abhijeet alte - intro to or-ms

Operations Research BY ABHIJEET ALTE : 

Operations Research BY ABHIJEET ALTE ABHI

welcome : 

welcome Welcome to Operations Research Course Course will run for 28 sessions(each session – 1.3hrs) Performance of students will be evaluated internally through - internal assessment (quiz/tests/assignments/midterm – 50 %) - final examination (50 %) ABHI

Road map : 

Road map Introduction to Management Science Introduction to LP LP- sensitivity analysis LP-applications Special LP models Network models ABHI

Road map (contd.) : 

Road map (contd.) PERT/CPM Simulation Decision analysis Case presentations ABHI

The importance of management science : 

The importance of management science Management science The discipline of applying advanced analytical methods to help make better decisions. Devoted to solving managerial-type problems using quantitative models Applications of management science Forecasting, capital budgeting, portfolio analysis, capacity planning, scheduling, marketing, inventory management, project management, and production planning. ABHI

Problem solving approaches : 

Problem solving approaches Managers tend to use a qualitative approach to problem solving when The problem is fairly simple. The problem is familiar. The costs involved are not great. Managers tend to use a quantitative approach when The problem is complex. The problem is not familiar. The costs involved are substantial. Enough time is available to analyze the problem. ABHI

Quantitative approach : 

Quantitative approach Mathematical tools have been used for thousands of years QA can be applied to a wide variety of problems Consider both Quantitative and Qualitative Factors One must understand: the specific applicability of the technique, its limitations and its assumptions ABHI

The Evolution of QA : 

The Evolution of QA 1990 1980 1970 1960 1950 1940 1930 1920 1910 1900 Expert Systems and Artificial Intelligence Decision Support Information System Goal Programming Decision Theory Network Models Dynamic Programming Game Theory Transportation Assignment Technique Inventory Control Queuing Theory Markov Analysis ABHI

Advantages of quantitative approach : 

Advantages of quantitative approach Directs attention to the essence of an analysis: to solve a specific problem. Improves planning which helps prevent future problems Results in more objective decisions than purely qualitative analysis. Incorporates advances in computational technologies to managerial problem-solving ABHI

Models : 

Models A Model An abstraction of reality. It is a simplified, and often idealized, representation of reality. Examples : an equation, an outline, a diagram, and a map By its very nature a model is incomplete. Provides an alternative to working with reality Symbolic models Use numbers and algebraic symbols Mathematical models Decision variables Uncontrollable variables ABHI

OR Models : 

OR Models OR models address three main questions: What are the decision alternatives>? Under what restrictions is the decision made? What is an appropriate objective criterion for evaluating the alternatives? ABHI

Example with limited alternatives : 

Example with limited alternatives Suppose you have a 5 week business commitment between Mumbai (MUM) and Delhi (DEL). You fly out of Mumbai on Monday and return on Wednesday. A regular round – trip ticket costs you Rs.10000, but a 20% discount is granted if the dates of the ticket include a weekend. A one way ticket either way costs 60% of the regular price. How should you buy the tickets for the 5 wk period? ABHI

alternatives : 

alternatives Three alternatives are considered: Buy five regular MUM-DEL-MUM. Buy one MUM-DEL, four DEL-MUM-DEL that cover weekends and one DEL-MUM. Buy one MUM-DEL-MUM to cover Monday of the first week and Wednesday of the last week and four DEL-MUM-DEL to cover the remaining trips. Each ticket in this alternative spans a weekend. ABHI

Objective criterion : 

Objective criterion Objective criterion to evaluate the alternatives will be to compare prices of these tickets. The alternative that yields the lowest cost is the best. Alt 1 cost = 5*10000 = 50000. Alt 2 cost = 0.60*10000+4*(0.8*10000) +0.60*10000 = 44000. Alt 3 cost = 5*0.8*10000 = 40000. The above example illustrates the three main components of an OR model. ABHI

Another example : 

Another example However, situations may differ leading to different constructions of each component. Consider forming a maximum area rectangle out of a piece of wire of length L inches. Here the number of alternatives is not finite, since the length and width of the rectangle can assume infinite possibilities. Hence the alternatives of the problem are identified by defining the length and width as continuous (algebraic) variables. ABHI

Piece of wire : 

Piece of wire Let l = length of the rectangle in cms w = width of the rectangle in cms. Based on the above definitions the restrictions can be expressed verbally as Length of rectangle + width of rectangle = half of the length of the wire. Length and width cannot be negative. We may write these conditions algebraically as 2(l+w) = L l>= 0, w>=0. ABHI

Piece of wire - formulation : 

