1. Introduction of Quantitative Technique 1 INTRODUCTION OF QUANTITATIVE TECHNIQUE INTRODUCTION People have been using mathematical tools to help solve problems for thousands of year; however, the formal study and application of quantitative techniques to practical decision making is largely a product of the 20th century. The technique we study in quantitative analysis have been applied successfully to an increasingly wide variety of complex problem in business, government, healthcare, education, and many other areas. To get aware of the mathematics of how a particular quantitative technique works; one must also be familiar with the limitations, assumption and specific applicability of the technique. The successful use of quantitative techniques usually results in a solution that is timely, accurate, flexible, economical, reliable, an easy to understand and use. WHAT IS QUANTITATIVE TECHNIQUE? Quantitative technique is the scientific approach to managerial decision making. The approach starts with data. Like raw material for a factory, these data are manipulated or processed into information that is valuable to people making decision. This processing and manipulating of raw data into meaningful information is the heart of quantitative technique. e.g., we can use quantitative technique to determine how much our investment will be worth in the future when deposited at a bank at a given interest rate for a certain number of years. Quantitative technique can also be used in computing financial ratio for the balance sheets for several companies whose stock we are considering. DEFINITION Different mathematician have given various definition of operation research. Operation research is a scientific method of producing executive departments with a quantitative basis for decision regarding the operation under their control -P.M. MORSE & G.E.KIMBALL Operation research is a act of giving bad answers to problems which otherwise have worse answers -T.S. SAATY Operation research is a scientific research an approach to problem solving for executive management -H.M.WAGNER

2. Introduction of Quantitative Technique 2 Operation research has been described as a method, an approach, set of techniques a team activity, a combination of many disciplines, an extension of particular disciplines (mathematics engineering economics), a new discipline, vocation, even a religion. It is perhaps some of all this things. -S.L.COOK THE QUANTITATIVE TECHNIQUE APPROACH CHARACTERISTICS OF QUANTITATIVE TECHNIQUE 1. Flexibility:-Models should be capable of adjustment with new formulas without having any significant changes in its frame. 2. Less no of variable: - The number of variable in a model should not be very large but at the time no variable selected to any important factors should be left ones. 3. Less time: - A model should not take much time in construction. ADVANTAGES OF QUANTITATIVE TECHNIQUES 1. It describes the problem much more concisely. 2. It provides systematic and logical approach to the problem. Acquiring Input Data Testing the Solution Analyzing the Results Implementing the Results Defining the problem Developing a Model Developing a Solution 3. Introduction of Quantitative Technique 3 3. It indicates the scope and limitation of the problem. 4. It enables the use of high power mathematical tools to analyze the problems. 5. It helps in finding avenues for new research and improvement in a system. TECHNIQUES A brief description of some of the commonly used techniques is given below. Details are covered in relevant chapters of this book. Some of the techniques which is covered in this book are:- LINEAR PROGRAMMING: It is an allocation technique where the objective function is linear and the constraints are modeled as linear inequalities. Example- Graphical Representation, Simplex Method etc. TRANSPORTATION MODEL: A special case of linear programming which matches sources of supply to destinations on cost or distance considerations. For example; movement of raw