It had been a dream come true for Ash Briggs, a struggling artist living in the...

     

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It had been a dream come true for Ash Briggs, a struggling artist living in the San Fran- cisco Bay Area. He had made a trip to the corner grocery store late one Friday after- noon to buy some milk, and on impulse, he had also purchased a California lottery ticket. One week later, he was a millionaire. Ash did not want to squander his winnings on materialistic, trivial items. Instead he wanted to use his money to support his true passion: art. Ash knew all too well the difficulties of gaining recognition as an artist in this postindustrial, technological soci- ety where artistic appreciation is rare and financial support even rarer. He therefore de- cided to use the money to fund an exhibit of up-and-coming modern artists at the San Francisco Museum of Modern Art. Ash approached the museum directors with his idea, and the directors became ex- cited immediately after he informed them that he would fund the entire exhibit in ad- dition to donating $1 million to the museum. Celeste McKenzie, a museum director, was assigned to work with Ash in planning the exhibit. The exhibit was slated to open one year from the time Ash met with the directors, and the exhibit pieces would re- main on display for two months. Ash began the project by combing the modern art community for potential artists and pieces. He presented the following list of artists, their pieces, and the price of dis- playing each piece' to Celeste. Artist Piece Description of Piece Price Colin Zweibell "Perfection" A wire mesh sculpture of $300,000 the human body "Burden" A wire mesh sculpture of a mule $250,000 "The Great Equalizer" A wire mesh sculpture of a gun $125,000 Rita Losky "Chaos Reigns" A series of computer-generated drawings $400,000 "Who Has Control?" A computer-generated drawing intermeshed with lines of computer $500,000 code "Domestication" A pen-and-ink drawing of a house $400,000 Norm Marson "Innocence" "Aging Earth" A pen-and-ink drawing of a child $550,000 A sculpture of trash covering a larger globe $700,000 "Wasted Resources" A collage of various packaging materials $575,000 Candy Tate "Serenity" An all blue watercolor painting $200,000 "Calm Before the Storm" A painting with an all blue $225,000 watercolor background and a black watercolor center Robert Bayer "Void" An all black oil painting $150,000 "Sun" An all yellow oil painting $150,000 David Lyman "Storefront Window" A photo-realistic painting of a $850,000 jewelry store display window "Harley" A photo-realistic painting of a Harley-Davidson motorcycle $750,000 Angie Oldman "Consumerism" A collage of magazine advertisements $400,000 "Reflection" A mirror (considered a sculpture) $175,000 "Trojan Victory" A wooden sculpture of a condom $450,000 'The display price includes the cost of paying the artist for loaning the piece to the museum, transporting the piece to San Francisco, constructing the display for the piece, insuring the piece while it is on display, and transporting the piece back to its origin. Artist Rick Rawls Piece "Rick" Description of Piece A photo-realistic self-portrait (painting) Price $500,000 "Rick II" A cubist self-portrait (painting) $500,000 "Rick III" An expressionist self-portrait (painting) $500,000 Bill Reynolds "Beyond" A science fiction oil painting depicting Mars colonization $650,000 "Pioneers" An oil painting of three astronauts $650,000 aboard the space shuttle Bear Canton "Wisdom" A pen-and-ink drawing of an $250,000 Apache chieftain "Superior Powers" A pen-and-ink drawing of a $350,000 traditional Native American rain dance "Living Land" An oil painting of the Grand Canyon $450,000 Helen Row "Study of a Violin" A cubist painting of a violin $400,000 "Study of a Fruit Bowl" A cubist painting of a bowl of fruit $400,000 Ziggy Lite "My Namesake" A collage of Ziggy cartoons $300,000 "Narcissism" A collage of photographs of Ziggy Lite $300,000 Ash Briggs "All That Glitters" A watercolor painting of the Golden Gate Bridge $50,000* "The Rock" "Winding Road" A watercolor painting of Alcatraz $ 50,000 A watercolor painting of Lombard Street $ 50,000 "Dreams Come True" A watercolor painting of the San Francisco Museum of Modern Art $ 50,000 *Ash does not require personal compensation, and the cost for moving his pieces to the museum from his home in San Francisco