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Introduction to Management Science and Linear Programming
The document outlines the problem-solving process in management science and operations research, emphasizing decision-making steps such as defining problems, evaluating alternatives, and implementing solutions. It discusses quantitative and qualitative analysis, various modeling techniques, and introduces linear programming as a method for decision optimization in business scenarios. Additionally, examples illustrate how linear programming can solve practical problems in production scheduling, investment portfolio selection, and resource allocation.
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Introduction to Management Science and Linear Programming
- 1. Dr. Kishore Morya (DoI REALLY Have to Know This Stuff?) An Introduction to Management Science/Operations Research
- 2. Problem Solving Problem solvingis a process of identifying a difference between the actual and desired state of affairs and then taking action to resolve the difference Problem should be important enough to justify cost, time, and efforts of careful analysis
- 3. Steps in ProblemSolving • Identify and define the problem • Determine the set of alternative solutions • Determine the criterion or criteria that will used to evaluate the alternatives • Evaluate the alternatives • Choose an alternative • Implement the skeleton alternative • Evaluate the results to determine whether a satisfactory solution has been obtained
- 4. Decision Making Decision Makingis the term generally associated with the first five steps of the problems solving process First step is to identify and define the problem Decision Making ends with the choosing of an alternative, which the act of making decision
- 5. Example • Assume thatyou want to get employed. You like a position that will lead to a satisfying career. • Suppose that your job research has resulted in offers from companies in - Rochester – New York - Dallas – Texas - Greensboro – North Carolina - Pittsburgh - Pennsylvania
- 6. Alternatives for yourdecision problems • Accept the position in Rochester • Accept the position in Dallas • Accept the position in Greensboro • Accept the position in Pittsburgh
- 7. Determining the Criteria •Starting Salary • Potential of Advancement • Location of the Job One criterion – Single-Criterion Decision Problems More than one criterion – Multi-criteria Decision Problems
- 8. Evaluate Each ofthe Alternatives Alternatives Starting Salary Potential for Advancement Job Location Rochester $38500 Average Average Dallas $36000 Excellent Good Greensboro $36000 Good Excellent Pittsburgh $37000 Average Good You are now ready to make choice! Are you?
- 9. The Choice isDifficult The criteria are probably not all equally important, and no alternative is “best” with regards to all criteria If you select any choice, let it be alternative 3 Alternative 3 thus referred to as the decision
- 10. Quantitative Analysis andDecision Making Prob- • Define the Problem • Identify the Alternatives -lem • Determine the criteria • Evaluate the Alternatives Sol- • Choose an Alternative -vi • Implement the Decision • Evaluate the Results -ng • Results are according to expectation, stop the process • Otherwise go back to the first stage Decision Making Decision
- 11. Quantitative analysis andDecision Making Define the problem Identify the alternati ves Make the Decision Summary and Evaluation Quantitative Analysis Qualitative Analysis Determin e the Criteria Structuring the problem Analysing the problem
- 12. Model Development • IconicModel - Physical replicas of real model, a scale model of an airplane is a representation of a real airplane • Analog Model - Do not have the physical appearance as the object being modeled • Mathematical Model - Representation of a problem by a system of symbols and mathematical relationships or expressions
- 13. Model… • Objective Function P= 10x x represents the quantity sold $10 is the profit per unit sold P is the total profit • Constraints - 5 hours are required to produce each unit - 40 hours of production time is available Production Time constraint is given by 5x ≤ 40
- 14. Complete Mathematical Model MaximizeP = 10x s. t. 5x ≤ 40 constraints x ≥ 0
- 15. Model of Cost,Revenue, Profit, • Cost and Volume Model C(x) = 3000 + 2x x = production volume in units C(x) = total cost of producing x units • Revenue and Volume Model R(x) = 5x x = sales volume in units R(x) = total revenue associated with selling x units • Profit and Volume Model P(x) = R(x) – C(x) = 5x – (3000 + 2x) = - 3000 + 3x
- 16. Management Science Techniques 1.Linear Programming 2. Decision Analysis 3. Forecasting 4. Stock Control 5. Simulation 6. Network Modelling 7. Waiting Line or Queuing Models 8. Goal Programming 9. Dynamic Programming 10.Markov Process Models
- 17. Linear Programming Linear Programmingis a problem solving approach developed to help managers make decisions e.g. GE Capital uses Linear Programming to help determine optimal lease structuring. Marathon Oil Company uses Linear Programming for Gasoline blending and to evaluate the economics of new terminal or pipelines
- 18. Situations where LPcan be used • A manufacturer wants to develop a production schedule and an inventory policy that will satisfy sales demand in future periods. Ideally, the schedule and policy will enable company to satisfy demand and at the time minimize the total production and inventory costs • A financial analyst must select an investment portfolio from a variety of stock and bond investment alternatives. The analyst would like to establish the portfolio that maximizes the return on investment
- 19. • A marketingmanager wants to determine how best to allocate a fixed advertising budget among alternative advertising media such as radio, television, newspaper, and magazine. The manager would like to determine the media mix that maximizes advertising effectiveness. • A company has warehouses in number of locations throughout Unites States. For a set of customer demand, the company would like to determine how much each warehouses should ship to each customer so that transportation costs are minimized.
