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**Readings**• Readings • Chapter 5 • Advanced Linear Programming Applications**Overview**• Overview**Tool Summary**Tool Summary • Do not make integer restrictions, and maybe the solution are integers. • Second Example: IMD = number of Indianapolis-Memphis-Discount seats • Use compound variables: • Second Example: IMD = number of Indianapolis-Memphis-Discount seats • Constrain a weighted average with a linear constraint: • Third Example: Constrain the weighted average risk factor to be no greater than 55: 60X1 + 70X2 + 75X3 + 20X4 + 30X5 + 22X6 + 50X7 + 10X9 < 55(X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10) • Interpret the unrealistic assumptions needed for a linear formulation: • First Example: Assume the set of alternative inputs and outputs for a high school is convex. Thus if 100 teaching hours gets 20 more students admitted to college, then 10 teaching hours gets at least 2 admitted. • Second Example: Assume demand has only two values. • Third Example: Assume risk is a linear function of investment shares.**Tool Summary**Tool Summary • Critique the unrealistic or approximate variables used because data on better variables may not be available: • First Example: High school output is measured by only three variables. • Average SAT Scores, even though maximizing those scores is not the same as maximizing learning. • The number of High School Graduates, even though there is no accounting for learning beyond a minimal level. • The number of College Admissions, even though there is no accounting for the quality or selectivity of the colleges. • First Example: High school input is measured by only three variables. • Senior Faculty • Budget ($100,000's) • Senior Enrollments, even though there is no measure for the quality of those students before their senior year.**Data Envelopment Analysis**• Data Envelopment Analysis**Data Envelopment Analysis**Overview Data Envelopment Analysis measures the relative efficiency of operating units with the same goals and objectives, and the same types of resources. Data Envelopment Analysis applies to fast-food outlets within the same chain, to hospitals, banks, courts, schools, and so on.**Data Envelopment Analysis**Overview • Data Envelopment Analysis creates a fictitious composite unit made up of an optimal weighted average (w1, w2,…) of existing units. • An individual unit, k, can be compared by determining E, the fraction of unit k’s input resources required by the optimal composite unit to achieve k’s goals and objectives. • If E < 1, unit k is less efficient than the composite unit, and is deemed relatively inefficient. • If E = 1, there is no evidence that unit k is inefficient, but one cannot conclude that k is best without further information (such as the value of 1 point higher Average SAT Score compared with 1 more College Admission in the following example).**Data Envelopment Analysis**Overview Min E s.t. Weighted outputs > Unit k’s output (for each measured output) Weighted inputs <E [Unit k’s input] (for each measured input) Sum of weights = 1 E, weights > 0**Data Envelopment Analysis**Question: The Langley County School District is trying to determine the relative efficiency of its three high schools. In particular, it wants to evaluate Roosevelt High. The district is evaluating performances on SAT scores, the number of seniors finishing high school, and the number of students admitted to college [outputs] as a function of the number of teachers teaching senior classes, the prorated budget for senior instruction, and the number of seniors enrolled [inputs]. • InputRooseveltLincolnWashington Senior Faculty 37 25 23 Budget ($100,000's) 6.4 5.0 4.7 Senior Enrollments 850 700 600 • OutputRooseveltLincolnWashington Average SAT Score 800 830 900 High School Graduates 450 500 400 College Admissions 140 250 370**Data Envelopment Analysis**Answer: • Define the decision variables E = Fraction of Roosevelt's input resources required by the composite high school w1 = Weight applied to Roosevelt's input/output resources by the composite