Piece of wire - formulation We now define the remaining component of objective of the problem: Maximize z=lw, where z is defined as the area of the rectangle. The complete model becomes Maximize z=lw, Subject to 2(l+w) = L l, w >=0. The optimal solution of this model is l=w = L/4, which means we have to construct a square. ABHI

OR model Format : 

OR model Format An OR model is usually presented in the following format: Maximize or minimize objective functionSubject toconstraints A solution of the model is feasible if it satisfies all the constraints. It is optimal if, in addition to being feasible, it yields the best (maximum or Minimum) value of the objective function. In the tickets example, there are three feasible alternatives sand the third one yields the optimal solution. In the rectangle example, a feasible alternative must satisfy the condition l+w = L/2 with l and w assuming non – negative values. This leads to infinity of feasible solutions and the optimal solution is determined by an appropriate mathematical tool (in this case, differential calculus). ABHI

Quality and sub optimality : 

Quality and sub optimality OR models are designed to optimize a specific objective criterion subject to a set of constraints. However the quality of the resulting solution depends on the accuracy of the model in representing the real system. For example, in the tickets problem, if one is not able to identify all the dominant alternatives, then the resulting solution is only optimal relative to the choices represented in the model. To be specific, if alt 3 is left out of the model, and then the resulting optimal solution would call for spending Rs.44000 to purchase the tickets, which provides only a sub optimal solution to the problem. Hence it can be concluded that the optimal solution is best only for that model. If the model represents the real system reasonably well, then its solution is optimal also for the real situation. ABHI

Phases of an OR study : 

Phases of an OR study The principal phases of implementing OR in practice include definition of the problem construction of the model solution of the model validation of the model Implementation of the solution. Of all the 5 phases, only phase 3 is well defined and the easiest to implement in an OR study because it deals with mostly precise mathematical models. The implementation of the remaining phases is more of an art than science. ABHI

Deterministic versus probabilistic models : 

Deterministic versus probabilistic models Deterministic models Used for problems in which information is known with a high degree of certainty. Used to determine an optimal solution to the problem. Probabilistic models Used when it cannot be determined precisely what values (requiring probabilities) will occur (usually in the future). ABHI

Fig -1 the management science approach : 

Fig -1 the management science approach ABHI

Excel spreadsheet : 

Excel spreadsheet ABHI

Functions screen : 

Functions screen ABHI

Add- in options : 

Add- in options ABHI

Break-even analysis : 

Break-even analysis Breakeven analysis (cost-volume analysis) Is concerned with the interrelationship of costs, volume (quantity of output or sales), and profit. The Break-Even Point (BEP) The volume for which total revenue and total cost are equal. The dividing line between profit and loss; sales higher than the break-even point will result in a profit, while sales that is lower than the break-even point will result in a loss. Where you get “out of the red.” ABHI

Break-even analysis : 

Break-even analysis Breakeven analysis (cost-volume analysis) Is concerned with the interrelationship of costs, volume (quantity of output or sales), and profit. Components of Break-Even Analysis Volume: the level of output of a machine, department, or organization, or the quantity of sales. Revenue: the income generated by the sale of a product. Total revenue = revenue per unit (selling price per unit) multiplied by units (volume) sold. Costs: costs that must be taken into account Fixed costs are not related to the volume of output. Variable costs increase and decrease with output ABHI

Assumptions of B/E analysis : 

Assumptions of B/E analysis The revenue per unit is the same for all volumes. The variable cost per unit is the same for all volumes. Fixed cost is the same for all levels of volume. Only one product is involved. All output is sold. All relevant costs are accounted for, and correctly assigned to either the fixed cost category or the variable cost category. ABHI

Fig: total revenue increases linearly as volume increases : 

Fig: total revenue increases linearly as volume increases ABHI

Fig: Fixed costs : 

Fig: Fixed costs ABHI

Fig: total variable cost : 

Fig: total variable cost ABHI

Fig: total cost : 

Fig: total cost ABHI

Fig: profit and B/E point : 

Fig: profit and B/E point ABHI

example : 

example ABHI

BEP in Excel : 

BEP in Excel ABHI

Goal seek input screen : 

Goal seek input screen Goal seek output screen ABHI

Breakeven analysis examples : 

Breakeven analysis examples 14.06 ET – Tata Motors bounced back smartly from its Rs. 500 cr.loss in 2000 by dramatically reducing the fixed costs in the business and thus lowering the breakeven so that the company could be insulated from the cyclical nature of business. ABHI

Breakeven analysis examples : 

Breakeven analysis examples ET 14.06 -Vishal retail to launch Rs. 6 per litre water. at present packaged water is sold in the Rs. 10-12 per ltr range and the pricing of the product by vishal, if it clicks will force key players to revisit their pricing. Cost of production is usually low but transport cost and marketing cost is high. ABHI

Breakeven analysis examples : 

