BIRLA INSTITUTE OF TECHNOLOGY MESRA, RANCHI

Department of Management(MBA Semester II)

Q1.Discuss the role and scope of quantitative methods for scientific decision-making in a business environment?Ans.Quantitative methods are always important in making important business decisions. Therefore, there are various quantitative subjects which have been introduced. For example, Total Quality Management and Quantitative methods and Techniques are two subjects which have a lot of quantitative methods to solve various problems and facilitate decision making process. For example Regression analysis can help the company to overview the previous trends in the sales of the company and decide the budgeted sales for the new year.

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FeaturesInterdisciplinary approach: Interdisciplinary is essential because while attempting to solve a complex management problem one person may not have the complete knowledge of all its aspects such as economical, social, political.Methodological approach: Operations Research is the application of scientific methods, techniques and tools to problems involving the operations of systems so as to provide those in control of operations research with optimum solutions to the problem.Holistic Approach: While arriving at a decision, an operations research team examines the relative importance of all conflicting and multiple objectives and the validity of claims of various departments of the organization from the perspective of whole organization .Objectivistic approach: An Operations Research approach seeks to obtain an optimal solution to the under analysis for these, measure of desirability is defined, based on the objective of the Organization

Application and scope of Operations researchSome of the industrial/government/business problems which can be analysed by OR Approach have arranged by functional areas as followsFinance and Accounting Dividend Policies Investment and portfolio management Auditing Balance sheet Cash flow Analysis Claim and complaint procedure and Public accounting Break even analysis, capital Budgeting, cost allocation and control, financial planning Establishing cost for By products and developing standard cost

Marketing Selection of Productmix Marketing and Export Planning Advertising, Media Planning, selection and effective Packing Alternatives Sales effort allocation and assignment Best time to launch a new product Predicting customer Loyalty

Purchasing, Procurement and Exploration Optimal Buying and reordering with or without price quantity discount Transportation Planning Replacement polices Bidding Policies Vendor AnalysisProduction management Facilities Planning Location and Size of warehouse or new plant Distribution centres and retail outlets

Manufacturing Aggregate Production Plaiing Assembly Line Blending Purchasing and Inventory control Maintenance and project schedulingPersonnel Management Selection of suitable personnel. Recruitment of employees. Assignment of jobs. Skills balancing. Research and Development Project selection. Control of R&D projects.Government Economic Planning Natural resources Social Planning Energy

Q2. Discuss advantages and limitation of Operations Research?Ans.Advantages Better Control:The management of large organizations recognize that it is a difficult and costly affair to provide continuous executive supervision to every routine work. An O.R. approach may provide the executive with an analytical and quantitative basis to identify the problem area. The most frequently adopted applications in this category deal with production scheduling and inventory replenishment. Better Systems:Often, an O.R. approach is initiated to analyze a particular problem of decision making such as best location for factories, whether to open a new warehouse, etc. It also helps in selecting economical means of transportation, jobs sequencing, production scheduling, replacement of old machinery, etc. Better Decisions:O.R. models help in improved decision making and reduce the risk of making erroneous decisions. O.R. approach gives the executive an improved insight into how he makes his decisions. Better Co-ordination:An operations-research-oriented planning model helps in co-ordinating different divisions of a company.

Limitations Dependence on an Electronic Computer:O.R. techniques try to find out an optimal solution taking into account all the factors. In the modern society, these factors are enormous and expressing them in quantity and establishing relationships among these require voluminous calculations that can only be handled by computers. Non-Quantifiable Factors:O.R. techniques provide a solution only when all the elements related to a problem can be quantified. All relevant variables do not lend themselves to quantification. Factors that cannot be quantified find no place in O.R. models. Distance between Manager and Operations Researcher:O.R. being specialist's job requires a mathematician or a statistician, who might not be aware of the business problems. Similarly, a manager fails to understand the complex working of O.R. Thus, there is a gap between the two. Money and Time Costs:When the basic data are subjected to frequent changes, incorporating them into the O.R. models is a costly affair. Moreover, a fairly good solution at present may be more desirable than a perfect O.R. solution available after sometime. Implementation:Implementation of decisions is a delicate task. It must take into account the complexities of human relations and behaviour.Opportunities It compels the decision maker to quit explicit about his objectives assumption and his prospective to constraints It makes the decision maker consider very carefully just what variables influence decisions Quickly points out gaps in the data required to support workable solutions to a problem Its model can be solved by a computer

3. It is said that operations research increases the creative capabilities of a decision maker. Do you agree with this view? Defend your point of view with examples.Ans. The main responsibilities of operations management are to manage and operate as efficiently and effectively as possible with the given resources. With today's global market, and large-scale systems, achieving the optimum performance is a challenge. Many decision science tools are available for all levels of decision makers. Quantitative methods such as Operations Research (OR), which comprises of simulation, linear and nonlinear programming, queueing theory and stochastic modeling, are well-accepted techniques by both research and practice communities. Large profit organization such as Ford Motors (Chelst, Sidelko, Przebienda, Lockledge, & Mihailidis, 2001), Merrill Lynch (Atschuler et al., 2000), AT&T (Ambs et al., 2000) andJournal of Applied Business and EconomicsSamsung (Leachman, Kang, & Lin, 2002) reported millions dollars of savings with OR. OR has a strong presence in nonprofit organizations as well. US Army Recruiting Office (Knowles, Parlier, Hoscheit, & Ayer, 2002) and Warner Robins Air Logistics Center (Srinivasan 2006) won the worlds most prestigious award, 2006 Franz Eldelman Award for outstanding achievements of OR.

