LECTURE 2 Introduction to Operations Research SSC 2201

LECTURE 2 Introduction to Operations Research SSC 2201 Modeling Process Richard Ntwari Institute of Computer Science Mbarara University of Science and Technology (MUST) P. O. Box 1410, Mbarara, Uganda http: //www. must. ac. ug/ Email: rntwari@must. ac. ug

OPERATIONS RESEARCH MODELS • Imagine that you have a 5 -week business commitment between Fayetteville (FYV) and Denver (DEN). You fly out of Fayetteville on Mondays and return on Wednesdays. A regular round-trip ticket costs $400, but a 20% discount is granted if the dates of the ticket span a weekend. A one-way ticket in either direction costs 75% of the regular price. How should you buy the tickets for the 5 -week period? 30/10/2020 Richard Ntwari ICS - MUST 2

OPERATIONS RESEARCH MODELS • We can look at the situation as a decision-making problem whose solution requires answering three questions: • What are the decision alternatives? • Under what restrictions is the decision made? • What is an appropriate objective criterion for evaluating the alternatives? 30/10/2020 Richard Ntwari ICS - MUST 3

OPERATIONS RESEARCH MODELS Three alternatives are considered: • Buy five regular FYV-DEN-FYV for departure on Monday and return on Wednesday of the same week. • Buy one FYV-DEN, four DEN-FYV-DEN that span weekends, and one DENFYV. • Buy one FYV-DEN-FYV to cover Monday of the first week and Wednesday of the last week and four DEN-FYV-DEN to cover the remaining legs. All tickets in this alternative span at least one weekend. 30/10/2020 Richard Ntwari ICS - MUST 4

OPERATIONS RESEARCH MODELS Under what restrictions is the decision made: • The restriction on these options is that you should be able to leave FYV on Monday and return on Wednesday of the same week. Appropriate objective criterion for evaluating the alternatives: • An obvious objective criterion for evaluating the proposed alternative is the price of the tickets. • The alternative that yields the smallest cost is the best. 30/10/2020 Richard Ntwari ICS - MUST 5

OPERATIONS RESEARCH MODELS • Specifically, we have i. Alternative 1 cost = 5 X 400 = $2000 ii. Alternative 2 cost =. 75 X 400 + 4 X (. 8 X 400) +. 75 X 400 = $1880 iii. Alternative 3 cost = 5 X (. 8 X 400) = $1600 Thus, we should choose alternative 3. • Though the preceding example illustrates the three main components of an OR model-alternatives, objective criterion, and constraintssituations differ in the details of how each component is developed and constructed. 30/10/2020 Richard Ntwari ICS - MUST 6

BASIC OR CONCEPTS • "OR is the representation of real-world systems by mathematical models together with the use of quantitative methods (algorithms) for solving such models, with a view to optimizing. " • We can also define a mathematical model as consisting of: § Decision variables, which are the unknowns to be determined by the solution to the model. § Constraints to represent the physical limitations of the system § An objective function § An optimal solution to the model is the identification of a set of variable values which are feasible (satisfy all the constraints) and which lead to the optimal value of the objective function. 30/10/2020 Richard Ntwari ICS - MUST 7

MODELING • Modeling is a word used in many contexts and sciences. In this monograph we deal with mathematical modeling, the process of solving real-world problems using mathematical techniques. This process involves much more than just mathematics. • Indeed, many people state that, while solving mathematical models is a science, modeling is an art! With that they mean to say there is no theoretical foundation to modeling, as there is to model solving, but that good modeling needs a combination of talent and experience. 30/10/2020 Richard Ntwari ICS - MUST 8

Introduction and definitions • A model is a (simplified) description of a real-world process or phenomenon. This is often a written description, but it can also be a physical construction, of for example a building. The object which is to be modeled need not exist (yet). • A model is a description of a part of a system or process and its interaction with its environment that allows an analysis of certain aspects of that system or process. • Modeling is a methodology for problem solving in which the use of models plays a crucial role. • A mathematical model is a model in which the relations within the model are given in mathematical terms. 30/10/2020 Richard Ntwari ICS - MUST 9

