Exam 12: Advanced Optimization Applications

The composite unit in data envelopment analysis

Which of the following is incorrect?

For a typical nonlinear problem,duals price are relatively insensitive to small changes in right-hand side values.

MegaSports,Inc.produces two high-priced metal baseball bats,the Slugger and the Launcher,that are made from special aluminum and steel alloys.The cost to produce a Slugger bat is $100,and the cost to produce a Launcher bat is $120.We can not assume that MegaSports will sell all the bats it can produce.As the selling price of each bat model - Slugger and Launcher - increases,the quantity demanded for each model goes down. Assume that the demand,S,for Slugger bats is given by S = 640  4P_S and the demand,L,for Launcher bats is given by L = 450  3P_L where P_S is the price of a Slugger bat and P_L is the price of a Launcher bat.The profit contributions are P_SS  100S for Slugger bats and P_LL  120L for Launcher bats.Develop the total profit contribution function for this problem.

List several industries in which revenue management has been applied and list several different types of decisions being made with the aid of the methodology.

The interpretation of the dual price for nonlinear models is different than the interpretation of the dual price for linear models.

Native Customs sells two popular styles of hand-sewn footwear: a sandal and a moccasin.The cost to make a pair of sandals is $18,and the cost to make a pair of moccasins is $24.The demand for these two items is sensitive to the price,and historical data indicate that the monthly demands are given by S = 400  10P₁ and M = 450  15P₂ ,where S = demand for sandals (in pairs),M = demand for moccasins (in pairs),P₁ = price for a pair of sandals,and P₂ = price for a pair of moccasins.To remain competitive,Native Customs must limit the price (per pair)to no more than $60 and $75 for its sandals and moccasins,respectively.Formulate this nonlinear programming problem to find the optimal production quantities and prices for sandals and moccasins that maximize total monthly profit.

Mountainside State Park has four visitor centers.To study the operation of these centers,a DEA model has been developed that compares inputs (size,number of staff,weekly hours of operation)and outputs (% of visitors attending educational program,daily sales in gift shop).The computer solution is shown below.What can you conclude about the efficiency of the North center? LINEAR PROGRAMMING PROBLEM Min $1 \mathrm { E } + 0 \mathrm { wn } + 0 \mathrm { ws } + 0 \mathrm { we } + 0 \mathrm { ww }$ S.T. 1) $- 400 \mathrm { E } + 400 \mathrm { wn } + 1200 \mathrm { ws } + 2400 \mathrm { we } + 1500 \mathrm { ww } < 0$ 2) $- 3 E + 3 w n + 6 w s + 10 w e + 7 w w < 0$ 3) $- 56 \mathrm { E } + 56 \mathrm { wn } + 108 \mathrm { ws } + 92 \mathrm { we } + 108 \mathrm { ww } < 0$ 4) $- 49 \mathrm { E } + 49 \mathrm { wn } + 83 \mathrm { ws } + 56 \mathrm { we } + 72 \mathrm { ww } > 0$ 5) $- 38 \mathrm { E } + 38 \mathrm { wn } + 425 \mathrm { ws } + 1200 \mathrm { we } + 630 \mathrm { ww } > 0$ 6) $+ 1 w n + 1 w s + 1 w e + 1 w w = 1$ OPTIMAL SOLUTION Objective Function Value = 1.000 Variable Value Reduced Cost E 1.000 0.000 wn 1.000 0.000 ws 0.000 0.000 we 0.000 0.000 ww 0.000 0.000 Constraint Slack/Surplus Dual Price 1 0.000 0.008 2 0.000 0.000 3 0.000 0.001 4 0.000 -0.034 5 0.000 -0.013 6 0.000 -1.000

A DEA linear programming model involving 4 input measures and 3 output measures will have