Functional entities such as Industrial or Systems Engineering uses both methodologies to provide feasible alternatives for operations mangers to decide on. An important component of decision-making process is verifying and validating alternatives, which typically involve decision makers and engineers or analysts. Thus, high-level understanding of the tools or methodologies used for recommendations is essential in making the effective decision for achieving the organizations common goal of maximizing profit. In the following sections a brief background on Operations Research is discussed along with selected OR techniques including Linear Programming, Discrete Event Simulation and Queueing Theory, and a typical problem solving procedures for OR. Also, Success cases of OR from both practical and research perspectives are discussed. The paper is concluded with a brief summary of the materials discussed. The primary motivation and purpose of this literature is to disseminate knowledge; hence, neither academic nor research disclosures are conversed.

OPERATIONS RESEARCH

During World War II, a set of diversified scientists from England and the United States developed scientific methods of planning military logistics such as most economical method of disseminating resources to various war sites. The scientists developed a famous quantitative method for such operations and named it Operations Research (OR) or often referred as managerial or decision science (Hillier, 2005; Turner, Mize, Case, & Nazemetz, 1993). With its proven successes, OR spread to private sectors promptly. With rapid improvements in computer technology, to this date, OR is one of the most powerful decision making tools in the Operations Management and Industrial Engineering disciplines. Murty defines Operations Research as a discipline that deals with techniques for system optimization (1993). ORs primary objective is to find optimal or near optimal solution to complex business to engineering problems.

Operations Research techniques are used to answer the common managerial questions such as: How many and much resources are required to meet the key performance target? Which alternatives require minimum cost and generate maximum profit? What is the optimal resource schedule to minimize overhead cost? What is the maximum and minimal resource utilization level? Where are primary and secondary constraints or bottlenecks? What range of queue and process time is allowed to achieve goal? What is the current capacity and required capacity to meet the goal? What are the anticipated risks for accepting or making new product or model?

OR evaluates the system of interest globally. It considers all factors or all factors identified the decision makers. For example, to maximize or optimize production performance, characteristics of each or management identified areas such as rate of production per work station, rate of material usage per product or program, average time spent in transition and available capacity in terms of space are considered in the OR models. Viewing such system with the philosophy of local efficiency, the management will be only interested in maximizing production performance at a workstation, not entire work stations. Numerous OR techniques areavailable to use; however, the scope of this paper is limited to the following commonly used techniques: Linear (LP), Discrete Event Simulation (DES) and Queueing Theory (QT). Combination of such techniques in optimization studies, especially in practice. Discussions of LP, DES and QT techniques are in the following sections.

Linear Programming

Linear programming techniques are considered as mathematics based decision-making tool. Such technique requires two fundamental types of functions, objective and constraints, that is developed to generate closed-form solution. In a typical OR problem, the objective function, often expressed as Z, is formulated to determine the maximum profit while minimizing cost with given set of rules or constraints such as business policies, resource availability, preventative maintenance schedule, transportation distance or time and capacity.

Consider a bank that is expecting an increase of customers due to the new investment programs, which recently were introduced to the customers. Thus, the operations management team has a task to determine the number of resources, tellers specifically, to obtain to sustain the current customer satisfactory level for its 24-hours operations with minimal cost. The following set of data was requested by and provided to a cross-functional team responsible for the analysis. This is a modified version of Hillers (2005) the Union Airways Personnel Scheduling Problem.

Based on the problem description given by the management, the goal of this study is to minimize total daily personnel cost. Following is the mathematical expression of this problem statement with a brief description of each function.

Minimize Z = 160 X1 + 176 X2 + 192 X3 + 208 X4 + 224 X5 (Sum of daily cost)Subject to

X1 64 (8:00 A.M. 12:00 P.M.)X1 + X2 111 (12:00 P.M. 4:00 P.M.)X2 + X3 180 (4:00 P.M. 8:00 P.M.)X3+ X4 78 (8:00 P.M. 12:00 A.M.)X4 + X5 19 (12:00 A.M. 4:00 A.M.)X5 25 (4:00 A.M. 8:00 A.M.)andXi 0, for i = 1, 2, 3, 4, 5WhereXi = number of tellers assigned to shift iUsing the OR software program called, LINDO for computation, the following set of optimal solution is found: Z = 48768, (X1, X2, X3, X4, X5) = (64, 102, 78, 0, 25). The minimum operating cost to sustain the customer satisfactory level is $48,768 which includes the requirement of recruiting 64, 102, 78, 0 and 25 tellers for 1st, 2nd, 3rd, 4th and 5th shift, respectively. Note that the 4th shift requires no additional tellers since the time slot of 8 P.M to 12:00 A.M. and 12 A.M. to 4 A.M. for 3rd and 5th shift, respectively, cover the time slot of 4th shift, 8:00 P.M. to 4:00 A.M.

As mentioned above, application of LP is abundant. LP models reality with determined, expected or most-likely values. In contrast, Stochastic Programming (SP) uses law of probability to model the randomness of reality. Though Stochastic Programming may represent the reality more closely, but the solution generated may or may not be more accurate than LP generated solution (Murty, 1995). Accuracy of solutions depends on the accurate representation of the accurately collected past data. LP solution may be more accurate, if the data is approximated more realistically than SP, vice versa. SP models are more complicated and time consuming than LP models since SP tries to model all possible or variable events. LP models are less time consuming than SP due to its deterministic nature. LP, however, allows decision makers to perform sensitivity analysis by creating multiple variable scenarios to see the more accurate insights of the system. Choice of modeling techniques depends on the complexity of the problem and economic constraints. As a decision maker, knowing the difference of the modeling techniques allows confidence in making the economically feasible and effective decisions.