materials from different sources to manufacturing plants at different locations based on availability of raw materials at various sources, the requirements at different plants and the cost of transportation involved. ASSIGNMENT MODEL: A special case of transportation model where the aim is to assign a number of origins to the same number of destinations at a minimum total cost. For example; assigning of man/machines to same number of jobs /tasks. REPLACEMENT MODEL: These models deal with the formulation of appropriate replacement policies when some items have to be replaced due to obsolescence or wear and tear. Replacement models also deal with equipments and items which fail completely and instantaneously. QUEUING THEORY: It studies the random arrivals at servicing or processing facility of limited capacity. These models attempt to predict the behavior of waiting lines, i.e.; the time spend waiting for a service. The technique is descriptive and describes behavior that can be expected given certain parameters. It is not prescriptive in nature and does not offer an optimal solution. The models deal with the tradeoffs between cost of providing service and the value of time spent in waiting for a service. DECISION THEORY: Decision situations can be classified into deterministic or certainty, probabilistic or risk and uncertainty. Decision making under certainty can be dealt with by various optimization techniques. Decision theory deals largely with decision making under risk where the probabilities of certain conditions occurring (such as demand for an item) are predicted and various options assessed based on these probabilistic values. In situations of uncertainty there can be no specific approach. A set of decision rules can be applied and 4. Introduction of Quantitative Technique 4 insight gained into the decision makers style of functioning. This is particularly applicable to studying a competitors style of decision making and then predicting how he would react to a certain condition so as to gain advantage for oneself. GAME THEORY: This deals with decision making under conditions of competition. Its assumptions currently restrict its usage. APPLICATION & SCOPE OF QUANTITATIVE TECHNIQUES Some of the industrial, government, business problems which can be analyzed by or approach have been arranged by functional areas as follows: ----- Finance and accounting: Dividend policies, investment and portfolio management, auditing, balance sheet and cash flow analysis Break-even analysis, capital budgeting, cost allocation and control, and financial planning Claim and complaint procedure, and public accounting Establishing costs for by-products and developing standards costs Marketing: Selection of product-mix, marketing and export planning Best time to launch a new product Sales effort allocation and assignment Predicting customer loyalty Advertising, media planning, selection and effective packing alternatives Purchasing, procurement and exploration: Optimal buying and recording under price quantity discount Bidding policies Transportation planning Vendor analysis Replacement policies Production management: Facilities planning Location and size of warehouse or new plant, distribution centers and retail outlets Logistics, layout and engineering design Transportation, planning and scheduling 5. Introduction of Quantitative Technique 5 Manufacturing Aggregate production planning, assembly line, blending, purchasing and inventory control Employment, training, layoffs and quality control Allocating R&D budgets most effectively Maintenance and project scheduling Maintenance policies and preventive maintenance Maintenance crew size and scheduling Project scheduling and allocation of resources Personal management: Manpower planning, wage/salary administration Negotiation in a bargaining situation Designing organization structures more effectively Skills and wages balancing Scheduling of training programmes to maximize skill development and retention Techniques and general management: Decision support systems and MIS; forecasting Making quality control more effective Project management and strategic planning Government: Economic planning, natural resources, social planning and energy Urban and housing problems Militar, police, pollution, control, etc. Limitation of Quantitative Techniques 1. All the problems cannot be converted into numerical values. So it is not solving by Quantitative technique. 