19 Optimization Analysis

Ms.Vinodini Kapoor

1. Learning Outcome:

After completing this module the students will be able to:

• Understand the basic concept of optimization.
• Understand the various components that form the optimization function.
• List the various complicating factors in optimization.
• Understand the importance of optimization.
• Analyze the optimization modeling process.
• Discuss classification of the optimization model.

2. Introduction

Fundamentally, the word optimization refers to the most effective use of resources. Resources at the organizational level may refer to the 5 M’s namely men, material, machinery, money and minutes (time). The ultimate aim for every organization is to utilize these resources diligently and yield best results. Hence, the concept of optimization is linked to achieving result with the most efficient use of resources while keeping in mind the limiting conditions or constraints attached. This implies the boundary to which the use of each resource is confined. It removes redundant processes and focuses on relevant ones. Thereby maximizing the potential of each activity. To simplify further, maximization implies to attain the highest or maximum result but with regard to its underlying cost or expense. The practice of optimization may also focus on minimizing the cost structure and overheads by restricting the amount of resources to what is necessary.

In the area of information systems, the decisions support systems (DSS) play a paramount role. They support activities that facilitate decision making. These are very useful in analytical modeling where mathematical equations help to set limits to a system. The DSS is able to analyze complex business data and find solutions to problems with ease.

The four basic types of analytical modeling supported by DSS are what-if analysis, goal seeking analysis, sensitivity analysis and optimization analysis (refer to exhibit 3).

Since optimization analysis is a technique used to find ‘optimum value’ for a target variable under given circumstances, the target or final outcome is not fixed. Rather, the target needs to be arrived at after taking into consideration the constraints involved in achieving the result. In this analysis, one or more variables are changed after taking the constraints into account until the best alternative or the optimal value is found. This method is widely used for making decisions related to optimum utilization of resources in an organization.

Exhibit 3 gives a broad overview of the sequential process that is involved in the process of optimization, from idea generation to support in decision making. During optimization analysis, the values for one or more variables are changed in a repetitive manner keeping in mind the specific constraints, until the best values for target variable are found. To exemplify, one can determine the highest level of production that can be achieved by varying job assignments to workers, in terms of man-hours, considering that some workers are skilled and their job assignment cannot be changed.

In a business quotation to be given to a vendor, the margin or discount can be repeatedly changed to see the affect on the final price of the solution.

Optimization analysis when complex or dealing with more than one variable requires special purpose software and techniques like linear programming. For instance, optimization analysis is used for finding out the appropriate marketing expenditure on advertisement and promotions with the given constraints of budget and use of print media. In this case, Solver, which is a what-if analysis tool in MS Excel, is useful to conduct the analysis.

An organization needs to ensure the right people handle the right task. They should be doing the task assigned in the right time frame. Information exchange is necessary between the stake holders of an organization. These include internal workforce, vendors, suppliers and clients. However, it is equally essential that the right information should be available to the right audience rather than transferring any business related data in bulk. Hence, lays the need for optimization.

3. Components of optimization

The purpose of optimization is to develop the optimal solution to a problem given a set of restrictions and assumptions.

The three components of optimization are depicted in Fig 1 and explained in detail below:

1. Decision Variable – The decision maker has control over this variable. This measures quantitatively the resources allocated or activities to be performed.In the simplest way, a decision variable determines what decision is to be taken? For example, in figure 2, there are three car manufacturing units located in Manesar, Haryana. They have to distribute the manufactured products between five warehouses in Delhi itself. This could be done in 3*5=15 ways. Among these 15 ways, each one is representative of the products produced by the manufacturing firm.

Each plant caters to 4 different types of cars being manufactured. To optimize the shipping plan over the next 6 months, the analysis will yield 3*4*5*6 = 360 decision variables. It can be seen how adding further detail like product type can elaborate the analysis. The modeling techniques help to decide how much detail is relevant to the analysis. The next step lies in defining the objective and outlining the constraints.

2. Constraint – After the decision variable is outlined, the next phase includes highlighting the limiting factors. Constraints refer to a limitation or a bottleneck imposed on the working of a system. A system consists of interrelationships and interdependencies between various components. Therefore, there lies a need to consider the limitations in a real world scenario. Such constraints are common in situations concerning budgetary forecasts, demand, production, storage, pricing etc. A constraint is defined by computing values using decision variables. It is equally, important to express a proper limit to the value for better understanding such as (<=, =, >=). Constraints are of three types namely equality, inequality and integer constraints. In case of equality the quantities assume the same value (A=B) while on the other hand inequality implies the values that are not equal to each other (such as A ≠ B or A<B or A>B). The integer constraints in case of linear programming assume only integer or whole values.