Simulation

A simple definition of simulation is an imitation or mimic of a system. There are two main types of simulation modeling techniques; Discrete Event and Continuous. Discrete event simulation (DES) is an event driven simulation. In other words, DES models chorological sequence of independent events. Referring to the bank example, arrival of a call, agent answering the call, system dropping a call or customer abandoning a call is discrete events. In opposition, continuous simulation models continual events where events change continuously not in increment. Examples for continuous systems include chemical or fluid flows, stock market prices, conveyors to move parts, etc.Simulation is an excellent communication tool. Unlike the Linear Programming or Stochastic Programming, simulation graphically represents a system along with visible display of relative numerical results. In a typical facility layout studies, decision makers will be provided with a set of graphical representation of potential layouts for the facility with people or parts moving through the system. A typical simulation of paperwork process will have process maps or flow charts as simulation layout and graphically show the different paperwork moving through the system.

As in LP and SP, DES uses estimated data; however, it does not find optimal solutions. Instead of finding the most achievable production rate with a given resources, DES finds expected or average production rate. When setting a goal for a system, LP or Stochastic Programming is ideal; however, simulation is more suitable for measuring on-going production performance. Almost all simulation software packages have accompanying optimization software; SimRunner for ProModel and OptQuest for Arena. Certain decision science skills are required for simulation modeling. Computer programming skill is required to build a simulation model using a software package. To optimize the model, basic optimization techniques are required to interpret results. As simulation modeling requires some level of computer programming skills, simulation optimization packages such as SimRunner and OptQuest also require optimization techniques. Awareness and basic knowledge of such optimization techniques along with simulation can play a positive role in making more realistic and economically feasible solutions.

Queueing Theory

Waiting is a simple definition of queue. Bank has a queue of customers for service, parts are in queue to be processed, grocery shoppers are in queue to pay, etc. Queueing or Queuing Theory (QT) is a mathematical or statistical study of waiting lines. A typical queueing system has three relative processes defined as arriving, waiting and servicing. With a set of common assumptions such as empty and full system, QT measures and reports performance of the system with the indicators such as average time a customer is expected to wait, probability of a customer waits more than certain hours or minutes in the queue, number of customers receiving services, etc.

QT uses Kendalls notations, A/B/C, where A, B, C represent arrival process, service process and number of servers, respectively. A and B can have Markov (M), Deterministic (D) or G (General). For example, M/M/1, which is the simplest queueing system, is interpreted as a system with arrival and service pattern of exponential or Markovian probability with one server. QT analyzes a system with Littles Theorem, N = T, where N = average number of customers in the system, = average arrival rate of customers and T = average service time. Based on Littles Theorem, different queueing systems have different mathematical formulas for calculation. Like Linear Programming, Stochastic Programming and Discrete Event Simulation, Queueing Theory also has a challenge of modeling reality accurately, which is common difficulty for mathematically restrictive approaches.

OPERATIONS RESEARCH METHODOLOGY

Depending on the complication of problem, the common business questions can be answered using qualitative tools such as fishbone diagrams, value stream maps and more. However, complex problems such as large-scale systems optimization problems, quantitative techniques combined with qualitative approaches are recommended and is common practice. A general procedure for solving Operations Research problems is as follows:Step 1 Define the problemOperations manager along with cross-functional team must define the problem. The problem definition should include symptoms and the systems objective.Step 2 Identify the decision variables and collect relevant dataDecision variables are parameters that can be controlled and affects performance and is often identified the management. In the bank example, a number of tellers per shift is a decision variable. Depending on the size of the problem and in reality, multiple decision variables involved in optimization studies.Step 3 Formulate a mathematical model of the system operations and goal or objectiveFunctional group such as Operations Analyst, Industrial and/or Systems Engineering are responsible for formulating the model. The mathematical model should read: maximize profit or minimize cost subject to set of parameters. In our bank example, the mathematical model is built to minimize cost based on the number of tellers assigned per shift.Step 4 Solve the model for an optimum solution and alternativesThough manual computation of optimization problems is possible, however, such a method is not efficient and unrealistic especially for the real-world problems that have multiple objectives and decision variables. In large problems, the computation can take more than 24hours easily. Thus, Software programs such as LINDO API, LINGO, ILOG and Microsoft Office Excel Solver, are used to solve real world problems which often is large and complex.Step 5 Perform sensitivity analysesIn this step, active participation from operations managers is required. Management presents series of what-if questions and series of sensitivity analyses are performed using the model by the modelers. In the bank example above, management can inquire about the level of impact on cost if more or less tellers are assigned per shift.Step 6 Update the model based on the managements prescription and decisionsAfter reviewing the results of various what-if scenarios, management needs to make a decision(s) then the optimization model is updated accordingly.Step 7 ImplementThis is one of the most important steps where the decisions determined by the model are implemented. Hence, active participation from the management is crucial in achieving the goal.

OPTIMIZATION IN PRACTICE

OR has strong presence in industries such as financial planning, health care, telecommunication, military, manufacturing and public services (Hillier, 2005). Ford Motors used OR for new design verification and reported annual saving of $250 million. The worlds largest manufacturer of digital integrated circuits, Samsung reduced production cycle time from more than 80 days to less than 30 days capturing additional $1 billion in sales revenue (Leachman et al., 2002). Merrill Lynch, brokerage and lending service provider, the Management Science Group developed optimization models to seize marketplace and reported savings of $80 million (Altschuler et al., 2002). In 2001, Continental Airlines reported savings of $40 million for major disruptions and leading five airlines in recovering operations after September 11 terrorist attack (Yu, Arguello, Song, & McCowan, 2003).

Nonprofit organizations such as The US Army reported savings of $204 million from a $1 billion recruiting program (Knowles et al., 2002). Warner Robins Air Logistics Center received the 2006 Franz Eldelman Award for its outstanding OR practice adding the centers annual revenue of $49.8 million (Srinivasan 2006). In addition to these companies, organizations including, but not limited to: GM, Athens 2004 Olympic games Organizing Committee, IBM, Motorola, Phillips, Waste Management, UPS, Texas Childrens Hospital, GE, Hewitt Packard, National Car Rental Systems, Harris Corporation and Proctor and Gamble have publicly disclosed significant achievements using OR techniques.

