? Operational Research? Other terms used are?

Management Science,? Industrial Engineering,? Decision Sciences.? The multiplicity of names comes primarily from the different academic departments that have hosted courses in this field. The subject is frequently referred to simply as? Problem Formulation motivation, short- and long-term objectives, decision variables, control parameters, constraints ; Mathematical Modeling representation of complex systems by analytical or numerical models, relationships between variables, performance metrics ; Data Collection model inputs, system observations, validation, tracking of performance metrics ; Solution Methods optimization, stochastic processes, simulation, heuristics, and other mathematical techniques ; Validation and Analysis model testing, calibration, sensitivity analysis, model robustness ; and Interpretation and Implementation solution ranges, trade-offs, visual or graphical representation of results, decision support systems.

These steps all require a solid background in mathematics and familiarity with other disciplines such as physics, economics, and engineering , as well as clear thinking and intuition. The mathematical sciences prepare students to apply tools and techniques and use a logical process to analyze and solve problems. OR became an established discipline during World War II, when the British government recruited scientists to solve problems in critical military operations.

Mathematical methods were developed to determine the most effective use of radar and other new defense technologies at the time. OR groups were later formed in the U. There are now many OR departments in industry, government, and academia throughout the world. Examples of where OR has been successful in recent years are the following:. Airline Industry routing and flight plans, crew scheduling, revenue management ; Telecommunications network routing, queue control ; Manufacturing Industry system throughput and bottleneck analysis, inventory control, production scheduling, capacity planning ; Healthcare hospital management, facility design ; and Transportation traffic control, logistics, network flow, airport terminal layout, location planning.

There are many mathematical techniques that were developed specifically for OR applications. These techniques arose from basic mathematical ideas and became major areas of expertise for industrial operations. One important area of such techniques is optimization. Many problems in industry require finding the maximum or minimum of an objective function of a set of decision variables, subject to a set of constraints on those variables. Typical objectives are maximum profit, minimum cost, or minimum delay.

Frequently there are many decision variables and the solution is not obvious. Techniques of mathematical programming for optimization include linear programming optimization where both the objective function and constraints depend linearly on the decision variables , non-linear programming non-linear objective function or constraints , integer programming decision variables restricted to integer solutions , stochastic programming uncertainty in model parameter values and dynamic programming stage-wise, nested, and periodic decision-making.

Another area is the analysis of stochastic processes i. Many real-world problems involve uncertainty, and mathematics has been extremely useful in identifying ways to manage it. Modeling uncertainty is important in risk analysis for complex systems, such as space shuttle flights, large dam operations, or nuclear power generation. In fact, the problem we see here today is a simplified version of a problem I covered in a DETC conference paper that I published a few years back.

They have an exclusive deal with Gallmart to supply the retail giant with 10, units over the next several years. The hot tub shells are made using injection-molding, in which molten plastic is squirted into metal molds at high pressure, and then allowed to cool. Once the shell has cooled, assembly workers finish the product by attaching the hoses and motors and installing insulation. Being a small company, Hot Bod doesn't have their own factory - they will have to rent space from the Berry Plastics Corporation.

Berry Plastics also requires that you specify how much storage space you will need in your contract - you have to pay for the space up front even if you're not using it the entire time. Assembly is a slow and delicate process, so Hot Bod workers can only finish about 10 tubs per day. The injection-molding machine on the other hand is very fast, but there is a catch - each time they start up the machine to make a fresh batch of hot tub shells, they have to carefully heat up and calibrate the machine, and get approved by the plant safety manager.

So, there is a tradeoff: if they run many small batches, they will incur a lot of start-up costs, but if they run fewer, larger batches, they will have to pay for more space to store the shells while they're waiting to be assembled. If Hot Bod wants to minimize their total production cost while meeting their quota of 10, units, how many tub shells should they mold at a time? How many batches in total will they run?

What will be their total production cost? Wow, that's a lot of information. As usual, you should first read the problem to get a general picture of the scenario I'll wait.

Once you've read it through once, we can focus on the sentences at the end of the problem and see what they're actually asking for:. In many cases, there are two or more variables in the problem. In the garden store example again, the length and width of the enclosure are both unknown. If there is an equation that relates the variables we can solve for one of them in terms of the others, and write the objective function as a function of just one variable.

Equations that relate the variables in this way are called constraint equations. The constraint equations are always equations, so they will have equals signs. For the garden store, the fixed area relates the length and width of the enclosure.

This will give us our constraint equation. Here is a link to the examples used in the videos in this section: Applied Optimization Examples. The manager of a garden store wants to build a square foot rectangular enclosure on the store's parking lot in order to display some equipment. The objective function is the cost function, and we want to minimize it. As it stands, though, it has two variables, so we need to use the constraint equation.

Since this is the only critical point in the domain, this must be the global minimum. The dimensions of the enclosure that minimize the cost are 20 feet by 30 feet. When trying to maximize their revenue, businesses also face the constraint of consumer demand. While a business would love to see lots of products at a very high price, typically demand decreases as the price of goods increases. In simple cases, we can construct that demand curve to allow us to maximize revenue.

What price should she sell the tickets at to maximize her revenue? The problem provides information about the demand relationship between price and quantity — as price increases, demand decreases. We need to find a formula for this relationship. To investigate, let's calculate what will happen to attendance if we raise the price:.