Introduction and definitions • An optimization model seeks to find values of the decision variables that optimize (maximize or minimize) an objective function among the set of all values for the decision variables that satisfy the given constraints. 30/10/2020 Richard Ntwari ICS - MUST 10

Classification of Problems • So far we gave definitions of a model and of modeling, and we discussed the various steps of a modeling project. Here we classify the different business problems that we encounter. • We will see for which type of problems modeling can help us find a solution. In the next section we take a closer look at the structure of a mathematical model. 30/10/2020 Richard Ntwari ICS - MUST 11

i. Programmed and un programmed problems • The most important classification is that between programmed and un programmed problems also called structured and unstructured. Problems are called programmed if they are repetitive and routine, to the extent that a definite procedure has been worked out for handling them“. Problems which are not programmed are called un programmed • There are several reasons for that, among which are the fact that for repetitive problems data is often available, and that programmed problems are usually easier to model. 30/10/2020 Richard Ntwari ICS - MUST 12

ii. Strategic, tactical and operational decisions A second way to classify problems is by its level of management decision. - Strategic decisions, dealing with the determination of long-range goals and the means to achieve these. - Managerial or tactical decisions, concerning the realization of the long-term goals and the management of the resources; - Operational decisions, dealing with the short-time planning. • Of course, top management is concerned with strategic decisions, and low level management or the production employees themselves are concerned with the short-term planning. We see that the lower the decision level, the shorter the horizon over which the decision takes effect and thus low level decisions are more often repetitive. 30/10/2020 Richard Ntwari ICS - MUST 13

iii. Internal and external coordination • We can also classify problems by the fact whether they have only internal or also external aspects. Thus problems that are concerned with external coordination are related to the way the firm should react to changes from the outside. Examples are the behavior of competitors (new products or prices), changes to the workforce due to union and government decisions, etc. Internal coordination is concerned with problems that exist totally within the interior of the firm. External coordination problems are very often at the strategic level, internal problems at the tactical/operational level. Sometimes this is even used as a definition of strategic and tactical/operational. 30/10/2020 Richard Ntwari ICS - MUST 14

iv. Classification by objective • Problems can also be classified by the goal which is to be achieved. This objective can range from an educated guess to the solution of a certain problem to a computer system that takes automatically and independently its decisions. • Even if the goal is to find an optimal decision, it can be better to build a tool able to answer “what if" questions by which the managers involved can find the solution by experimentation. The goal should be taken into account when modeling. It is clear that a modeling approach is more successful in situations where a detailed solution is wanted. 30/10/2020 Richard Ntwari ICS - MUST 15

iv. Classification by objective Example. • To fulfill a bottleneck analysis in a production system a simple spreadsheet model may suffice. • To produce an optimal production schedule one often has to solve a complicated model. 30/10/2020 Richard Ntwari ICS - MUST 16

v. Design and control problems • A final distinction is between design and control problems. • Design problems are those related to setting up a system or process, control problems are those dealing with operating systems or processes. • Design problems are often at the strategic or tactical level, control problems are often operational. 30/10/2020 Richard Ntwari ICS - MUST 17

v. Design and control problems • Example. In a distribution setting, choosing the location of warehouses is a design problem at the strategic level. Selecting the daily routes from the warehouse to the customers is a control problem at the operational level. Setting up the information and other systems to be able to select these routes in an efficient way involves decisions at the tactical level. 30/10/2020 Richard Ntwari ICS - MUST 18

vi. Planning and scheduling • What are called control problems above can be split up in planning and scheduling problems. Planning is concerned with the long-term control • issues, often at the tactical level. Scheduling deals with the operational short-term control. Sometimes the word control is also used to indicate the activity consisting of checking whether the foreseen plans or schedules are met and taking the appropriate actions when necessary. 30/10/2020 Richard Ntwari ICS - MUST 19

vi. Planning and scheduling • Example. In this Part, we make the distinction between production planning and production control. Production planning deals with building production plans taking into account various constraints. Production scheduling deals with the actions that allow the production system to keep to the production plans. 30/10/2020 Richard Ntwari ICS - MUST 20

