1 / 30

Management Science/ Operations Research

Management Science/ Operations Research.

feo
Download Presentation

Management Science/ Operations Research

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Management Science/Operations Research

  2. Subjects to be covered –1. Decision Making Under Uncertainty/ Risk, Marginal Analysis, Decision Trees, Game Theory2. Forecasting3. Linear Programming, LP Extensions4. Transportation/Assignment Problems5. Network Models, PERT/CPM6. Queuing7. Simulation, Markov

  3. Linear ProgrammingObjectives:1. Introduce LP as an aid to decision making and discuss its main characteristics.2. Demonstrate graphical method in solving LP problems as well as attendant technical issues in its use.3. Discuss the simplex method under both the minimization and maximization environments.

  4. Linear Programming Examples: 1. Bank – allocate funds to achieve highest possible return  liquidity limit set by CB  flexibility to meet withdrawal demands by depositors 2. Manufacturer – maximize profits, produce mix of tables & chairs  limits on production time for departments: assembly/finishing  demand of tables/chairs from customers 3. Socio-civic organization – hi protein food mix at lowest cost  choose from 10 possible ingredients  minimize cost

  5. Linear Programming LP – mathematical technique  find best use  organization’s resources Linear:direct proportionality of relationship of variables Programming:make schedules or plans of activities to undertake in the future LP defined:planning by use of linear relationship of variables involved Major Characteristics of LP Problem: 1. Firm  objective to achieve  expressed as a function  could be maximize or minimize

  6. Linear Programming Major Characteristics of LP Problem: 2. Alternative courses of action, one of w/c will achieve objective 3. Resources must be limited nonnegativity constraint  limits actions of decision maker  decision not violate constraints 4. Able to express objective function & limitations (constraints) as mathematical equations or inequalities;must be linear equations/inequalities Max/min objective function Subject to { constraints { x>=0, y>=0

  7. Linear Programming Two Methods of Solutions: • Graphical method – used for 2 or 3 variables 2. Simplex method – handles problems w/ any number of variables

  8. Linear Programming Graphical Method Steps to follow in graphical method: 1. Represent the unknown in the problem 2. Tabulate data about unknown 3. Formulate objective function/constraints 4. Graph constraints; solve for coordinates at point of intersection of lines 5. Substitute coordinates at vertices of feasible region in objective function 6. Formulate decision  select optimal value

  9. Linear ProgrammingMargan Furniture makes 2 products: tables & chairs, which needs processing in assembly & finishing departments. Assembly is available for 60 hours/production period; finishing for 48 hours. To finish one table requires 4 hours in assembly and 2 hours in finishing. A chair requires 2 hours in assembly and 4 hours in finishing. One table contributes P180 to profit; a chair P100. Determine the number of tables and chairs to produce per production period to maximize profit.

  10. Linear Programming Technical Issues 1. Extreme points – if LP has solution, found in one of extreme points or corners of the feasible region 2. Infeasibility – no solution satisfies all the constraints 3. Unboundedness –feasible region extends indefinitely > problem not correctly formulated > no real situation permits management an infinitely large solution or profit 4. Redundancy – constraint w/c does not affect feasible region 5. Alternative Optima – any point on the extreme line bordering the feasible region may produce optimal solution

  11. LP Application – Media Selection Pleasant Tours promote trips from a large Midwestern city to resorts in the Bahamas. The agency has budgeted up to $8,000 per week for local advertising. The money is to be allocated among four promotional media: TV spots, newspaper ads, and two types of radio ads. Pleasant’s goal is to reach the largest possible high-potential audience through the various media. The following table presents the number of potential tourists reached by making use of an ad in each of the four media. It also provides the cost per ad placed and the maximum number of ads that can be purchased per week.

  12. LP Application – Media Selection Audience/ Cost per Maximum ads Medium Ad Ad ($) per week TV spot 5,000 800 12 (1 minute) Daily newspaper 8,500 925 5 (full-page ad) Radio spot (30 sec, 2,400 290 25 prime time Radio spot (1 min, 2,800 380 20 afternoon

  13. LP Application – Media Selection Pleasant’s contractual arrangements require that at least five radio spots be placed each week. To ensure a broad-scoped promotional campaign, management also insists that no more than $1,800 be spent on radio advertising every week. Determine the number of ads to be placed in each media to maximize audience coverage.

  14. Transportation and Assignment ModelsObjectives:1. Structure special LP problems using transportation and assignment models.2. Use northwest corner, stepping stone, Vogel’s approximation in solving transport problems.3. Handle unbalanced transport and assignment problems.

  15. Transportation ProblemThe RGV Gravel Supply Co. has received a contract to supply gravel to three new road projects located at three different locations. Project A needs 174 truckloads, B needs 204 truckloads, and C needs 143 truckloads.The company has three gravel warehouses located in three different places:Warehouse 1 has 158 truckloads available; Warehouse 2 has 184;Warehouse 3 has 179.