Breakeven analysis examples Competition eats into telecom operators’ margins. Gross revenues per min -India – 60 paise -China – 1.30 Rs. -UK – 12 Rs. -Asia – 4 Rs. -Japan & Korea – 8 Rs. ABHI

Breakeven analysis examples : 

Breakeven analysis examples Mint 15.01- AL puts off expansion plan. The move comes on the back of rising interest rates, which have reduced demand for commercial vehicles in the current financial year. AL had originally planned to add 100000 units to its existing capacity of 84000 units by the first half of 2010 – in two locations, Chennai and Uttarakhand.AL now says it will only go ahead with the plan at Uttarakhand, but keep the 50000 unit expansion in Chennai for 2012. ABHI

assignments : 

assignments Bring any breakeven analysis examples from the business world appearing in news reports and discuss in the next class. Interpret the breakeven examples. Solve the River Crossing Problem. ABHI

River crossing problem : 

River crossing problem Amy, Jim, John, and Kelly are standing on the east bank of a river and wish to cross to the west side using a canoe. The canoe can hold at most two individuals at a time. Amy, being the most athletic, can cross the river in one minute. Jim, John, and Kelly would take 2, 5, and 10 minutes respectively. If two people are in the canoe, the slower person dictates the crossing time. The objective is for all four individuals to be on the other side of the river in the least time possible. Identify at least two feasible plans for crossing the river (remember, the canoe is the only mode of transportation and it cannot be shuttled empty). Define the criterion for evaluating the alternatives. What is the smallest time for moving all four individuals to the other side of the river? ABHI

Simple breakeven analysis : 

Simple breakeven analysis Sale price rs. 100 per unit. Variable cost rs. 50 per unit. Contribution rs. 50 per unit. Fixed expenses rs. 5000 per month Breakeven quantity 100 per month. Let us look at a bit more complicated example. ABHI

Multiproduct breakeven analysis : 

Multiproduct breakeven analysis Sale price prod A - rs.100 per unit. Sale price prod B – rs. 80 per unit. V cost prod A – rs. 50 per unit. V cost prod B – rs. 20 per unit. Fixed cost rs. 5500 per month This situation requires simplifying assumption of sales ratio of products. ABHI

Multi product analysis : 

Multi product analysis ABHI

constrained optimization : 

constrained optimization Firms must choose the product mix which maximizes the total profits. Usually, we assume that only unit-based variable costs are relevant to the product mix decision. Thus, assuming that non-unit level costs are the same for any product mix, a manager needs to choose the mix alternative which will maximize total contribution margin. If the firm’s resources are unlimited, then the product mix decision is simple – produce infinite quantity of each product. Unfortunately, this is not the case. the limitations on the resources are called constraints. ABHI

Example of constrained optimization : 

Example of constrained optimization A company produces two types of machine parts:X and Y with unit contribution margins of rs.300 and rs.600 respectively. Assuming that the company can sell all that is produced, we may argue that only part Y should be produced since it has the largest contribution margin. But this may not necessarily the best. ABHI

Role of constraints : 

Role of constraints The constraints may be internal – limiting factors of the resources or External – such as market demand. Although resources and demand may be limited, certain mixes may not meet all the demand or use all the resources. Constraints whose limited resources are not fully utilised by a product mix are loose constraints. If a product mix uses all of the limited resources of a constraint, then that constraint is a binding constraint. Selection of the optimal mix can be significantly affected by the relationship of the constrained resources to the individual products. ABHI

Example of one binding internal constraint : 

Example of one binding internal constraint Assume that each part must be drilled by a machine. the company owns three drilling machines that provide a total of 120 drilling hours per week. Part X requires 1 hour of drilling and part Y- 3 hours. Assuming no other binding constraints, what is the optimal product mix of parts? ABHI

Analysis –one internal constraint : 

Analysis –one internal constraint ABHI

Additional external binding constraint. : 

Additional external binding constraint. An analysis leads us to the conclusion that it is better to produce only Part X. Contribution margin per unit is not the critical concern, the contribution per unit of scarce resource is the deciding factor. This factor can also be used to identify the optimal product mix when a binding external constraint exists. Assume also that the company can sell at most 60 units of part X and 100 units of part Y. The internal constraint allows the company to produce 120 units of part X, but this is no longer a feasible choice since only 60 units of X can be sold. ABHI

Analysis with external binding constraint : 

Analysis with external binding constraint ABHI

Multiple binding constraints : 

Multiple binding constraints It is possible for a company to have more than one binding constraint. All firms face multiple constraints : limitations of materials, limitations of labour inputs, limited machine hours and so on. The solution of the product mix problem in the presence of multiple binding constraints is considerably more complicated and requires the use of a specialized mathematical technique called linear programming. ABHI