2. Not understandable to everyone who are not aware of the technique of Quantitative technique. 3. The some of the technique of the Quantitative technique are very complex in solving 4. Mathematical model: as most of the operation research techniques are mathematical in nature. It is therefore necessary to put the problem in the term 6. Introduction of Quantitative Technique 6 of mathematical model. In many situations it is not possible to represent the problem in mathematical form and hence operation research techniques cannot be applied. 5. Expensive: As application of operation research for the problem solving requires service of specialist and the use of computer. Therefore it is expensive to use operation research technique for solving day to day problems. 6. Imperfection of solution: By operation research technique we cannot obtain the perfect answer to our problems. But only the quality of the solution is improved from worse to bad answer. 7. The techniques used must be applicable to the problem and must reflect the purpose and scope of the problem. 7. Assignment Problems 7 ASSIGNMENT PROBLEMS INTRODUCTION In the world of trade Business Organizations are confronting the conflicting need for optimal utilization of their limited resources among competing activities. The course of action chosen will invariably lead to optimal or nearly optimal results. The assignment problem is a special case of transportation problem in which the objective is to assign a number of origins to the equal number of destinations at the minimum cost (or maximum profit). Assignment problem is one of the special cases of the transportation problem. It involves assignment of people to projects, jobs to machines, workers to jobs and teachers to classes etc., while minimizing the total assignment costs. One of the important characteristics of assignment problem is that only one job (or worker) is assigned to one machine (or project). Hence the number of sources are equal the number of destinations and each requirement and capacity value is exactly one unit. Although assignment problem can be solved using either the techniques of Linear Programming or the transportation method, the assignment method is much faster and efficient. This method was developed by D. Konig, a Hungarian mathematician and is therefore known as the Hungarian method of assignment problem. In order to use this method, one needs to know only the cost of making all the possible assignments. Each assignment problem has a matrix (table) associated with it. Normally, the objects (or people) one wishes to assign are expressed in rows, whereas the columns represent the tasks (or things) assigned to them. The number in the table would then be the costs associated with each particular assignment. It may be noted that the assignment problem is a variation of transportation problem with two characteristics.(i)the cost matrix is a square matrix, and (ii)the optimum solution for the problem would be such that there would be only one assignment in a row or column of the cost matrix . Application Areas of Assignment Problem Though assignment problem finds applicability in various diverse business situations, we discuss some of its main application areas: (i) In assigning machines to factory orders. (ii) In assigning sales/marketing people to sales territories. (iii) In assigning contracts to bidders by systematic bid-evaluation. (iv) In assigning teachers to classes. (v) In assigning accountants to accounts of the clients. 