Q4. What is linear programming? What are its major assumptions and limitations?Ans.Linear programming models consist of an objective function and the constraints on that function. A linear programming model takes the following form:Objective function:Z = a1X1 + a2X2 + a3X3 + . . . + anXnConstraints:b11X1 + b12X2 + b13X3 + . . . + b1nXn < c1b21X1 + b22X2 + b23X3 + . . . + b2nXn < c2 . . bm1X1 + bm2X2 + bm3X3 + . . . + bmnXn < cmIn this system of linear equations, Z is the objective function value that is being optimized, Xi are the decision variables whose optimal values are to be found, and ai, bij, and ci are constants derived from the specifics of the problem.Linear Programming Assumptions The constraints and objective function arelinear. This requires that the value of the objective function and the response of each resource expressed by the constraints isproportionalto the level of each activity expressed in the variables. Linearity also requires that the effects of the value of each variable on the values of the objective function and the constraints areadditive. In other words, there can be no interactions between the effects of different activities; i.e., the level of activityX1should not affect the costs or benefits associated with the level of activityX2. Divisibility-- the values of decision variables can be fractions. Sometimes these values only make sense if they are integers; then we need an extension of linear programming called integer programming.

Certainty-- the model assumes that the responses to the values of the variables are exactly equal to the responses represented by the coefficients.

Data-- formulating a linear program to solve a problem assumes that data are available to specify the problemLimitations of Linear ProgrammingLinear programming is applicable only to problems where the constraints and objective function are linear i.e., where they can be expressed as equations which represent straight lines. In real life situations, when constraints or objective functions are not linear, this technique cannot be used. Factors such as uncertainty, weather conditions etc. are not taken into consideration.

There may not be an integer as the solution, e.g., the number of men required may be a fraction and the nearest integer may not be the optimal solution.

i.e., Linear programming techqnique may give practical valued answer which is not desirable.

Only one single objective is dealt with while in real life situations, problems come with multi-objectives.

Parameters are assumed to be constants but in reality they may not be so.

Q5. Linear programming is one of the most frequently and successful applied operations research technique to managerial decision. Elucidate this statement with some examples.

Ans.INTRODUCTION TO LINEAR PROGRAMMINGA Linear Programmingmodelseeks to maximize or minimize a linear function, subject to a set of linear constraints. The linearmodelconsists of the following components: A set ofdecision variables, xj. An objective function, cj xj. A set of constraints, aij xj 0, then the corresponding variable enters the basis.

Step-7: Compute the ratio {X B / Entering column} and choose the minimum of these ratios. The row which is corresponding to this minimum ratio is called leaving row. The common element which is in both entering column and leaving row is known as the leading element or key element or pivotal element of the table.Step-8: Convert the key element to unity by dividing its row by the leading element itself and all other elements in its column to zeros by using elementary row transformations.Step-9: Go to step-5 and repeat the computational procedure until either an optimum solution is obtained or there is an indication of an unbounded solution.Q10. What do you mean by an optimal basic feasible solution to a linear programming problem?Ans.The basic feasible solution which optimizes (maximizes or minimizes) the objective function value of the given LP problem is called an optimum basic feasible solution. The terms basic solution and basic feasible solution are very important parts of the standard vocabulary of linear programming, we now need to clarify their algebraic properties. For the augmented form of the example, notice that the system of functional constraints has 5 variables and 3 equations, so Number of variables _ number of equations _ 5 _ 3 _ 2.This fact gives us 2 degrees of freedom in solving the system, since any two variables can be chosen to be set equal to any arbitrary value in order to solve the three equations in terms of the remaining three variables.1 The simplex method uses zero for this arbitrary value. Thus, two of the variables (called the nonbasic variables) are set equal to zero, and then the simultaneous solution of the three equations for the other three variables (called the basic variables) is a basic solution. These properties are described in the following general definitions.A basic solution has the following properties:1. Each variable is designated as either a nonbasic variable or a basic variable.2. The number of basic variables equals the number of functional constraints (now equations).Therefore, the number of nonbasic variables equals the total number of variablesminus the number of functional constraints.3. The nonbasic variables are set equal to zero.4. The values of the basic variables are obtained as the simultaneous solution of the systemof equations (functional constraints in augmented form). (The set of basic variablesis often referred to as the basis.)5. If the basic variables satisfy the nonnegativity constraints, the basic solution is a BFsolution.xj _ 0, for j _ 1, 2, . . . , 5.

EXTREME POINTS AND OPTIMAL SOLUTIONS MULTIPLE OPTIMAL SOLUTIONSThere may be more than one optimal solutions. However, the condition is that the objective function must be parallel to one of the constraints. If a weightage average of different optimal solutions is obtained, it is also an optimal solution.

Q11.Define the dual of a linear programming problem. State the functional properties of duality.Ans.The term `dual in general sense implies two or double. But in the context of linear programming, duality implies that each linear programming problem can be analyzed in two different ways but having equivalent solutions. Each linear programming problem (both maximization and minimization) stated in its original form has associated with another linear programming problem called dual linear programming problem or in short dual. Which is unique, based on the same data. In general, it is immaterial which of the two problems is called primal or dual, since the dual of the dual primal.For example, if the primal is concerned with maximizing the contribution from the three products A, B, and C and from the three departments X, Y, and Z, then the dual will be concerned with minimizing the costs associated with the time used in the three departments to produce those three products. An optimal solution to the dual problem provides aShadow Priceof the time spent in each of the three departments.functional properties of dualityFrom an algorithmic point of view, solving the primal problem with the dual simplex method is equivalent to solving the dual problem with the primal simplex method. So, these are following functional properties of duality: When the primal has n variables and m constraints, the dual has m variables and n constraints. The constraints for the primal are all less than or equal to, while the constraints for the dual are all greater than or equal to. The objective for the primal is to maximize, while the objective for the dual is to minimize. All variables for either problem are restricted to be nonnegative. For every primal constraint, there is a dual variable. Associated with the ith primal constraint is dual variable pi the dual objective. Function coefficient for pi is the right-hand side of the ith primal constraint, bi. For every primal variable, there is a dual constraint. Associated with primal variable xj is the jth dual constraint whose right-hand side is the primal objective function coefficient cj. The number a ij is, in the primal, the coefficient of xj in the ith constraint, while in the dual, a ij is the coefficient of pi in the jth constraint.