Considering the example stated above, for the movement of products out of the warehouse, for each time period, there is a balance constraint stating,Ending inventory = Beginning inventory + products received – products shipped.This implies, the closing inventory of one time period becomes the beginning inventory for the other.

3. Objective Function – It outlines the main aim of the model. The target is to obtain the maximum or minimum values of the function under a set of constraints using linear programming. Linear programming is a computation technique for determining the maximum or minimum values from a mathematical model that is stated in the form of linear equations.

The statement of an optimization problem is stated as:

For X = f[ x1, x2, x3 …………….xn], the lowest values of f(X) is subject to constraints i.e.,ai(X) <= 0,  where i = 1, 2, 3 …….m bj(X) = 0, where j= 1,2,3……..p where f(X) is called the objective function and ai(X) and bj(X) are known as inequality and equality constraints respectively.

3.1 Complicating factors in optimization

Though optimization techniques might be very useful, yet it might become difficult to solve these. Some of these problems are discussed as under:

• There could be a case of more than one decision variable. In firms with larger product portfolio, effect of each product on the overall profit is to be considered as compared to firms that are offering a single product.
• There could be complex nature of the relationships between the decision variables and the associated outcome. For example, in a government policy decision on opening bank accounts with zero balance, it may become difficult to derive a direct relation between the fresh enrollments and its impact on bank deposits. There could be a change due to availability of bio metric machines for identity verification, awareness among locals and readiness for people to join.
• There can be limiting conditions or constraints being imposed on decision variables. An organization needs to work under controlled conditions by using its manpower, material, money diligently. These are decision variables over which it can exercise control. The end motive of any organization is profit maximization and cost minimization.
• When each activity is known to lead to a certain outcome, there is less uncertainty. However, the presence of uncertainty arises when constraints are imposed on resources to be used. It is important to conduct risk assessment in such a case.

For example, if we wish to design the fuel efficient high mileage bike, the objective may be to maximize the engine efficiency. The engine may be required to provide a specific power output with an upper limit on the amount of engine emission. Certain parameters such as ignition ability, quick acceleration, over size and wider tyre will serve as constraints for optimization. The design variables that are allowed to be changed during optimization may be the compression ratio, fuel mixture ratio, etc.

4. Why is optimization necessary?

The concept of optimization can be applied to a number of fields. It can help develop better range of products, financial portfolios, analysis of expenditure, resource allocation, budgets etc.

Optimization is the act of achieving the best possible result under given limiting factors. The applications can be seen in multi faceted disciplines such as design, construction, maintenance etc where engineers have to take decisions. The goal of all such decisions is to maximize benefits with optimum resources. This is evident from the basic principle that every business operates with the underlying objective of profit maximization while minimizing overhead costs. The effort or the benefit can be usually expressed as a function of certain design variables. Hence, optimization is the process of finding the conditions that give the maximum or the minimum value of a function.

The concept of optimization helps to solve a number of technical and design related problems such as design considerations for aircrafts, trajectories for aerospace missions, shortest path, minimum processing time, minimizing signal delay, optimal design of electric networks.

The necessity and importance of optimization can be observed since it helps find the answer that yields the best result–the one that attains the highest profit, output or happiness, or the one that achieves the lowest cost, waste or discomfort.

A number of factors can be listed (Fig 3.) that makes optimization analysis a key tool in mathematical programming.

• Cost reduction – Proper planning and automation of processes removes duplication of time and It also helps to remove redundancies that may be present. Reengineering products as per changing business and customer needs lead to optimization thus removing and reducing unnecessary overheads.
• Safety & error reduction – Management best practices and quality control ensures process to be mitigated with parameters that ensure quality with minimal defects and correction at each step. This promises safety as per international standards and error reduction.
• Time efficiency – Removal of repetitive activities and planning is an essential step in ensuring time efficiency.
• Reproducibility – Documenting procedures and archiving essential steps in the development and production cycle ensures the knowledge behind it is not lost. Also, as experts move from organizations, their intelligence and competence can be utilized by organizations by ensuring effective product and service documentation. Optimization at this level ensures reproducibility and rebuilding to be a less time consuming activity from a business standpoint.
• Innovation & efficiency – Optimization is the diligent method of ensuring business effectiveness in terms of resourced employed and the final outcome. These efforts help organizations achieve competent outcomes and high returns on investment—while delivering their mission in unprecedented business climates.

Optimization necessitates the need for efficient use of resources like money, time, machinery, staff, inventory and more. Optimization techniques are a powerful set of tools that are important in efficiently managing an enterprise’s resources and thereby maximizing shareholder’s wealth. This is well explained in exhibit 5.