8. Assignment Problems 8 Format of Assignment Problems In assigning police vehicles to patrolling area. Job Persons j1 j2 ------ jn I1 X11 X12 ----- X1n I2 X21 X22 ----- X2n --- ----- ---- ----- ----- In ----- ---- ----- Xnn Cij is the cost of performing jth job by ith worker. Xij is the number ith individual assigned to jth job. Total cost = X11 * C11 + X12 * C12 + ----- + Xnn * Cnn. Solution Methods There are various ways to solve assignment problems. Certainly it can be formulated as a linear program, and the simplex method can be used to solve it. In addition, since it can be formulated as a network problem, the network simplex method may solve it quickly. However, sometimes the simplex method is inefficient for assignment problems (particularly problems with a high degree of degeneracy). The Hungarian Method used with a good deal of success on these problems and is summarized as follows. Step 1. Determine the cost table from the given problem. (i) If the no. of sources is equal to no. of destinations, go to step 3. (ii) If the no. of sources is not equal to the no. of destination, go to step 2. Step 2. Add a dummy source or dummy destination, so that the cost table becomes a square matrix. The cost entries of the dummy source/destinations are always zero. Step 3. Locate the smallest element in each row of the given cost matrix and then subtract the same from each element of the row. Step 4. In the reduced matrix obtained in the step 3, locate the smallest element of each column and then subtract the same from each element of that column. Each column and row now have at least one zero. Step 5. In the modified matrix obtained in the step 4, search for the optimal assignment as follows: 9. Assignment Problems 9 (a) Examine the rows successively until a row with a single zero is found. In rectangle this row () and cross off (X) all other zeros in its column. Continue in this manner until all the rows have been taken care of. (b) Repeat the procedure for each column of the reduced matrix. (c) If a row and/or column has two or more zeros and one cannot be chosen by inspection then assign arbitrary any one of these zeros and cross off all other zeros. Step 6. If the number of assignment () is equal to n (the order of the cost matrix), an optimum solution is reached. If the number of assignment is less than n(the order of the matrix), go to the next step. Step7. Draw the minimum number of horizontal and/or vertical lines to cover all the zeros of the reduced matrix. Step 8. Develop the new revised cost matrix as follows: (a) Find the smallest element of the reduced matrix not covered by any of the lines. (b) Subtract this element from all uncovered elements and add the same to all the elements laying at the intersection of any two lines. Step 9. Go to step 6 and repeat the procedure until an optimum solution is attained. 10. Assignment Problems 10 See diagrammatic Representation of Hungarian Approach 11. Assignment Problems 11 MINIMIZATION PROBLEM (BALANCED) Example 1: A Company has 5 machines and 5 jobs. The relevant cost matrix is given below: Find the assignment that minimizes the total cost: Machines Jobs J1 J2 J3 J4 J5 M1 10 4 5 3 11 M2 13 11 9 12 10 M3 12 3 10 1 9 M4 9 1 11 4 8 M5 8 6 7 3 10 , Solutions: Step 1: First we have to check that the given matrix is square matrix or not. Here the given matrix is square matrix. Step 2: Subtract least entry of each row from all the entries of that row. The first reduced cost matrix will be as given below: J1 J2 J3 J4 J5 M1 7 1 2 0 8 M2 4 2 0 3 1 M3 11 2 9 0 8 M4 8 0 10 3 7 M5 5 3 4 0 7 Then in the above matrix subtract least entry of each column from all entries of that column. The second reduced cost matrix or the total opportunity cost matrix will be as follows: J1 J2 J3 J4 J5 M1 3 1 2 0 7 M2 0 2 0 3 0 M3 7 2 9 0 7 M4 4 0 10 3 6 M5 1 3 4 0 6 12. Assignment Problems 12 Step3: For testing the optimality we draw minimum number of straight lines to cover all the zeros. Since three lines cover all the zeros, which is not equal to the matrix