Q12.Explain the primal dual relationship?Ans.In the linear programming, in the primal problem, from each sub-optimal point that satisfies all the constraints, there is a direction orsubspaceof directions to move that increases the objective function. Moving in any such direction is said to remove "slack" between the candidate solution and one or more constraints. An "infeasible" value of the candidate solution is one that exceeds one or more of the constraints.In the dual problem, the dual vector multiplies the constants that determine the positions of the constraints in the primal. Varying the dual vector in the dual problem is equivalent to revising the upper bounds in the primal problem. The lowest upper bound is sought. That is, the dual vector is minimized in order to remove slack between the candidate positions of the constraints and the actual optimum. An infeasible value of the dual vector is one that is too low. It sets the candidate positions of one or more of the constraints in a position that excludes the actual optimum.

If primalThen Dual

(i) Objective is to maximize(i) Objective is to minimize

(ii) Variable x(ii) Constraint j

(iii) Constraint i(iii) Variables y

(iv) Variables x unrestricted in sign(iv) Constraint j is = type

(v) Constraint I is = type(v) Variables y is unrestricted in sign

(vi) Type constraints (vi) Type constraints

(vii) X is unrestricted in sign (vii) jth constraints is an equation

13.What is duality? What is the significance of dual variables in a LP model?Ans.Duality implies that each linear programming problem can be analyzed in two different ways but having equivalent solutions. Each linear programming problem (both maximization and minimization) stated in its original form has associated with another linear programming problem called duality.The - Dual formulation can be derived from the same data from which the primal was formulated. The Dual formulated can be solved in the same manner in which the Primal is solved since the Dual is also a LP formulation. The Dual can be considered as the 'inverse' of the Primal in every respect. The column coefficients in the Primal constrain become the row co-efficient in the Dual constraints. The coefficients in the Primal objective function become the right hand side constraints in the Dual constraints. The column of constants on the right hand side of the Primal constraints becomes the row of coefficients of the dual objective function. The direction of the inequalities is reversed. If the primal objective function is a 'Maximization' function then the dual objective function is a 'Minimization' function and vice-versa.Significance of dual variables The importance of the dual LP problem is in terms of the information which it provides about the value of the resources. The economic analysis in this chapter is concerned in deciding whether or not to secure more resources and how much to pay for these additional resources.The significance of the study of dual is as follows: The dual variables provide the decision-maker a basis for deciding how much to pay for additional units of resources. The maximum amount that should be paid for one additional unit of a resource is called its shadow price. The total marginal value of resources equals the optimal objective function value. The dual variables equal the marginal value of resources.

Q14. Write a short note on Sensitivity analysis?Ans.Sensitivity analysis is the study of sensitivity of the optimal solution of an LP problem due to discrete variations (changes) in its parameters. The degree of sensitivity of the solution due to these variations can range from no change at all to a substantial change in the optimal solution of the given LP problem. Thus, in sensitivity analysis, we determine the range over which the LP model parameters can change without affecting the current optimal solution. For this, instead of resolving the entire problem as a new problem with parameters, we may consider the original optimal solution as an initial solution for the purpose of knowing the ranges both lower and upper, within which a parameter may assume a value. In other words, Sensitivity analysis is a systematic study of how sensitive (duh) solutions are to (small) changes in the data.For example, if you think that the price of your primary output will be between Rs.100 and Rs.120 per unit, you can solve twenty dierent problems (one for each whole number between Rs.100 and Rs.120).1 This method would work, but it is inelegant and (for large problems) would involve a large amount of computation time. (In fact, the computation time is cheap, and computing solutions to similar problems is a standard technique for studying sensitivity in practice.) The process of studying the sensitivity of the optimal solution of an LP model is also called post optimality analysis because it is done after an optimal solution, assuming a given set of parameters, has been obtained for the model.Different categories of parameter changes in the original LP model are as follow:-

Profit or cost per unit associated with both basic and non-basic decision variables (coefficient in the objective function). Availability of resources (right hand side constants). Consumptions of resources per unit of decision variables (coefficients of decision variables on the left hand side of constraints) Addition of a new variable to the existing list of variables in LP problem. Addition of a new constraint to the original LP problem constraints.

Q15. What is the role of sensitivity analysis in Linear Programming? Under what circumstances is it needed, and under what conditions do you think it is not necessary?Ans. Linear programming determines the optimal solution, but the dynamic nature of the values we input to determine optimal solution leads to the problem of uncertainty. The effect of these input can be determine by Sensitivity analysis. Sensitivity analysis can be defined as the study of knowing the effect on optimal solution of the LP model due to variations in the input coefficients (also called parameters) one at a time. Sensitivity analysis allows us to determine how sensitive the optimal solution is to changes in data values.Sensitivity analysis can be used in the following conditions:1. If the objective function changes, how does the solution change?2. If resources available change, how does the solution change?3. If a constraint is added to the problem, how does the solution change?