- 20. Commonality • Maximizing • Minimizing Somequantity Objective of LP is to Maximize or Minimize some quantity.
- 21. Maximizing • Par, Inc.is a small manufacturer of golf equipment and supplies. • Management has decided to move in to the market for medium and high priced golf bags • Par’s distributor is enthusiastic about the new product line and has agreed to buy all the golf bags Par produces over the next three months. After a thorough investigation of the steps involved in manufacturing a golf bag, managers determined that each golf bag produced will require the following operations
- 22. Par, Inc. • Cuttingand Dying the Material • Sewing • Finishing (inserting umbrella holder, club separators, etc) • Inspection and Packaging
- 23. Production Requirements pergolf bag Department Production time (Hours) Total Time Available Standard Bag Deluxe Bag (Hours) Cutting and Dying 7/10 1 630 Sewing ½ 5/6 600 Finishing 1 2/3 708 Inspection and Packaging 1/10 1/4 135 Profit Contribution Standard Bag $10/unit Deluxe Bag $ 9/unit
- 24. Problem Formulation • Understandingthe Problem Thoroughly • Describe the Objective • Describe each constraint - Constraint 1 - Constraint 2 - Constraint 3 - Constraint 4
- 25. • Define thedecision variables • Write the objective in terms of the decision variables • Write the constraint in terms of the decision variables Mathematical Statement of the Par, Inc. Problem
- 26. Max 10S +9D subject to (s.t.) 7/10 S + 1 D ≤ 630 1/2 S + 5/6 D ≤ 600 1 S + 2/3 D≤ 708 1/10 S + 1/4 D ≤ 135 S, D ≥ 0
- 27. A Simple MinimizationProblem • M & D Chemicals produces Product A & Product B that are sold to companies manufacturing both soaps and laundry detergents. Based on an analysis of current inventory levels and potential demand for the coming month, M & D chemical management specified that the combined production for Product A and B must total at least 350 gallons. Separately, a major customer’s order for 125 gallons of product A must also be satisfied
- 28. • Product Arequires 2 hrs of processing time per gallon and product B requires 1 hr of processing time per gallon. For the coming month, 600 hrs of processing time are available. • M & D’s objective is to satisfy these requirements at a minimum total production cost. Production costs are $2 per gallon for product A and $3 per gallon for product B.
- 29. • Objective Function Min.Z = 2x1 + 3x2 Constraints s. t. x1 ≥ 125 Customer Order’s Constraint x1 + x2 ≥ 350 Production Constraint 2x1 + x2 ≤ 600 Processing Time Constraint x1, x2 ≥ 0
- 30. • Infeasibility No solutionto the linear programming problem satisfies all the constraints, including the non-negativity conditions. Graphically, infeasibility means that a feasible region does not exist; that is no points satisfy all the conditions • Unbounded The solution to a maximizing linear programming problem is unbounded if the value of the solution may be made infinitely large without violating any of the constraints. For a minimization problem, the solution is unbounded if the value may be infinitely small.