size (number of row= number of column number of draw line). Step4: Select the smallest entry from all the entries which are not covered by a straight line (here it is 1). Subtract this smallest entry from all the uncovered entries and add it to all those entries which are at the intersection of two lines and other covered entries remain unchanged. The revised reduced cost matrix is given below: J1 J2 J3 J4 J5 M1 2 0 1 0 6 M2 0 2 0 4 0 M3 6 1 8 0 6 M4 4 0 10 4 6 M5 0 2 3 0 5 Again we see that only four straight lines are required to cover all the zeros of the revised cost matrix, therefore optimal assignment cannot be made at this stage. Repeating the step 4 we get the following revised cost matrix: J1 J2 J3 J4 J5 M1 1 0 0 0 5 M2 0 3 0 5 0 M3 5 1 7 0 5 M4 3 0 9 4 5 M5 0 3 3 1 5 Since 5 straight lines (equal to the number of rows and columns) are required to cover all the zeros, optimal assignment can be made at this stage. It is given below: 13. Assignment Problems 13 J1 J2 J3 J4 J5 M1 1 0 0 5 M2 0 3 0 5 M3 5 1 7 5 M4 3 9 4 5 M5 3 3 1 5 Step5: Select one row containing exactly one zero and surrounded it by Here we select row no. 3. We can also select 4 or 5. Cut all the zeros of that column (If has). Machine1 Job3 Cost Rs 5 Machine2 Job5 Cost Rs 10 Machine3 Job4 Cost Rs 1 Machine4 Job2 Cost Rs 1 Machine5 Job1 Cost Rs 8 Total 25 Q.1) Unbalanced Minimization: A B C D E F I 17 25 11 08 16 31 II 23 13 44 16 19 17 III 32 19 31 28 12 25 IV 26 24 27 21 29 07 V 28 21 19 45 23 43 Ans. (STEP-I) The number of Column are less than number of Row, therefore we add one Dummy in Column with relative cost zero. The assignment problem is given below: 0 0 0 0 0 14. Assignment Problems 14 Balancing: A B C D E F I 17 25 11 08 16 31 II 23 13 44 16 19 17 III 32 19 31 28 12 25 IV 26 24 27 21 29 07 V 28 21 19 45 23 43 Dummy 0 0 0 0 0 0 (STEP-II) Subtract least entry of each Row from all the entries of that Row. The matrix will be as given below: Row minimization: A B C D E F I 09 17 03 00 08 23 II 10 00 31 03 07 04 III 20 07 19 16 00 13 IV 19 17 20 14 22 00 V 09 02 0 26 04 24 Dummy 00 00 00 00 00 00 (STEP-III) Subtract least entry of each Column from all the entries of that Column. The matrix will be as given below: Column minimization: A B C D E F I 09 17 03 00 08 23 II 10 00 31 03 07 04 III 20 07 19 16 00 13 IV 19 17 20 14 22 00 V 09 02 0 26 04 24 Dummy 00 00 00 00 00 00 15. Assignment Problems 15 (STEP-IV)Draw minimum line & covers maximum zero: A B C D E F I 09 17 03 00 08 23 II 10 00 31 03 07 04 III 20 07 19 16 00 13 IV 19 17 20 14 22 00 V 09 02 00 26 04 24 Dummy 00 00 00 00 00 00 (STEP-V) Make box to assign the particular jobs: A B C D E F I 09 17 03 08 23 II 10 31 03 07 04 III 20 07 19 16 13 IV 19 17 20 14 22 V 09 02 26 04 24 Dummy 00 00 00 00 00 00 I=D=08 II=B=13 III=E=12 IV=F=07 V=C=19 Dummy=A=00 Total= 59(Ans.) 00 00 00 00 00 16. Assignment Problems 16 Example 2: Company XYZ has 5 jobs. 5 people applied for those jobs. The company needs to find the minimum salary taken for those jobs. Expected salary of those jobs of person A, B, C, D, E is given below as matrix form: Person Jobs (in thousand) I II III IV V A 4 3 1 5 2 B 7 4 2 10 5 C 8 7 4 6 4 D 3 5 8 7 9 E 5 6 3 8 10 Condition a) A cannot get job 5. b) Job 3 get only C. c) E cannot get any job. Step 2: Since A cannot get job no. 5, therefore we have to put at job 5 of A. C get job no. 3, therefore we have to remove row of C and job no. 3 column from the matrix. E cannot get any job; therefore we have to remove all the row of E. It is shown in table below: Person Jobs I II IV V A 4 3 5 B 7 4 10 5 D 3 5 7 9 Step2: Here the matrix is unbalanced so we have to take dummy person as a row. New matrix as follows: Person Jobs I II IV V A 4 3 5 B 7 4 10 5 D 3 5 7 9 Dm 0 0 0 0 17. Assignment Problems 17 Row Minimization: Person Jobs I II IV V A 1 0 2 B 3 0 6 1 D 0 2 4 6 Dm 0 0 0 0 Column Minimization: Person Jobs I II IV V A 1 0 2 B 3 0 6 1 D 0 2 4 6 Dm 0 0 0 0 Step3: Select the smallest entry from all the entries