UNIVERSITY OF SAN CARLOSP. del Rosario Street

Cebu City, Philippines 6000

COLLEGE OF COMMERCEDepartment of Accountancy

Quantitative Techniques in Business Management(BA46)

MWF 3:30 – 4:30 PM

Problem No. 25 Page 176Applied Management Science 2nd Edition

(By Lawrence and Pasternack)

Flaire Belle A. Cabrera flairey@yahoo.com

Mr. Pepito EchavezInstructor

September 8, 2010 I. Problem

Application of Linear and Integer Programming Models

ZOO DESIGN.The San Diego Wild Animal Park has won countless awards for its design and concepts and its record of successfully breading many endangered species. Now an investment group wishes to bring a similar attraction to the Orlando, Florida, area. The group has secured and plans to develop a 350-acre parcel of land not too far from Disneyworld, Universal Studios, and other central Florida attractions.

The animal park can be thought of as being divided into seven general areas:

Zoo habitat attractions Show areas where animal shows will be seen throughout the

day Restaurant areas Retail establishments Maintenance areas “Green” areas – consisting of parks and other required green

spaces. Walkways and service roads that intermingle throughout the

park

The following is a list of zoning agency and other conditions that must be met by zoo planners:

Each acre devoted to habitat areas is expected to generate $1000 per hour in gross profit to the park and is to be surrounded by .03 acre of green area. At least 40% of the park will consist of habitat areas (not including the required green areas).

Each acre is devoted to show areas is expected to generate $900 per hour in gross profit to the park and is to be surrounded by .40 acre of green areas. At least 5% of the park will consist of show areas (not including the required green areas).

Combined, the habitat and show areas (excluding corresponding green areas) should not account for more than 70% of the park. Also, the show areas (including corresponding green areas) should not represent more than

20% of the combined acreage for habitat and show areas (including their corresponding green areas).

At least 25% of the park that is not dedicated to habitat and show areas (not including their required green areas) should be green areas.

Maintenance facility space is required as follows: .01 acre for each acre of habitat, .10 acre for every acre of shows, .08 acre for every acre of restaurants, .06 acre for every acre of retail establishments, .02 acre for every acre of green space and .04 acre for every acre of walkways/roads.

Restaurants will average .25 acre. Each must be surrounded by .15 acre of green space. It is estimated that each restaurant will generate $800 per hour in gross profit. The park should contain between 20 and 30 restaurants.

Retail stores will average .20 acre and will be surrounded by .10 acre of green areas. Each store will generate approximately $750 per hour in gross profit. The park should contain between 15 and 25 stores but there should at least as many restaurants as retail stores.

At least 10 acres of the park should be walkways and service roads. Adjoining each walkway and service road must be green areas equal to 25% of the corresponding walkway/service area.

At least 100 acres of the animal park should be park areas, which are green areas not required by the habitat and show areas

The park will be open 10 hours per day 365 days per year. It has fixed daily operating expenses of $2000,000.

Create an optimal design for the park that will maximize total hourly gross profit. The design should indicate the number of acres devoted to zoo habitat, show attractions, maintenance, walkways, and park areas, It should also detail the number and acreage required for restaurant and retail store areas and summarize the total green space in the animal park. What would be the total annual net profit of this design?

II. Definition of the Problem

Let Xz = zoo habitatLet Xs = show areaLet Xra = restaurant area

Let Xre = retailLet Xg = green areas (parks and other required green spaces)Let Xma = maintenanceLet Xw = walkways

III. Formulation

Objective FunctionMaximize total hourly gross profit = 1000Xz + 900Xs + 3200XrA + 3750Xre – 2000000

Subject to:

At least 40% of the park will consist of habitat areas (not including the required green areas).

(1)Xz>= 140

At least 5% of the park will consist of show areas (not including the required green areas).

(2)Xs >= 17.5

Combined, the habitat and show areas (excluding the corresponding areas should not account for more than 70% of the park.