Modelling Phases 1. Definition of the problem: This phase involves defining the scope of the problem under investigation. • This function should be carried out by the entire OR team. The aim is to identify three principal elements of the decision problem: i. description of the decision alternatives, ii. determination of the objective of the study, and iii. specification of the limitations under which the modeled system operates. 30/10/2020 Richard Ntwari ICS - MUST 21

DATA COLLECTION AND ANALYSIS • Gathering and preparing data is an important activity in a modeling project. First we have to realize the importance of obtaining correct data. Garbage in| garbage out" is not without reason a well known phrase. Getting good input for your model is more than doing statistics well: above all it is acquiring, measuring and estimating data correctly and then using it correctly. 30/10/2020 Richard Ntwari ICS - MUST 22

Modelling Phases 2. Construction of the model: • This phase entails an attempt to translate the problem definition into mathematical relationships. If the resulting model fits one of the standard mathematical models, such as linear programming, we can usually reach a solution by using available algorithms. • Alternatively, if the mathematical relationships are too complex to allow the determination of an analytic solution, the OR team may opt to simplify the model and use a heuristic approach, or they may consider the use of simulation, if appropriate. 30/10/2020 Richard Ntwari ICS - MUST 23

Modelling Phases 3. Solution of the model. • This phase is by far the simplest of all OR phases because it entails the use of well-defined optimization algorithms. An important aspect of the model solution phase is sensitivity analysis. • It deals with obtaining additional information about the behaviour of the optimum solution when the model undergoes some parameter changes. Sensitivity analysis is particularly needed when the parameters of the model cannot be estimated accurately. In these cases, it is important to study the behaviour of the optimum solution in the neighbourhood of the estimated parameters. 30/10/2020 Richard Ntwari ICS - MUST 24

Modelling Phases 4. Validation of the model: • Model validity checks whether or not the proposed model does what it purports to do-that is, does it predict adequately the behaviour of the system under study? Initially, the OR team should be convinced that the model's output does not include "surprises. “ • In other words, does the solution make sense? Are the results intuitively acceptable? On the formal side, a common method for checking the validity of a model is to compare its output with historical output data. The model is valid if, under similar input conditions, it reasonably duplicates past performance. 30/10/2020 Richard Ntwari ICS - MUST 25

VERIFICATION AND VALIDATION • The term verification and validation are often used in a simulation context, but they can and should be used for any type of model. Verification means confirming that the implementation of the model is correct; validation means that it is checked that the model outcomes correspond with those of the system up to a certain extent. As the system need not exist already this is not always possible. It is always possible however to check parts of the model, or to predict outcomes in some other way. • Validation not always means comparing a model with a system; it can also be the case that a model is validated with another model, one which has more detail for example. 30/10/2020 Richard Ntwari ICS - MUST 26

Modelling Phases 5. Implementation of the solution: • Implementation of the solution of a validated model involves the translation of the results into understandable operating instructions to be issued to the people who will administer the recommended system. The burden of this task lies primarily with the OR team. 30/10/2020 Richard Ntwari ICS - MUST 27

SUB-OPTIMIZATION • Modeling is always a compromise between the scope of the model and the complexity. If the model is too complex to solve satisfactorily then decreasing the model scope is an option. This has the risk that the influence on system parts that are not modeled is ignored. This influence can be important enough to change the proposed solution to the problem. • Another possibility for checking the influence of a small scope optimization procedure on larger scope issues is using simulation as a large scope model. Thus the small scope optimization is validated by a large scope simulation. Often however the influence of sub optimization on other systems is hard to quantify, and modeling cannot help us to assess the consequences. 30/10/2020 Richard Ntwari ICS - MUST 28

END • Questions? 30/10/2020 Richard Ntwari ICS - MUST 29