which are not covered by a straight line (here it is 1). Subtract this smallest entry from all the uncovered entries and add it to all those entries which are at the intersection of two lines and other covered entries remain unchanged. The matrix is given below: Person Jobs I II IV V A 0 0 1 B 2 0 5 0 D 0 3 4 6 Dm 0 0 0 0 Here no. of line = matrix size, therefore optimal assignment can be made at this stage. Person Jobs I II IV V A 0 0 1 B 2 0 5 0 D 0 3 4 6 Dm 0 0 0 0 18. Assignment Problems 18 The solution is: A Job2 = 3 B Job5 = 5 C Job3 = 4 D Job1 = 3 Dm Job2 =0 Total = 15 MAXIMIZATION Maximization Case Question: Four sales men are to be assigning to four sales territories (1 to each). Estimates of the sales revenues in thousand of rupees for each sales man are as under: Sales territories Salesman T1 T2 T3 T4 S1 25 38 43 20 S2 45 12 19 4 S3 43 16 29 24 S4 9 40 45 44 You are required: To obtain the optimal assignment pattern that maximizes our sales revenue. Solution: Step 1- As given matrix gives revenues which are to be maximized. In order to use minimization technique obtain relative loss matrix by subtracting all the revenues from maximum revenue i.e. 45 Relative loss matrix 20 7 2 25 0 33 26 41 2 29 16 21 36 5 0 1 19. Assignment Problems 19 Step 2- We have to check that the given matrix is square or not. Here the matrix is square. Step 3- Subtract least entry of each row from all entries of that row to obtain row reduce matrix. It is given below: 18 5 0 23 0 33 26 41 0 27 14 19 36 5 0 1 Step 4- In the above matrix subtract least entry of each column from all the entries of that column to obtain total reduce matrix. Try to cover all the zeros of this matrix by using minimum number of lines. It is shown below- 18 0 0 22 0 28 26 40 0 22 14 18 36 0 0 0 Since the number of covering lines is 3 which is less than size of the matrix optimal solution cannot be obtained. Subtract minimum uncovered element (14 in this case) from all the uncovered entries and add it to all those entries which are at the intersection of two lines. Draw minimum number of lines to cover all the zeros of this new matrix. 32 0 0 22 0 14 12 26 0 8 0 4 50 0 0 0 Since number of covering lines is 4, which is equal to size of the matrix. Optimal assignment can be made. It is given in the following table- 20. Assignment Problems 20 32 0 0 22 0 14 12 26 0 8 0 4 50 0 0 0 Thus optimal assignment is as follows: Salesman Sales Territory Sales revenue S1 T2 38000 S2 T1 45000 S3 T3 29000 S4 T4 44000 1,56,000 MAXIMIZATION UNBALANCED Ques.1. There is 5 jobs for product selling and only 4 executives applied for it. You have to find maximum selling out of them and assign the job. A B C D E I 26 33 14 53 27 II 37 17 22 08 11 III 55 13 24 41 12 IV 42 38 32 27 49 Sol. Here we have to find maximum profit so we have to reduce all the values from the maximum value. For maximization to minimization. So the new table is A B C D E I 29 22 41 02 28 II 18 38 33 47 44 III 00 42 31 14 43 21. Assignment Problems 21 IV 13 17 23 28 06 This is not a square matrix so we have to balance it with the help of dummy row. A B C D E I 29 22 41 02 28 II 18 38 33 47 44 III 00 42 31 14 43 IV 13 17 23 28 06 Dm 00 00 00 00 00 Row minimization A B C D E I 27 20 39 00 26 II 00 20 15 29 26 III 00 42 31 14 43 IV 07 11 17 22 00 Dm 00 00 00 00 00 Colum minimization A B C D E I 27 20 39 00 26 II 00 20 15 29 26 III 00 42 31 14 43 IV 07 11 17 22 00 Dm 00 00 00 00 00 22. Assignment Problems 22 Draw minimum line for covers maximum zeros A B C D E I 27 20 39 00 26 II 00 20 15 29 26 III 00 42 31 14 43 IV 07 11 17 22 00 Dm 00 00 00 00 00 Numbers of lines is not equal to numbers of row or Colum. Then we have to find least number(11) of free numbers and subtract it from free numbers and add with there when lines are intersects and dont do any things where lines are pass. And again draw the minimum lines which covers maximum zeros. Free numbers are those which are not in lines. A B C D E I 27 09 28 00 26 II 00 09 04 29 26 III 