Hervis Car Rental in Austin,TX has 50 high-performance Shelby-H Mustangs in its rental fleet.These cars will be in greater demand than usual during the last weekend in July when the Central Texas Mustang Club holds its annual rally in Austin.At times like this,Hervis uses a revenue management system to determine the optimal number of reservations to have available for the Shelby-H cars. Hervis has agreed to have at least 60% of its Shelby-H Mustangs available for rally attendees at a special rate.Although many of the rally attendees will request a Saturday and Sunday two-day package,some attendees may select a Saturday only or a Sunday only reservation.Customers not attending the rally may also request a Saturday and Sunday two-day package,or make a Saturday only or Sunday only reservation.Thus,six types of reservations are possible.The cost for each type of reservation is shown here. Two-Day Package Saturday Only Sunday Only Rally \ 125 \ 75 \ 65 Regular \ 150 \ 85 \ 75 The anticipated demand for each type of reservation is as follows: Two-Day Package Saturday Only Sunday Only Rally 20 10 15 Regular 10 20 25 Hervis Car Rental would like to determine how many Shelby-H Mustangs to make available for each type of reservation in order to maximize total revenue. a.Define the decision variables. b.Formulate a linear programming model for this revenue management application.

The output shows the solution to a DEA model where facilities in Seaview (S),Farmington (F),Lewiston (L),and San Domingo (D)are compared.The inputs,in order,are number of machines,size of work force,and goodness of location.The outputs,in order,are production,quality rating,and on-time completion percentage.The model examines the efficiency of Lewiston. Min $0 \mathrm {~S} + 0 \mathrm {~F} + 0 \mathrm {~L} + 0 \mathrm { D } + 1 \mathrm { E }$ S.T. 1) $1 \mathrm {~S} + 1 \mathrm {~F} + 1 \mathrm {~L} + 1 \mathrm { D } = 1$ 2) $5 \mathrm {~S} + 22 \mathrm {~F} + 36 \mathrm {~L} + 15 \mathrm { D } - 36 \mathrm { E } < 0$ 3) $400 \mathrm {~S} + 1500 \mathrm {~F} + 3150 \mathrm {~L} + 1060 \mathrm { D } - 3150 \mathrm { E } < 0$ 4) $24 \mathrm {~S} + 13 \mathrm {~F} + 32 \mathrm {~L} + 17 \mathrm { D } - 32 \mathrm { E } < 0$ 5) $800 \mathrm {~S} + 2900 \mathrm {~F} + 1860 \mathrm {~L} + 1700 \mathrm { D } + 0 \mathrm { E } > 1860$ 6) $95 \mathrm {~S} + 92 \mathrm {~F} + 83 \mathrm {~L} + 94 \mathrm { D } + 0 \mathrm { E } > 83$ 7) $83 \mathrm {~S} + 85 \mathrm {~F} + 90 \mathrm {~L} + 91 \mathrm { D } + 0 \mathrm { E } > 90$ OPTIMAL SOLUTION Objective Function Value = 0.510 Variable Value Reduced Cost S 0.000 0.385 F 0.167 0.000 L 0.000 0.490 D 0.833 0.000 E 0.510 0.000 Constraint Slack/Surplus Dual Price 1 0.000 1.365 2 2.208 0.000 3 474.479 0.000 4 0.000 0.031 5 40.000 0.000 6 10.667 0.000 7 0.000 -0.021 a.Is the Lewiston plant efficient? Why or why not? If not, which plants should it emulate in order to improve? b.How much more production does the composite facility provide than the Lewiston site? c.What is the quality rating for the composite facility?

In data envelopment analysis,the percentage of an individual operating unit's resources that are available to the composite operating unit is the

Revenue management methodology enables an airline to maximize the number of full-fare seats it sells on each flight.

The goal of portfolio models is to create a portfolio that provides the best balance between risk and return.

In the case of functions with multiple local optima,most nonlinear optimization software methods can get stuck and terminate at a local optimum.

Provide several examples of both nonlinear objective functions and nonlinear constraints.

If the coefficient of each squared term in a quadratic function is positive,the function is

A feasible solution is a global optimum if there are no other feasible points with a better objective function value in the feasible region.

The objective for the Markowitz portfolio model is to minimize portfolio return variance.

Revenue management methodology was originally developed for