is minimal. The cost of displaying his pieces therefore only includes the cost of constructing the display and insuring the pieces. Ash possesses certain requirements for the exhibit. He believes the majority of Americans lack adequate knowledge of art and artistic styles, and he wants the exhibit to educate Americans. Ash wants visitors to become aware of the collage as an art form. but he believes collages require little talent. He therefore decides to include only one collage. Additionally, Ash wants viewers to compare the delicate lines in a three- dimensional wire mesh sculpture to the delicate lines in a two-dimensional computer- generated drawing. He therefore wants at least one wire mesh sculpture displayed if a computer-generated drawing is displayed. Alternatively, he wants at least one com- puter-generated drawing displayed if a wire mesh sculpture is displayed. Furthermore, Ash wants to expose viewers to all painting styles, but he wants to limit the number of paintings displayed to achieve a balance in the exhibit between paintings and other art forms. He therefore decides to include at least one photo-realistic painting, at least one cubist painting, at least one expressionist painting, at least one watercolor paint- ing, and at least one oil painting. At the same time, he wants the number of paintings to be no greater than twice the number of other art forms. Ash wants all his own paintings included in the exhibit since he is sponsoring the exhibit and since his paintings celebrate the San Francisco Bay Area, the home of the exhibit. Ash possesses personal biases for and against some artists. Ash is currently hav- ing a steamy affair with Candy Tate, and he wants both of her paintings displayed. Ash counts both David Lyman and Rick Rawls as his best friends, and he does not want to play favorites among these two artists. He therefore decides to display as many pieces from David Lyman as from Rick Rawls and to display at least one piece from each of them. Although Ziggy Lite is very popular within art circles, Ash believes Ziggy makes a mockery of art. Ash will therefore only accept one display piece from Ziggy, if any at all. Celeste also possesses her own agenda for the exhibit. As a museum director, she is interested in representing a diverse population of artists, appealing to a wide audi- ence, and creating a politically correct exhibit. To advance feminism, she decides to include at least one piece from a female artist for every two pieces included from a male artist. To advance environmentalism, she decides to include either one or both of the pieces "Aging Earth" and "Wasted Resources." To advance Native American rights. she decides to include at least one piece by Bear Canton. To advance science, she de- cides to include at least one of the following pieces: "Chaos Reigns," "Who Has Con- trol," "Beyond," and "Pioneers." Celeste also understands that space is limited at the museum. The museum only has enough floor space for four sculptures and enough wall space for 20 paintings, col- lages, and drawings. Finally, Celeste decides that if "Narcissism" is displayed, "Reflection" should also be displayed since "Reflection" also suggests narcissism. Please explore the following questions independently except where otherwise in- dicated. (a) Ash decides to allocate $4 million to fund the exhibit. Given the pieces available and the specific requirements from Ash and Celeste, formulate and solve a BIP model to maximize the number of pieces displayed in the exhibit without exceeding the budget. How many pieces are displayed? Which pieces are displayed? (b) To ensure that the exhibit draws the attention of the public, Celeste decides that it must in- clude at least 20 pieces. Formulate and solve a BIP model to minimize the cost of the ex- hibit while displaying at least 20 pieces and meeting the requirements set by Ash and Ce- leste. How much does the exhibit cost? Which pieces are displayed? (c) An influential patron of Rita Losky's work who chairs the Museum Board of Directors learns that Celeste requires at least 20 pieces in the exhibit. He offers to pay the minimum amount required on top of Ash's $4 million to ensure that exactly 20 pieces are displayed in the ex- hibit and that all of Rita's pieces are displayed. How much does the patron have to pay? Which pieces are displayed? It had been a dream come true for Ash Briggs, a struggling artist