Question 16. Explain in brief the three methods of initial feasible solution for transportation problem.Ans.Initial Basic Feasible Solution is the condition of obtaining the solution by allocation of resources to the rows and the column with the non-negative values There are three different methods to obtain the initial basicFeasible solution those are mentioned below:(I) North-West corner rule(II) Lowest cost entry method(III) Vogels approximation method Northwest corner ruleThe major advantage of the northwest corner rule method is that it is very simple and easy to apply. Its major disadvantage, however, is that it is not sensitive to costs and consequently yields poor initial solutions.Lowest cost entry methodIn this method, allocations are made on the basis of economic desirability. Least cost method finds a better starting solution by the concentrating on the cheapest routes. This method assigns as much as possible to the variable with the smallest unit transportation cost in the entire tableau. Cross out the satisfied row or column and amounts of supply and demand are adjusted accordingly. Again we search for the next lowest unit transportation cost and the process continues till total supply and demand is exhausted.Vogels approximation method (unit cost penalty method) (VAM)VAM is an improved version of the least-cost method that generally, but not always, produces better starting solutions. VAM is based upon the concept of minimizing opportunity (or penalty) costs. The opportunity cost for a given supply row or demand column is defined as the difference between the lowest cost and the next lowest cost alternative. This method is preferred over the methods discussed above because it generally yields, an optimum, or close to optimum, starting solutions. Consequently, if we use the initial solution obtained by VAM and proceed to solve for the optimum solution, the amount of time required to arrive at the optimum solution is greatly reduced.Q17. State the transportation problem. Describe clearly the steps involved in solving it?Ans. Northwest corner ruleThe steps involved in determining an initial solution using northwest corner rule are as follows:Step1. Write the given transportation problem in tabular form (if not given).Step2. Go over to the north-west corner of the table. Suppose it is the (i, j)th cell.Step3. Allocate min (ai, bj) to this cell. If the min (ai , bj) = ai, then the availability of the ith origin is exhausted and demand at the jth destination remains as bj-ai and the ith row is deleted from the table. Again if min (ai, bj) = bj, then demand at the jth destination is fulfilled and the availability at the ith origin remains to be ai-bj and the jth column is deleted from the table.Step4. Repeat steps 2, 3 until all origins are exhausted and all demands are fulfilled.

Lowest cost entry methodIn this method, allocations are made on the basis of economic desirability. The steps involved in determining an initial solution using least-cost method are as follows:Step1. Write the given transportation problem in tabular form (if not given).Step2. Choose the cell with minimum cost. If it is not unique, anyone can be chosen. Suppose it is the (i, j)th cell.Step3. Allocate min (ai, bj) to this cell. If the min (ai , bj) = ai, then the availability of the ith origin is exhausted and demand at the jth destination remains as bj-ai and the ith row is deleted from the table. Again if min (ai, bj) = bj, then demand at the jth destination is fulfilled and the availability at the ith origin remains to be ai-bj and the jth column is deleted from the table.Step4. Repeat steps 2, 3 until all origins are exhausted and all demands are fulfilled. The process must end as .

Vogels approximation method (unit cost penalty method) (VAM)The steps involved in determining an initial solution using VAM are as follows:Step1. Write the given transportation problem in tabular form (if not given).Step2. Compute the difference between the minimum cost and the next minimum cost corresponding to each row and each column which is known as penalty cost.Step3. Choose the maximum difference or highest penalty cost. Suppose it corresponds to the ith row. Choose the cell with minimum cost in the ith row. Again if the maximum corresponds to a column, choose the cell with the minimum cost in this column.Step4. Suppose it is the (i, j)th cell. Allocate min (ai, bj) to this cell. If the min (ai , bj) = ai, then the availability of the ith origin is exhausted and demand at the jth destination remains as bj-ai and the ith row is deleted from the table. Again if min (ai, bj) = bj, then demand at the jth destination is fulfilled and the availability at the ith origin remains to be ai-bj and the jth column is deleted from the table.Step5. Repeat steps 2, 3, and 4 with the remaining table until all origins are exhausted and all demands are fulfilled.Q18. State the transportation problem. Describe clearly the steps involved in solving it?Ans. Northwest corner ruleThe steps involved in determining an initial solution using northwest corner rule are as follows:Step1. Write the given transportation problem in tabular form (if not given).Step2. Go over to the north-west corner of the table. Suppose it is the (i, j)th cell.Step3. Allocate min (ai, bj) to this cell. If the min (ai , bj) = ai, then the availability of the ith origin is exhausted and demand at the jth destination remains as bj-ai and the ith row is deleted from the table. Again if min (ai, bj) = bj, then demand at the jth destination is fulfilled and the availability at the ith origin remains to be ai-bj and the jth column is deleted from the table.Step4. Repeat steps 2, 3 until all origins are exhausted and all demands are fulfilled.

Lowest cost entry methodIn this method, allocations are made on the basis of economic desirability. The steps involved in determining an initial solution using least-cost method are as follows:Step1. Write the given transportation problem in tabular form (if not given).Step2. Choose the cell with minimum cost. If it is not unique, anyone can be chosen. Suppose it is the (i, j)th cell.Step3. Allocate min (ai, bj) to this cell. If the min (ai , bj) = ai, then the availability of the ith origin is exhausted and demand at the jth destination remains as bj-ai and the ith row is deleted from the table. Again if min (ai, bj) = bj, then demand at the jth destination is fulfilled and the availability at the ith origin remains to be ai-bj and the jth column is deleted from the table.Step4. Repeat steps 2, 3 until all origins are exhausted and all demands are fulfilled. The process must end as .