5. Optimization Modeling Process

The process of optimization has to be carried over a particular time period as per business needs. For this, the most important task is problem definition. This requires first describing a problem and then highlighting a solution. The regular up gradation of the system is necessary to incorporate changes wherever needed. It is necessary to have feedback and control loops between various steps.

The optimization process is shown in figure 4. Let us try to understand each of the steps one by one.

1. Mathematical formulation of the problem: This stage refers to framing the mathematical formulation algorithm. This must be reflective of the problem definition and should exclude the redundant information.
2. Finding an optimal solution: This stage refers to consideration of all factors and formulating the solution algorithm for proper implementation.
3. Managerial interpretations of the optimal solution: The next stage after formulating the algorithm is the software package to use. The management intervention and help of consultants is essential here. The result of the optimization process should be easy to interpret by the decision maker.
4. Post- solution analysis: The post solution analysis stage refers to timely updating of the optimization solution. As business needs change, conditions under which the optimization solution was found may change. The decision maker should closely observe changes with respect to changing business needs.
5. The importance of feedback and control: Feedback is of paramount importance in any optimization solution. An optimum strategy is what takes into account all updated factors of the changing business environment.

Classification of the Optimization Model

The optimization model can be classified in a number of ways. Different types of categorization techniques are mentioned in Fig 5.

Continuous optimization versus discrete optimization – It is easier to solve problems concerning continuous optimization. This is because the function can yield results for every given point over an interval or its neighborhood. Discrete values on the other hand, take input of only discrete set of integers. Exhibit 6 shows one such case for discrete and continuous optimization in context of visualization of storylines depicting social interactions. What is seen is that the discrete method showcases a graphic interpretation through alignment of entities and crossings, while the continuous technique optimizes it and makes it more consistent.

• Unconstrained optimization versus constrained optimization – An important aspect of the unconstrained optimization is that the objective function has no limiting conditions and holds true for all values over a specified range. On the other hand, constrained optimization refers to the limits that are imposed on the objective function. Constraints could be of various types such as linear, non linear or convex. Exhibit 7 exemplifies the way limits are imposed on function f(x,y).

Exhibit 8 explained various techniques in case of constrained optimization. Some of which include basic linear programming, integer programming and multiple function programming.

• None, one or many objectives – Optimization problems vary in sense of having either one or more objective function. However, in some cases there would be none at all. Most optimization problems have a single optimization function. From multiple objective functions one infers that a decision is dependent on multiple distinct complexities. In such cases, a weighted combination is formed for ease of computation. Multi objective optimization occurs in many instances where the decisions have to take into account conflicting objectives. Fields such as supply chain, manufacturing, design etc are common examples.
• Deterministic Optimization versus Stochastic Optimization – Deterministic optimization implies that the data corresponding to the situation is known clearly. Stochastic optimization on the other hand, takes into account the uncertainty to be incorporated. In this case, for future time periods, data cannot always be accurately forecasted as in case of discounts, product demand, meteorological forecasts etc. Fig 6 highlights a number of models for deterministic and stochastic optimization techniques.

Deterministic Models	Stochastic Models
Linear Programming Chains	Discrete Time Markov
Network Optimization	Continuous Time Markov Chains
Integer Programming	Queuing
Non Linear Programming	Decision Analysis

7. Summary

The aim of every decision maker in an organization is to find the best solution to a problem. The ‘best solution’ in this regard has various attributes. An optimized solution is most accurate, profitable, cost effective and timely implemented. At the organizational level, those who manage and control systems of men, machines, markets and money face the problem of continuous improvement or rather optimizing business performance. Any project in the organizational climate is itself a system which comprise a network of elements (tasks) that are interconnected and interdependent, that work together to achieve a specific goal. The problem may either be reducing operational cost on one hand and maintaining competent service levels on the other. To maintain profits without increasing overhead cost within limits imposed by government policies and regulations. Thus, optimization is focused on ensuring overall product quality without reducing any other attribute. Simulation techniques and mathematical computational models help to optimize functions and procedures and facilitate decision making. Optimization models as part of decision support systems (DSS) help find the best fit solution in a quantitative manner. Researchers have made it possible to possible to analyze even multiple variable problems using optimization software’s such as CPLEX and OSL to find near optimal solutions. These solutions can easily be obtained on desktops that use commercially available software packages that run optimization algorithms.

To build optimization based DSS, management information systems develop interactive graphic user interface for its software. This is integrated with database management software to work on multiple data sets till the best fit solution is obtained. Optimization analysis is not limited to business information systems but has numerous applications. Engineering design, statistics, market forecasting, budget allocation to name a few.

Web Resources

• http://www.solver.com/examplesoptimizationproblems
• http://home.ubalt.edu/ntsbarsh/opre640a/partviii.htm