living in the San Fran- cisco Bay Area. He had made a trip to the corner grocery store late one Friday after- noon to buy some milk, and on impulse, he had also purchased a California lottery ticket. One week later, he was a millionaire. Ash did not want to squander his winnings on materialistic, trivial items. Instead he wanted to use his money to support his true passion: art. Ash knew all too well the difficulties of gaining recognition as an artist in this postindustrial, technological soci- ety where artistic appreciation is rare and financial support even rarer. He therefore de- cided to use the money to fund an exhibit of up-and-coming modern artists at the San Francisco Museum of Modern Art. Ash approached the museum directors with his idea, and the directors became ex- cited immediately after he informed them that he would fund the entire exhibit in ad- dition to donating $1 million to the museum. Celeste McKenzie, a museum director, was assigned to work with Ash in planning the exhibit. The exhibit was slated to open one year from the time Ash met with the directors, and the exhibit pieces would re- main on display for two months. Ash began the project by combing the modern art community for potential artists and pieces. He presented the following list of artists, their pieces, and the price of dis- playing each piece' to Celeste. Artist Piece Description of Piece Price Colin Zweibell "Perfection" A wire mesh sculpture of $300,000 the human body "Burden" A wire mesh sculpture of a mule $250,000 "The Great Equalizer" A wire mesh sculpture of a gun $125,000 Rita Losky "Chaos Reigns" A series of computer-generated drawings $400,000 "Who Has Control?" A computer-generated drawing intermeshed with lines of computer $500,000 code "Domestication" A pen-and-ink drawing of a house $400,000 Norm Marson "Innocence" "Aging Earth" A pen-and-ink drawing of a child $550,000 A sculpture of trash covering a larger globe $700,000 "Wasted Resources" A collage of various packaging materials $575,000 Candy Tate "Serenity" An all blue watercolor painting $200,000 "Calm Before the Storm" A painting with an all blue $225,000 watercolor background and a black watercolor center Robert Bayer "Void" An all black oil painting $150,000 "Sun" An all yellow oil painting $150,000 David Lyman "Storefront Window" A photo-realistic painting of a $850,000 jewelry store display window "Harley" A photo-realistic painting of a Harley-Davidson motorcycle $750,000 Angie Oldman "Consumerism" A collage of magazine advertisements $400,000 "Reflection" A mirror (considered a sculpture) $175,000 "Trojan Victory" A wooden sculpture of a condom $450,000 'The display price includes the cost of paying the artist for loaning the piece to the museum, transporting the piece to San Francisco, constructing the display for the piece, insuring the piece while it is on display, and transporting the piece back to its origin. Artist Rick Rawls Piece "Rick" Description of Piece A photo-realistic self-portrait (painting) Price $500,000 "Rick II" A cubist self-portrait (painting) $500,000 "Rick III" An expressionist self-portrait (painting) $500,000 Bill Reynolds "Beyond" A science fiction oil painting depicting Mars colonization $650,000 "Pioneers" An oil painting of three astronauts $650,000 aboard the space shuttle Bear Canton "Wisdom" A pen-and-ink drawing of an $250,000 Apache chieftain "Superior Powers" A pen-and-ink drawing of a $350,000 traditional Native American rain dance "Living Land" An oil painting of the Grand Canyon $450,000 Helen Row "Study of a Violin" A cubist painting of a violin $400,000 "Study of a Fruit Bowl" A cubist painting of a bowl of fruit $400,000 Ziggy Lite "My Namesake" A collage of Ziggy cartoons $300,000 "Narcissism" A collage of photographs of Ziggy Lite $300,000 Ash Briggs "All That Glitters" A watercolor painting of the Golden Gate Bridge $50,000* "The Rock" "Winding Road" A watercolor painting of Alcatraz $ 50,000 A watercolor painting of Lombard Street $ 50,000 "Dreams Come True" A watercolor painting of the San Francisco Museum of Modern Art $ 50,000 *Ash does not require personal compensation, and the cost for moving his pieces to the museum from his home in San Francisco is minimal. The cost of displaying his pieces therefore only includes the cost of constructing the display and insuring the pieces. Ash possesses certain requirements for the exhibit. He believes the majority