Vogels approximation method (unit cost penalty method) (VAM)The steps involved in determining an initial solution using VAM are as follows:Step1. Write the given transportation problem in tabular form (if not given).Step2. Compute the difference between the minimum cost and the next minimum cost corresponding to each row and each column which is known as penalty cost.Step3. Choose the maximum difference or highest penalty cost. Suppose it corresponds to the ith row. Choose the cell with minimum cost in the ith row. Again if the maximum corresponds to a column, choose the cell with the minimum cost in this column.Step4. Suppose it is the (i, j)th cell. Allocate min (ai, bj) to this cell. If the min (ai , bj) = ai, then the availability of the ith origin is exhausted and demand at the jth destination remains as bj-ai and the ith row is deleted from the table. Again if min (ai, bj) = bj, then demand at the jth destination is fulfilled and the availability at the ith origin remains to be ai-bj and the jth column is deleted from the table.Step5. Repeat steps 2, 3, and 4 with the remaining table until all origins are exhausted and all demands are fulfilled.Q19. What is an assignment problem?Give two applications.

Ans.

An assignment problem is a specific case of transportation problem whichrefers to the minimization of total cost or the maximization of total profit ofallocation by assigning a number of resources to an equal number ofactivities. The problem of assignment arises because available resources such asmen,machines,etc have varying degrees of efficiency for performing a differentactivities. Therefore, cost, profit or time of performing the different activities isdifferent. Thus the problem is:How should the assignments be made so as to optimizethe given objective.

Some of the problems where the assignment technique may be useful are: Assignment of workers to machines, salesmen to different sales areas, clerks to various check-outcounters, classes to rooms, etc. The assignment problem is a special type of linearprogramming problem where assignees are being assigned to perform tasks. Forexample, the assignees might be employees who need to be given work assignments.Assigning people to jobs is a common application of the assignment problem. However,the assignees need not be people. They also could be machines, or vehicles, or plants, oreven time slots to be assigned tasks. The first example below involves machines beingassigned to locations, so the tasks in this case simply involve holding a machine. Asubsequent example involves plants being assignedproducts to be produced.

Example

The JOB SHOP COMPANY has purchased three new machines of different types. There are four available locations in the shop where a machine could be installed. Some of these locations are more desirable than others for particular machines because of their proximity to work centers that will have a heavy work flow to and from these machines. (There will be no work flow between the new machines.) Therefore, the objective is to assign the new machines to the available locations to minimize the total cost of materials handling. The estimated cost in dollars per hour of materials handling involving each of the machines for the respective locations. Location 2 is not considered suitable for machine 2, so no cost is given for this case.

Applications of assignment problem

The linear programming model encompasses a wide variety of specific types of problems. The general simplex method is a powerful algorithm that can solve surprisingly large versions of any of these problems. However, some of these problem types have such simple formulations that they can be solved much more efficiently by streamlined algorithms that exploit their special structure. These streamlined algorithms can cut down tremendously on the computer time required for large problems, and they sometimes make it computationally feasible to solve huge problems. This is particularly true for the two types of linear programming problems studied in this chapter, namely, the transportation problem and the assignment problem. Both types have a number of common applications, so it is important to recognize them when they arise and to use the best available algorithms. These special-purpose algorithms are included in some linear programming software packages.This problem has the interpretation of minimizing the cost for the flow of goods through a network. A streamlined version of the simplex method called the network simplex method is widely used for solving this type of problem, including its various special cases.

A supplementary chapter on the books website,, describes various additional special types of linear programming problems. One of these, called the transshipment problem, is a generalization of the transportation problem which allows shipments from any source to any destination to first go through intermediate transfer points. Since the transshipment problem also is a special case of the minimum cost flow problem. Much research continues to be devoted to developing streamlined algorithms for special types of linear programming problems, including some not discussed here. At the same time, there is widespread interest in applying linear programming to optimize the operation of complicated large-scale systems. The resulting formulations usually have special structures that can be exploited. Being able to recognize and exploit special structures is an important factor in the successful application of linear programming.

Q20. Give the mathematical formulation of an assignment problem. How does it differ from a transportation problem?

Ans. The assignment problem is a special type of transportation problem, where the objective is to minimize the cost or time of completing a number of jobs by a number of persons. In other words, when the problem involves the allocation of n different facilities to n different tasks, it is often termed as an assignment problem. The model's primary usefulness is for planning. The assignment problem also encompasses an important sub-class of so-called shortest- (or longest-) route models. The assignment model is useful in solving problems such as, assignment of machines to jobs, assignment of salesmen to sales territories, travelling salesman problem, etc. It may be noted that with n facilities and n jobs, there are n! possible assignments. One way of finding an optimal assignment is to write all the n! possible arrangements, evaluate their total cost, and select the assignment with minimum cost. But, due to heavy computational burden this method is not suitable. Formulation of an assignment problem Suppose a company has n persons of different capacities available for performing each different job in the concern, and there are the same number of jobs of different types. One person can be given one and only one job. The objective of this assignment problem is to assign n persons to n jobs, so as to minimize the total assignment cost. The cost matrix for this problem is given below:Table: Cost MatrixResources(Workers)Activities(Jobs)J1 J2 . . . . J nSupply

W1W2...WnC11 C12 . . . . C 1nC21 C22 . . . . C 2n . . . . . . . . .Cn1 Cn2 . . . . C n n11...1

Demand 1 1 . . . . 1n

Let Xi j denote the assignment of worker i to job j such that xij = 1 if job j is performed by worker i

0 otherwise

for i = 1, 2, ..., n and j = 1, 2, ..., nIn the above table, cij is the cost of performing jth job by ith worker. The optimization model isMinimize c11x11 + c12x12 + ------- + cnnxnnsubject toxi1 + xi2 +..........+ xin = 1 i = 1, 2,......., nx1j + x2j +..........+ xnj = 1 j = 1, 2,......., n xij = 0 or 1In Sigma notationMinimize Z = cijxij subject to xij = 1for i = 1, 2, ....., n (resource availability) xij = 1 for j = 1, 2, ....., n (activity requirement} xij = 0 or 1 for all i and jAssumptions Number of jobs is equal to the number of machines or persons. Each man or machine is assigned only one job. Each man or machine is independently capable of handling any job to be done. Assigning criteria is clearly specified (minimizing cost or maximizing profit).

