Operations Research: Solving WWII Problems with Linear Programming and Duality Theory, Study notes of Linear Programming

Download Operations Research: Solving WWII Problems with Linear Programming and Duality Theory and more Study notes Linear Programming in PDF only on Docsity!   “Chapter-2(8th*proof)” — 2011/4/29 — 15:41 — page 19 — #1       Techniques of Operations Research C H A P T E R 2 2.1 INTRODUCTION The term, Operations Research was first coined in 1940 by McClosky and Trefthen in a small town called Bowdsey of the United Kingdom. This new science came into existence in a military context. During World War II, military management called on scientists from various disciplines and organised them into teams to assist in solving strategic and tactical problems, relating to air and land defense of the country. Their mission was to formulate specific proposals and plans for aiding the Military Commands to arrive at decisions on optimal utilisation of scarce military resources and efforts and also to implement the decisions effectively. This new approach to the systematic and scientific study is operational research. Hence, OR can be associated with “an art of winning the war without actually fighting it”. 2.2 SCOPE OF OPERATIONS RESEARCH There is a great scope for economists, statisticians, administrators and the technicians working as a team to solve problems of defense by using the OR approach. Besides this, OR is useful in the various other important fields like: 1. Agriculture 2. Finance 3. Industry 4. Marketing and Personnel management 5. Production Management 6. Research and Development etc. 2.3 VARIOUS TECHNIQUES OF OPERATIONS RESEARCH 1. Linear Programming Problem • Simplex Algorithm • Two-Phase Simplex Method • Duality Theory   “Chapter-2(8th*proof)” — 2011/4/29 — 15:41 — page 21 — #3       Techniques of Operations Research 21 2.4.2 Applications • Optimising refinery operations. • Supply distribution marketing (SDM) system. • Diet problem. • Work scheduling problem. • Capital budgeting problem. • Short-term financial planning. • Blending problems. • Productions process models. • Inventory model, etc. 2.4.3 Simplex Method Simplex method is an iterative procedure for solving Linear Programming Problem (LPP) in a finite number of steps. This method provides an algorithm which consists of moving from one vertex of the region of feasible solution to another in such a manner that the value of the objective function at the succeeding vertex is less or more as the case may be than at the previous vertex. This procedure is repeated and since the number of vertices is finite, the method leads to an optimal vertex in a finite number of steps or indicates the existence of unbounded solution. Definition Let Xb be a basic feasible solution to the LPP, Max Z = Cx Subject to Ax = b And X ≥ 0, such that it satisfies Xb = B−1b whereB is the basic matrix formed by the column of basic variables. The vector Cb = (Cb1, Cb2...Cbm) where Cij are components of C associated with the basic variables is called the cost vector associated with the basic feasible solution Xb. In the simplex method the following procedure is to be studied to solve the problem and this will be explained in detail in the next chapters: 1. Converting linear programming to standard form 2. Finding basic and non-basic variables 3. Determining the entering and leaving variable 4. Feasible solutions 5. Unbounded solutions and others The main feature of the simplex method is that it solves the LP in iterations. Each iteration moves the solution to a new corner point that has the potential to improve the value of the objective function. The process ends when no further improvements can be realised. The simplex method involves tedious and voluminous computations, which makes the computer an essential tool for solving LP problems. The computational rules of the simplex method are thus designed to facilitate automatic computations.   “Chapter-2(8th*proof)” — 2011/4/29 — 15:41 — page 25 — #7       Techniques of Operations Research 25 The assignment model is actually a special of the transportation model in which the workers represented the sources and its jobs represent the destinations. The supply amount at each desti- nation exactly equal one. The cost of transporting worker ‘i’ to the job ‘j’ is Cij . The assignment model can be solved directly as a regular transportation model. The fact that all supply and demand amount equal one has led to the development of a simple solution algorithm called the Hungarian method. The new method appears totally unrelated to the transportation model. The algorithm is actually rooted in the simplex method just as the transportation model is. A more efficient way of solving the assignment problem has been developed based on a mathematical property due to the Hungarian method. The basic principle of this method is that the optimal assignment is not affected if a constraint is added or subtracted from any row or column of the standard assignment of cost matrix. A näıve approach to solve this problem is to enumerate all possible assignments of jobs to machines. For each assignment the total cost may be computed, and the one with the least cost is picked as the best assignment. This will be an inefficient and expensive approach since the number of possible assignments is n! even for n=5, there are 120 possible assignments! It can be thought of formulating to linear programming problem. The assignment problem can be solved by the Hungarian Method. Step 1: In the original cost matrix, identify each row’s minimum and subtract it from all the entries of the row. Step 2: For the matrix resulting from step 1, identify each column’s minimum and subtract it from all the entries of the column. Step 3: Identify the optimal solution as the feasible assignment associated with the zero elements of the matrix obtained in step 2. The Assignment model also, can be solved by softwares like TORA, Excel Solver Solution, LINGO and LINDO and others. 