2. What is operation research? - The first formal activities of operation research (OR) where initiates in England during world war Π. - following the end of the war, the ideas advanced in military operations were adapted to improve efficiency and productivity in the civilian sector.

3. What is operation research? (continue) - Today, OR is a dominate and indispensable decision making tool. - A corner stone of OR is mathematical modeling.

4. Operation research models: - A decision making problem solution required identifying three components: 1- What are the decision alternatives? 2- Under what restrictions is the decision made? 3- What is an appropriate objective criterion for evaluating the alternatives that satisfy the restrictions?

5. Example: Mohammed would like to buy a fried oil to his family. His wife ask him only to buy sunflower oil or corn oil. Mohammed always would like to reduce the live expense so he always buy less price oil. Pleased help Mohammed to select from the following options? 1) Alkafif oil sunflower by 2.5 RO. 2) Afia oil corn by 3.25 RO. 3) Suhar oil online by 1.5 RO. 4) Almanar oil corn by 2 RO.

6. Example: - Imagine that you have a 5-week business commitment between Muscat and UAE. - You fly out of Muscat on Monday and return on Wednesday. - A regular round-trip ticket cost 400 RO, 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 5-week period?

7. Solution :- A) Alternatives: Three alternatives are considered:- 1- Buy five regular Muscat-UAE-Muscat. 2- Buy one Muscat-UAE, four UAE-Muscat-UAE that span weekend and one UAE-Muscat. 3- Buy one Muscat-UAE-Muscat to cover Monday of the first week and Wednesday of last week and four UAE-Muscat-UAE to cover the remaining legs ( each ticket in this alternatives span weekend).

8. B) The restriction on these option it that you should be able to leave Muscat on Monday and return on Wednesday of the same week for five weeks. C) An obvious objective criterion for evaluating the proposed alternatives is the price of the tickets. That alternative that yields the smallest cost is the best. Specifically, we have :- - Alternative 1 cost=5 * 400=2000. - Alternative 2 cost=0.75 * 400 + 4 * (0.8*400) + 0.75 * 400 =1880. - Alternative 3 cost=5 *(0.8 * 400) = 1600. Thus, you should choose alternative 3.

9. Example :- Forming maximum area rectangle out of a piece of wire of length l inches ( what should be the width and height of the rectangle )? - In contrast with the tickets example, the number of alternatives in this example is not finite, namely the width and height of the rectangle can assume infinite possibilities. - Let w= width of the rectangle in inches. h= height of the rectangle in inches. - The restrictions can be given by: 1. 2(w+h) =L 2. W ≥ 0, h ≥ 0 - The objective of problem: Maximize z= wh, subject to 2(w+h)=L, w,h ≥ 0 - The optimal solution of this model is w = h = L/4 which calls for consternating a square shape.

10. - The definition of the OR models consists: 1- The alternatives or the decision variables of the problem. 2- The decision variables are use to construct the objective function. 3- The constraints of model. - The OR model is usually organized in following general format; Maximize or minimize objective function subject to : constraints

11. Example :- Suppose that you have 50 RO and you wont to buy your requirement which include Kima, Kandora, Wizar, Sliper, bag within minimum cost. Configure the OR model for the above problem?

12. Solution :- -Let X1 =The price of Kima. X2 = The price of Kandora. X3 = The price of Wizar. X4 = The price of Sliper. X5 = The price of bag. Minimize z = X1 + X2 + X3 + X4 + X5 subject to X1 + X2 +X3 + X4 + X5 ≤ 50 X1, X2,X3, X4, X5 ≥ 0

13. Ex: Form an OR model for the following problem: “ If Someone want to buy a hose with minimum cost as he has only 55000 OR, the man want the area of the house should be not less than 500M 2 and the price of each meter also not much than 200 OR”?

14. EX: Formulate an OR model for the following problem: “someone wants to buy a computer with minimum cost and with the following features: -Processor speed higher than 2.5 GHz. -RAM Capacity Higher than 4 G bytes. -Screen Size Higher than 16 inches. The cost of the processor is 20 OR per GHz. The cost of Memory is 5 OR per G byte. The screen cost is 10 OR per Inches. The other parts cost is 50 OR. That man has only 300 OR to buy the computer”

15. A solutions of the model A – Is feasible if it satisfies all the constraints. B – It is optimal if in addition to being feasible, it yield the best ( maximum or minimum ) value of objective function.

16. solving the OR model: - In OR, we don’t have a single general technique that solve all mathematical model. - The most prominent OR techniques: 1- Linear programming: Designed for models with strict liner objective and constraints functions. 2- Integer programming: In which the variables assume integer values( ex: problems dealing with units like human, cars, and, rooms). 3- Dynamic programming: The original model can be decomposed into smaller sub problems. 4- Network programming: The problem can be modeled as network (ex: construct new roads net to connect a no.of cities) 5- Nonlinear programming: The function of the model are nonlinear.

17. Queuing and simulation model: - Queuing and simulation deals with study of waiting lines. - They are not optimization techniques; rather they are determine measures of performance of waiting lines, such as average waiting time in queue, average waiting time for service, and utilization of service facilities. - Queuing models utilize probability to analyze waiting lines.

18. Queuing and simulation model: (continue ) - Simulation estimates the measure of performance by imitating the behavior of the real systems. - The main difference between queuing and simulation is that queuing models are purely mathematical, that is limit their scope of applications. - Simulation, on the other hand, is flexible and can be used to analyze practically any queuing situation.

19. Art Of Modeling: - The following figure depicts the levels of abstraction that characterize the development of an OR model. - The assumed real world is abstracted from the real situation by concentrating on the dominate variables that controls the behavior of the real system. Real world Assumed Real world Model

20. Phases Of an OR study: -An OR study is root in team work, where both the OR analysts and the client works side by side. -The OR analysts with their expertise in modeling must by complemented with the experience and cooperation of the client for whom the study in being carried out. -The principle phases for implementing OR in practice include:-

21. Phases Of an OR study (continuo) A)Definition of the problem: problem definition involves defining the scope of the problem under investigation. This is a function that should be carried out by the entire OR team. B) Construction of the model: model construction entails translating the problem definition into mathematical relation ships(OR model) ( done by the OR analyst). C)Solution of the model: model solution by far, is the simplest of all OR phases because it entails the use of well-defined Optimization Algorithms. ( done by the OR analyst).

22. Phases Of an OR study (continuo) D) Validation of the model :- Model validation checks whether or not the proposed model dose what it is suppose to do. ( team work ) E) Implementation of the solution :- Implementation of solution of a validated model involves the translation of the resulting solution into operating instructions issued in understandable format that individuals who will administer the recommended system. ( team work ).