high school w2 = Weight applied to Lincoln’s input/output resources by the composite high school w3 = Weight applied to Washington's input/output resources by the composite high school • Define the objective function. Minimize the fraction of Roosevelt High School's input resources required by the composite high school: Min E**Data Envelopment Analysis**• Constrain the sum of the weights to one: (1) w1 + w2 + w3 = 1 • Constrain each output of the composite school to be at least Roosevelt’s: (2) 800w1 + 830w2 + 900w3> 800 (SAT Scores) (3) 450w1 + 500w2 + 400w3> 450 (Graduates) (4) 140w1 + 250w2 + 370w3> 140 (College Admissions) • Constrain the inputs used by the composite high school to be no more than the multiple, E, of the inputs available to Roosevelt: (5) 37w1 + 25w2 + 23w3< 37E (Faculty) (6) 6.4w1 + 5.0w2 + 4.7w3< 6.4E (Budget) (7) 850w1 + 700w2 + 600w3< 850E (Seniors)**Data Envelopment Analysis**Interpretation: The output shows that the composite school is made up of equal weights of Lincoln and Washington. Roosevelt is 76.5% efficient compared to this composite school when measured by High School Graduates (because of the 0 slack on this constraint (#3)). It is less than 76.5% efficient if output were only measured by SAT Scores and College Admissions (there is positive slack in constraints 2 and 4.)**Revenue Management**• Revenue Management**Revenue Management**Overview Revenue Management Problems are Resource Allocation Problems when inputs are fixed. Revenue Management Problems thus help airlines determine how many seats to sell at an early-reservation discount fare and many to sell at a full fare. Other applications include hotels, apartment rentals, car rentals, cruise lines, and golf courses.**Revenue Management**Answer: LeapFrogAirways provides passenger service for Indianapolis, Baltimore, Memphis, Austin, and Tampa. LeapFroghas two WB828 airplanes, one based in Indianapolis and the other in Baltimore. Each morning the Indianapolis based plane flies to Austin with a stopover in Memphis. The Baltimore based plane flies to Tampa with a stopover in Memphis. Both planes have a coach section with a 120-seat capacity, and they arrive in Memphis at the same time. LeapFrog uses two fare classes: a discount fare D class and a full fare F class. Leapfrog’s products, each referred to as an origin destination itinerary fare (ODIF), are listed on the next slide with their fares and forecasted demand. LeapFrog wants to determine how many seats it should allocate to each ODIF.**Revenue Management**Answer: Define 16 decision variables, one for each ODIF.For example, IMD = number of seats allocated to Indianapolis-Memphis-Discount class.**Revenue Management**Simplification:Although the Revenue Management problem is an Integer Linear Programming problem, it has a special form that allows it to be formulated without integer constraints, and the solutions turn out to be integers.**Revenue Management**• Define the objective function. Maximize total revenue: Max (fare per seat for each ODIF) x (number of seats allocated to the ODIF) Max 175IMD + 275IAD + 285ITD + 395IMF + 425IAF + 475ITF + 185BMD + 315BAD + 290BTD + 385BMF + 525BAF + 490BTF + 190MAD + 180MTD + 310MAF + 295MTF**Revenue Management**• Define the 4 capacity constraints, one for each flight leg: Indianapolis-Memphis leg (1) IMD + IAD + ITD + IMF + IAF + ITF < 120 Baltimore-Memphis leg (2) BMD + BAD + BTD + BMF + BAF + BTF < 120 Memphis-Austin leg (3) IAD + IAF + BAD + BAF + MAD + MAF < 120 Memphis-Tampa leg (4) ITD + ITF + BTD + BTF + MTD + MTF < 120**Revenue Management**• Define the first 8 demand constraints, one for each ODIF: (5) IMD <44 (6) IAD < 25 (7) ITD < 40 (8) IMF < 15 (9) IAF < 10 (10) ITF < 8 (11) BMD <26 (12) BAD < 50**Revenue Management**• Define the remaining 8 demand constraints, one for each ODIF: • (13) BTD < 42 (14) BMF < 12 (15) BAF < 16 (16) BTF < 9 • (17) MAD < 58 (18) MTD < 48 (19) MAF < 14 (20) MTF < 11**Revenue Management**Interpretation: Total revenue = $94,735.00, with the specified seat allocation.**BA 452**Quantitative Analysis End of Lesson A.11