(3)Xz + Xs <= 245

Also, the show areas (including the corresponding show areas) should not represent more than 20% of the combined acreage for habitat and show areas.

(4)-.206Xz + 1.12Xs <= 0

At least 25% of the park that is not dedicated to habitat and show areas (not including their required green areas) should be green areas.

(5)-.25Xs - .25Xz + Xg >= 87.5

Maintenance facility space is required as follows: .01 acre for each acre of habitat, .10 acre for every acre of shows, .08 acre for every acre of restaurants, .06 acre for every acre of retail establishments, .02 acre for every acre of green space and .04 acre for every acre of walkways/roads.

(6)Xma - .01Xz - .1Xs - .08Xra - .06Xre - .12Xg - .04Xw >= 0

Restaurants will average .25 acre. Each must be surrounded by .15 acre of green space.

(7)Xra <= 7.5(8)Xra >= 5

Retail stores will average .20 acre and will be surrounded by .10 acre of green areas.

(9)Xre <= 5(10) Xre >= 3

The park should contain between 15 and 25 stores but there should at least as many restaurants as retail stores.

(11) 4Xra - 5Xre >= 0

At least 10 acres of the park should be walkways and service roads.

(12) Xw >= 10

Adjoining each walkway and service road must be green areas equal to 25% of the corresponding walkway/service area.At least 100 acres of the animal park should be park areas, which are green areas not required by the habitat and show areas, restaurant and retail establishments and walkways/pathways.

(13) Xg - 0.03Xz - .4Xs -.6Xra - .5Xre - .25Xw >= 100(14) Xz + Xs + Xra + Xre + Xm + Xg + Xw <= 350

IV. Computer Solution

V. Final Answer

Number of acres devoted to:Zoo Habitat Areas = 158.1785 acresShow Areas = 17.5 acresMaintenance Areas = 20.40213 acresWalkways and Service Roads = 10 acres

Number and acreage required for:Retail Store Areas = 5 acres = 5 acres/.2 = 25 storesRestaurant Areas = 7.5 acres = 7.5acres/.25 = 30 restaurants

Summary of the total green area:Park areas = 105.1742 acresReq. green area for Zoo Areas = 0.03(158.1785) = 4.745355 acresReq. green area for Show Areas = .4(17.5) = 7 acresReq. green area for Restaurants = .6(7.5) = 4.5 acresReq. green area for Retail Stores = .5(5) = 2.5 acresReq. green area for Walkways = .25(10) = 2.5 acres

Total Green Areas = 131.4196 acres

Xz = 158.1785 x $1000 x 10 x 365Xs = 17.5 x $900 x 10 x 365Xre = 5 / .2 x $750 x 10 x 365Xra = 7.5 / .25 x $800 x 10 x 365

Hourly Income= Xz + Xs + Xre + Xra = (158.1785 x $1000 x 10 x 365) + (17.5 x $900 x 10 x 365) + ( (5/.2) x $750 x 10 x 365) + ( (7.5/ .25) x $800 x 10 x 365)= $216,678.47

Optimal Annual Profit= (Hourly income x 10 x 365) – (Fixed Daily Operating Exp x 365)= ($216,678.47 x 10 x 365) – ($2,000,000 X 365)= $60,876,415.5

VI. Sensitivity Analysis

VII. Conclusion and Recommendation

This is a very complicated, and difficult problem. I had a lot of difficulties in this as I had to consider zoning angency’s and other conditions for constructing the zoo in defining the constraints. I asked many people for their opinion on my problem; most of them agreed that my problem was hard to answer. They helped me formulate the constraints and understand the problem. I could not have done it on my own. I recommend that the person who is to answer this problem, should review and analyze carefully the problem. They should also understand the entirety of the problem, not just the specific points, because as I’ve learned in this problem, everything is not what it seems.

I have learned to be analytical and critical minded in approaching problems. I have also learned to be patient, and take things slow to fully comprehend this problem.