00 31 20 14 43 IV 07 00 06 22 00 Dm 11 00 00 11 11 Number of lines is not equals to row or column then we have to apply the same step. A B C D E I 27 05 24 00 22 II 00 05 00 29 22 III 00 27 16 14 39 IV 11 00 06 26 00 Dm 15 00 00 15 11 23. Assignment Problems 23 Candidate I assign the job D, II assign the job C, III Assign the job A, IV assign the job E and Dm will be assign the job B. So the minimum salaries which is given to employees is First option is I*D= 53 II*C= 37 III*A= 55 IV*E= 49 Dm*B= 0 194 So maximum sell is 194 Ans. 24. Assignment Problems 24 MULTIPLE CONDITIONED QUESTIONS If there is any unbalanced, multiple conditioned question, first we will minimize it (if question is maximization) then we will balance the question. After that fulfill the conditions and if needed balance the question again. Then after follow the general procedure. Q.1) Profit earned by different sales man in different territories are as follows:- TERRITORY SALES MAN T1 T2 T3 T4 T5 T6 A 31 16 14 13 15 30 B 25 19 18 17 19 26 C 38 17 22 21 23 22 D 15 22 26 25 27 18 E 14 23 30 29 31 14 CONDITIONS 1. C has to be assigned job T5 2. E should not get T2 & T6 3. D should not get any job SOLUTION Minimization and applying conditions. STEP-I TERRITORY SALES MAN T1 T2 T3 T4 T5 T6 A 31 16 14 13 15 30 B 25 19 18 17 19 26 C 38 17 22 21 23 22 D 15 22 26 25 27 18 E 14 30 29 31 25. Assignment Problems 25 STEP-II Balancing the problem TERRITORY SALES MAN T1 T2 T3 T4 T6 A 7 22 24 25 8 B 13 19 20 21 12 E 24 8 9 D1 0 0 0 0 0 D2 0 0 0 0 0 STEP-III Row minimization TERRITORY SALES MAN T1 T2 T3 T4 T6 A 0 15 17 18 1 B 1 7 8 9 0 E 16 0 1 D1 0 0 0 0 0 D2 0 0 0 0 0 STEP-IV Column minimization TERRITORY SALES MAN T1 T2 T3 T4 T6 A 0 15 17 18 1 B 1 7 8 9 0 E 16 0 1 D1 0 0 0 0 0 D2 0 0 0 0 0 26. Assignment Problems 26 STEP-V Assigning the job. SALES MAN TERRITORIES SALES PROFIT A T1 31 B T6 26 C T5 23 D NO JOB 00 E T3 30 D1 T2/T4 00 EXERCISES MINIMIZATION PROBLEMS 1. A company has 5 machines and 5 jobs. The revelant cost is given below Find the assignment that minimizes total cost. Machines JOBS J1 J2 J3 J4 J5 M1 10 4 5 3 11 M2 13 11 9 12 10 M3 12 3 10 1 9 M4 9 1 11 4 8 M5 8 6 7 3 10 Ans. Machine 1 JOB 3 COST RS.3 Machine 2 JOB 5 COST RS.1O Machine 3 JOB 4 COST RS.1 Machine 4 JOB 2 COST RS. 1 Machine 5 JOB 1 COST RS. 8 TOTAL. 25 27. Assignment Problems 27 2. A car hire company has one car at each of the five depots a,b,c,d,e. A customer requires car at each town vit. A,B,C,D and E. Distance (in kms) between deposits (origins) and towns (destination) are given in the following distance matrix. How should the cars be assigned to customers so as to minimize the total distance travelled. A B C D E A 160 130 175 190 200 B 135 120 130 160 175 C 140 110 155 170 185 D 50 50 80 80 110 E 55 35 70 80 105 Ans. A * e =200 km B * c =130 km C * b =110 km D * a =50 km E * d =80 km Total = 570 km MAXIMIZATION PROBLEMS 1. A marketing manager has 5 sales man & 5 sales area, considering the capabilities of the salesmen nature of areas, the maraketing manager estimates that sales per month (in thousands of rupees) for salesman in each area would be as follow: Salesman AREAS A1 A2 A3 A4 A5 S1 42 48 50 38 50 S2 50 34 38 31 36 S3 51 37 43 40 47 S4 32 48 51 46 46 S5 39 43 50 45 49 28. Assignment Problems 28 Final Optimal Assignment Ans. TOTAL COST 241000 S1=A2 S1=A2 S2=A1 S2=A5 S3=A5 OR S3=A1 S4=A3 S4=A4 S5=A4 S5=A5 < 2. Quantity of clothes of different brands sold in different cities per mnth are as follows Wrangler Pantaloon Pvogue Lee Levis Delhi 25 22 55 42 48 Gurgaon 78 41 40 46 41 Noida 18 33 52 50 37 Jaipur 32 18 30 37 20 Chandigarh 40 20 42 48 51 Find the showroom for different brands which should be opened in different cities. Ans = 254 Unbalance Assignment problem (Minimization) Q. 1) A company is face with the problem of assigning six different machines to Five different job. The cost are estimated as follows (in hundred of Rs.) Solve the problem assuming that the objective is to minimize the total