2.6.1 Applications • To minimise the total time required to complete the jobs. • To minimise cost assignment of persons to jobs. • To minimise time for arrival and departure of flights. The application of assignment technique is assigning people to tasks, airline departure and arrival of the flights and other areas. 2.7 QUEUING THEORY Queues are a part of everyday’s life. We all wait in queues to buy a movie ticket, to make a bank deposit, pay for groceries, mail a package, obtain a food in a cafeteria, to have ride in an amusement park and have become adjusted to wait but still get annoyed by unusually long waits. Queueing theory is the study of waiting in all these various activities. It uses queueing models to represent the various types of queueing systems that arise in practice. Formulas for each model indicate how the corresponding queueing system proceed, perform, including the average amount of waiting that will occur under a variety of circumstances. The queueing models are very helpful for determining   “Chapter-2(8th*proof)” — 2011/4/29 — 15:41 — page 26 — #8       26 Operations Research how to operate a queueing system in the most effective way if too much service capacity to operate the system involves excessive costs. The models enable finding an appropriate balance between the cost of service and the amount of waiting. The flows of materials through manufacturing operations through a sequence of processing stages represent other forms of queues. In this chapter we learn how to construct and solve equations that describe queueing behaviour for a wide variety of situations. We cannot exhaust even the elementary models, because the number of possible variations is enormous. In the terminology of queueing theory, the first proposal involves changing the parameter values of the models but does not involve structural changes. The queueing models are very helpful for determining how to operate a queueing system (as shown in Fig. 2.1) in the most effective way if too much service capacity to operate the system involves excessive costs. The models enable finding an appropriate balance between the cost of service and the amount of waiting. The queueing models can also be solved using software like TORA, Excel Solver Solution, LINGO and LINDO and others. Input Input Source Queue Discipline Service Mechanism Served Units Fig. 2.1 The basic structure of Queueing Model. The second proposal is for a two-server, single queue system and the third is for a two-server separated queue system. 2.7.1 Applications • Bank teller • Bank staffing • Barber shop • Tool centre • Service station. 2.8 GAME THEORY Life is full of conflicts and competition. The examples of conflicts include parlour games, military battles, political campaigns, advertising and marketing campaigns by competing business firms etc. Game theory deals with decision situation in which two intelligent opponents have competing objectives. In a game conflict, two opponents known as players will each have a number of alternatives or strategies. Associated with each pair of strategies is a pay-off that one player pays to the other, such games are known as two-person zero sum games because the gain by one player equals the loss of the other so that the sum of their net winning is zero. The basic characteristics of two-person zero sum games consider the game called evens and odds. In general a two-person game is characterised by 1. The strategies of player 1 2. The strategies of player 2 3. The pay-off table.   “Chapter-2(8th*proof)” — 2011/4/29 — 15:41 — page 27 — #9       Techniques of Operations Research 27 A primary objective of game theory is the development of rational criteria for selecting a strategy. The following two key assumptions are made: 1. Both players are rational 2. Both players choose their strategies solely to promote their own welfare. Game theory contrasts with decision analysis where the assumption is that the decision maker is playing a game with a passive opponent which chooses its strategies in some random fashion. 2.8.1 Applications • Constant sum TV game • Odds and evens • Coin toss game with bluffing • Prisoner’s dilemma • Advertising prisoner’s dilemma game • Chicken game. 2.9 NETWORK METHODS Management of big projects with a large number of activities pose complex problems in planning, scheduling and controlling, especially when the activities have to be performed in a specialised technological sequence. Fortunately, two operation research techniques, PERT (Program Evaluation and Review Technique) and CPM (Critical Path Method) are available to assist the project manager in carrying out these responsibilities. These techniques make heavy use of networks to help plan and display the coordination of all the activities. They also normally use a software package to deal with all the data needed to develop schedule information and then to monitor the progress of the project. Project management software, namely, MS Project is available for these purposes. Some recent surveys report that as much as 70% of the world’s mathematical programming problems can be represented by network related models. The applications of networks are: • Determination of the shortest route between two cities in a network of roads. • Determination of the time schedule for the activities of a construction project. • Determination of the time schedule from oil fields to refineries through a pipeline network. The solutions of these situations are accomplished through a variety of network optimization algorithms. The following are the algorithms: • Minimal spanning tree • Shortest route algorithm • Maximum flow algorithm • Minimum cost network algorithm. The Network model also, can be solved using software like TORA, Excel Solver Solution, LINGO and LINDO and others.