of Americans lack adequate knowledge of art and artistic styles, and he wants the exhibit to educate Americans. Ash wants visitors to become aware of the collage as an art form. but he believes collages require little talent. He therefore decides to include only one collage. Additionally, Ash wants viewers to compare the delicate lines in a three- dimensional wire mesh sculpture to the delicate lines in a two-dimensional computer- generated drawing. He therefore wants at least one wire mesh sculpture displayed if a computer-generated drawing is displayed. Alternatively, he wants at least one com- puter-generated drawing displayed if a wire mesh sculpture is displayed. Furthermore, Ash wants to expose viewers to all painting styles, but he wants to limit the number of paintings displayed to achieve a balance in the exhibit between paintings and other art forms. He therefore decides to include at least one photo-realistic painting, at least one cubist painting, at least one expressionist painting, at least one watercolor paint- ing, and at least one oil painting. At the same time, he wants the number of paintings to be no greater than twice the number of other art forms. Ash wants all his own paintings included in the exhibit since he is sponsoring the exhibit and since his paintings celebrate the San Francisco Bay Area, the home of the exhibit. Ash possesses personal biases for and against some artists. Ash is currently hav- ing a steamy affair with Candy Tate, and he wants both of her paintings displayed. Ash counts both David Lyman and Rick Rawls as his best friends, and he does not want to play favorites among these two artists. He therefore decides to display as many pieces from David Lyman as from Rick Rawls and to display at least one piece from each of them. Although Ziggy Lite is very popular within art circles, Ash believes Ziggy makes a mockery of art. Ash will therefore only accept one display piece from Ziggy, if any at all. Celeste also possesses her own agenda for the exhibit. As a museum director, she is interested in representing a diverse population of artists, appealing to a wide audi- ence, and creating a politically correct exhibit. To advance feminism, she decides to include at least one piece from a female artist for every two pieces included from a male artist. To advance environmentalism, she decides to include either one or both of the pieces "Aging Earth" and "Wasted Resources." To advance Native American rights. she decides to include at least one piece by Bear Canton. To advance science, she de- cides to include at least one of the following pieces: "Chaos Reigns," "Who Has Con- trol," "Beyond," and "Pioneers." Celeste also understands that space is limited at the museum. The museum only has enough floor space for four sculptures and enough wall space for 20 paintings, col- lages, and drawings. Finally, Celeste decides that if "Narcissism" is displayed, "Reflection" should also be displayed since "Reflection" also suggests narcissism. Please explore the following questions independently except where otherwise in- dicated. (a) Ash decides to allocate $4 million to fund the exhibit. Given the pieces available and the specific requirements from Ash and Celeste, formulate and solve a BIP model to maximize the number of pieces displayed in the exhibit without exceeding the budget. How many pieces are displayed? Which pieces are displayed? (b) To ensure that the exhibit draws the attention of the public, Celeste decides that it must in- clude at least 20 pieces. Formulate and solve a BIP model to minimize the cost of the ex- hibit while displaying at least 20 pieces and meeting the requirements set by Ash and Ce- leste. How much does the exhibit cost? Which pieces are displayed? (c) An influential patron of Rita Losky's work who chairs the Museum Board of Directors learns that Celeste requires at least 20 pieces in the exhibit. He offers to pay the minimum amount required on top of Ash's $4 million to ensure that exactly 20 pieces are displayed in the ex- hibit and that all of Rita's pieces are displayed. How much does the patron have to pay? Which pieces are displayed?

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To solve the given optimization problems we can use binary integer programming BIP models Lets start by formulating the BIP model for each question a Maximizing the Number of Pieces Displayed within t... View the full answer