cost. A1 A2 A3 A4 A5 M1 05 10 02 12 02 M2 04 10 03 14 06 M3 06 13 04 16 06 M4 07 14 04 18 09 M5 08 14 06 18 12 M6 12 18 10 20 12 29. Assignment Problems 29 Ans. Machine Cost Machine Cost M1 A5 02 M1 A5 02 M2 A1 04 M2 A4 14 M3 A4 16 M3 A1 06 M4 A3 04 M4 A3 04 M5 A2 14 M5 A2 14 M6 A6 00 M6 A6 00 Total=40 Total=40 Q. 2) Solve the following unbalance Assignment problem of the minimizing total time for doing all jobs. Jobs J1 J2 J3 J4 J5 O1 16 12 15 12 16 O2 12 15 18 17 17 O3 17 18 16 19 18 O4 16 12 13 14 15 O5 19 13 18 19 17 O6 14 17 14 16 18 Ans: Operation Job Time O1 J4 12 O2 J1 12 O3 J6 00 O4 J5 15 O5 J2 13 O6 J3 14 Total=66 30. Assignment Problems 30 Unbalance Assignment problem(Maximization) Q.1) Four different Airplanes are to be assigned to handle three cargo consignment with a view to maximize profit . The profit matrix is a follow in thousand of Rupees Cargo Consignment Airplane I II III W 08 11 12 X 09 10 10 Y 10 10 10 Z 12 08 09 Ans: W to III profit 12,000 W to III profit 12,000 X to II profit 10,000 X to Dummy profit 0 Y to Dummy profit 00 Y to II profit 10,000 Z to I profit 12,000 Z to I profit 12,000 Total=34,000 Total=34,000 Q.2) Solve the following assignment problem. The data give him the table refer to production in central unit. Machine A B C D I 10 05 07 08 II 11 04 09 10 III 08 04 09 07 IV 07 05 06 04 V 08 09 07 05 31. Assignment Problems 31 Ans. Operation Machine No. of Units 1 A 10 2 D 10 3 C 9 4 Dummy 0 5 B 9 Unit=38 CONDITION EXERCISE- Solve the following assignment problem in which 5 jobs and 7 peoples are given and some conditions are considered when jobs assign, the conditions are- Job 3 has to be assigned to B. Job 1 and 4 can not be assigned to G. D has to get job 5 1 2 3 4 5 A 36 33 25 30 26 B 42 25 35 26 45 C 54 50 52 35 28 D 28 35 40 28 36 E 20 28 32 40 52 F 26 30 48 25 27 G 24 28 22 26 30 ANSWER- 128 32. Transportation Problem 32 TRANSPORTATION PROBLEMS INTRODUCTION The Transportation model deals with situations where some commodity or product is distributed from multiple sources to multiple destinations. The model may be used to find the minimum transportation cost or the maximum profit, depending upon the amounts shipped from each source to each destination. The problem solution will be the optimum distribution scheme, showing exactly how much of the commodity should be transported via each possible route. The transportation algorithm discussed in this chapter is applied to minimize the total cost of transporting a homogeneous commodity from supply origins to demand destinations. However, it can also be applied to the maximization of some total value or utility, for example, financial resources are distributed in such a way that the profitable return is maximized. ADVANTAGES Proper utilization of resources take place without much of the losses. The selections and allocations of resources to their destinations become more accurate. This process helps in cost minimization and profit maximization, which major objective of organizations. It helps in planning and decision making. Some Important Terms or Definitions 1) Feasible solution-: Set of Non Negative values xij=1,2,3,4.m,j=1,2,3,4.n. which satisfy the following condition is called feasible solution. 2) Basic feasible solution-: a feasible solution with an allocation of (m+n-1) number of variables . xij,i=1,2,3..m, j=1,2,3,4.n. is called a basic feasible solution. 3) Optimum Solution-: A basic feasible solution of transportation problem which minimizes the total transportation cost or maximizes total revenue. 4) Rim requirement-: The quantity required or available are called rim requirement. 5) Balanced transportation problem-: If in any transportation problem total number of units available is equal to the total number of units required, then it is called balanced transportation problem. 33. Transportation Problem 33 STEPS OR PROCEDURE OF SOLVING TRANSPORTATION PROBLEMS Make Empty Cells(4) YES NO NO Calculate Value ofRi and Cj (5) Xij =TCij (Ri + Cj) YES Is Xij