The differences between AP and TP are the following:

1. TP has supply and demand constraints while AP does not have the same.2. The optimal test for TP is when all cell evaluation \s are greater than or equal to zero whereas in AP the number of lines must be equal to the size of matrix.3. A TP sum is balanced when demand is equal to supply and an AP sum is balanced when number of rows are equal to the number of columns.4. for AP. We use Hungarian method and for transportation we use MODI method5. In AP. We have to assign different jobs to different entities while in transportation we have to find optimum transportation cost.

Q21.Define competitive game, payoff matrix, pure and mixed strategies, saddle point, optimal strategies and rectangular(or two person zero- sum) game.Ans. 1.Two person game theory- To illustrate the basic characteristics of two-person, zero-sum games, consider the game called odds and evens. This game consists simply of each player simultaneously showing either one finger or two fingers. If the number of fingers matches, so that the total number for both players is even, then the player taking evens (say, player 1) wins the bet (say, $1) from the player taking odds (player 2). If the number does not match, player 1 pays $1 to player 2.Thus, each player has two strategies: to show either one finger or two fingers. In general, a two-person game is characterized by1. The strategies of player 12. The strategies of player 23. The payoff table2. Payoff matrix:- A quantitative measure of satisfaction a player gets at the end of the play in terms of gains or losses, when players select theirs particular strategies can be represented in the form of a matrix, called the payoff matrix.3. Competitive game:- The game refers to the general situation of complete and competition in which two or more competitors are involved in decision making activities in anticipation of certain outcomes over a period of times.4. Pure strategies:- It is the decision rule which is always used by the player to select the particular strategy. Thus, each player knows in advance of all strategies out of which he always selects only one particular strategy regardless of the other players strategies, and the objective of the players is to maximize gains or minimize losses.5. Mixed strategies:- Courses of action that are to be selected on a particular occasion with some fixed probability are called mixed strategies. Thus, there is a probabilistic situation and objective of the players is to maximize expected losses by making choice among pure strategies with fixed probabilities.6. Saddle point:- If the maxi min value equals the minimax value, then the game is said to have a saddle point.7. Optimal strategies

Q22. Explain: Minimax and Maximin principle used in the game theory.Maximin Principle- For a player A minimum value of each row represents the least gain (pay off) to him if he chooses his particular strategy . these are written in the matrix by row minima . He will then select the strategy that gives largest gain among the row minimum value. This choice of player A is called the maximin principle and the corresponding gain is called the maximin value of the game.Minimax Principle- For player B who assumes to be the loser, the maximum value in each column represents the maximum loss to him if he chooses his particular strategy. These are written in the payoff matrix by column minima. He will then select the strategy that wil give minimum loss among the column maximum values.This choice of player B is called minimax principle and the corresponding loss is the minimax value of the game.

Q23.Explain the following terms in PERT/CPMEARLIEST TIME: Earliest occurrence time of an event , i .It is the earliest time for an event to occur immediately after all the proceedings activity have been completed without delaying the entire project.LATEST TIME: Latest allowable time of an event, i . It is the latest time for an event to occur immediately without causing delay in already determined projects completion time.EVENT SLACK: The slack also known as float of an event is the difference between its latest occurrence time and its earliest occurrence time.i.e.Event float =Li-EiIt is a measure of how long an event can be delayed without increasing the project completion time.CRITICAL PATH: It is the sequence of critical activity that form a continuous path between the start of a project and its completion . It is critical in the sense that if any activity in this sequence in delayed, the completion of the entire project is delayed

Q24.What is a float? What are different types of float?Ans.The float or free time is the length of time to which a non-critical activity and/or an event can be delayed or extended without delaying the total project completion time.SLACK ON EVENT : It is the difference between its latest occurrence time and its earliest occurrence time. It is a measure of how long an event can be delayed without increasing the project completion time.SLACK ON ACTIVITY : The slack(float) on activity is the length of time available within the estimated time of of an non-critical activity.It tells how long an activity time may be increased without increasing project time.There are three types of slack:TOTAL FLOAT: It is the time by which an activity can be delayed if all its proceeding activities are completed at their earliest time and all successor activities can bedelayed until their latest permissible time.FREE FLOAT: Free float for a non-critical activity is defined as the time by which the completion of any activity can be delayed without causing any delay in its immediate succeeding activity.INDEPENDENT FLOAT: It is the amount of time available when preceeding activity are completed at their latest permissible times and all the following activity can still be completed at their earliest possible time.Q25.What is replacement? Describe some important replacement situation?Ans.The problem of replacement is felt when the job performing units such as men, machines, equipment, parts etc become less effective or useless due to either sudden or gradual deterioration in their efficiency, failure or frequent internal, maintenance and other overhead cost can be reduced.Some replacement situations are:Items like machines, vehicles, tyres etc. Whose efficiency deteriorates with age due to constant use and which needs increased operating and maintenance cost.Items such as light bulbs and tubes, electric motors, radio,television parts etc. Which do not give any indication of deterioration ith time but fail of a sudden and become completely useless.The existing working staff is an organisation gradually reduces due to retirement , death, retrenchment and other reasonsQ26.What is dual simplex algorithm? State various steps involved in the dual simplex algorithm.Ans.The dual simplex method is often used in situations where the primal problem has a number of equality constraints generating artificial variables in the l.p. canonical form. Like in the primal simplex method, the standard form for the dual simplex method assumes all constraints are