  16. Transportation Problem (Continued)The cost of transportation from the warehouse to the projects are:From warehouse 1 to Projects A, B, C: P4, P8, P8 per truckload respectively;From warehouse 2: P16, P24, P16 respectively;From warehouse 3: P8, P16, P24 respectively.The objective is to design a plan of distribution that will minimize the cost of transportation.

  17. Assignment ProblemFour engineers are to work on 4 projects of ABC Construction Co. The problem is to decide which engineer should be assigned to which project. Each engineer charges different fees on each project, due to distances of the projects and the complexity of the work. The cost of assigning particular engineers to particular projects are as follows:

  18. LINEAR PROGRAMMING EXTENSIONS Integer Programming Goal Programming Objective: Introduce the above LP extensions and solve applicable problems employing them.

  19. I. Integer ProgrammingDefined: An LP in which decision variables are required to take on integer (whole number) valuesExamples: Margan Furniture problem on tables and chairsassembly, finishing constraints x, y >= 0 and integral

  20. II. Goal ProgrammingDefined:A variation of the simplex algorithm which permits the decision maker to specify multiple objectives and to target goals in order of their priorities.Expands on LP’s limitation of being one dimensio-nal in objective function formulation.Organizations generally have several goals whichfrequently conflict with each other:e.g. Profit Service Revenue Cost

  21. II. Goal Programming (Continued)General procedure –Set estimated target for each goal Rank goals in order of importanceàGP formulation tries to minimize deviations from set targets Starts with most important goal until achievement of a less important goal would cause failure to achievea more important oneModels –1.Single-goal model2.Equally ranked multiple goals3.Priority-ranked multiple goals

  22. II. Goal Programming (Continued)Application – Harrison Electric Co.Original LP problemMax profit 7x + 6y Subject to 2x + 3y <= 12 (wiring hours) 6x + 5y <= 30 (assembly hours) x, y >= 0 where x - number of chandeliers y - number of ceiling fans produced

  23. Goal Programming ApplicationHarrison ElectricDeviational variables:d1 - = underachievement of profit target d2 + = overachievement of profit targetRestate problem as a single-goal programming model:Min under or overachievement of profit target (d1 -) + (d2 +)Subject to 7x + 6y + (d1-) - (d2+) = 30 (profit goal constraint) 2x + 3y <= 12 (wiring hours constraint) 6x + 5y <= 30 (assembly hours cons.)Extension to equally important multiple goals –Goal 1: produce as much profit above $30 as possible during production periodGoal 2: fully utilize available wiring department hours Goal 3: avoid overtime in assembly departmentGoal 4: meet contract req’t to produce at least 7 ceiling fans

  24. Goal Programming ApplicationHarrison ElectricRevised deviational variables (d1 -) = underachievement of the profit target (d1 +) = overachievement of the profit target (d2 -) = idle time in the wiring department (underutilizatiion) (d2 +) = overtime in the wiring department (overutilization) (d3 -) = idle time in the assembly department (underutilization) (d3 +) = overtime in the assembly department (overutilization) (d4 -) = underachievement of the ceiling fan goal (d4 +) = overachievement of the ceiling fan goal Revised objective function/constraints –Min total deviation (d1 -) + (d2 -) + (d3 +) + (d4 -)Subject to 7x + 6y + (d1 -) – (d1 +) = 30 (profit constraint) 2x + 3y + (d2 -) – (d2 +) = 12 (wiring hours constraints) 6x + 5y + (d3 -) – (d3 +) = 30 (assembly constraint) y + (d4 -) – (d4 +) = 7 (ceiling fan constraint) All x, y, di variables >=0

  25. Goal Programming ApplicationHarrison ElectricRanking goals with priority levelsGoalPriorityReach a profit as much above $30 as possible P1 Fully use wiring department hours available P2 Avoid assembly department overtime P3 Produce at least 7 ceiling fans P4Revised objective function with ranked goals –Min total deviation P1(d1 -) + P2(d2 -) + P3(d3 +) + P4(d4 -)Constraints remain the same

  26. NETWORK MODELS Learning Objectives: Discuss the methods of solution and applications of the three network models – o minimal-spanning tree, o maximal flow, and o shortest route.

  27. I. Minimal-Spanning-Tree Problem Description: Find a way to reach all the nodes in a networkfrom some particular node (a source) in such a way that the total length of all thebranches used is minimal. Example: Subdivision developer - laying out pipeline for water connectionComputer engineers laying out a local area network (LAN) connection in an office

  28. II. Maximal-Flow Problem Description: In a network with one source node (point of entry) and one output node (point of exit), maximal-flow problems seek to find themaximal flow (whether cars, planes, fluids, electricity) which can enter the network at thesource node,flow through it, and exit at theoutput node in a given period of time.

  29. Maximal-Flow Problems (continued)Example:EDSA MRT Project: choices – elevated street level underground

  30. III. Shortest Route Problem Description: Find shortest route from a sourceto destinationthrough a connecting network. Examples:1.  Tourist bus travel 2.  Chartered airplane flight navigation

More Related