Assigning Art

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? Decision Variable X1| 1| If “Perfection” of Colin Zweibell is displayed| | 0| otherwise| | | | X2| 1| If “Burden” of Colin Zweibell is displayed| | 0| otherwise| | | | X3| 1| If “The Great Equalizer” of Colin Zweibell is displayed| | 0| otherwise| | | |

X4| 1| If “Chaos Reigns” of Rita Losky is displayed| | 0| otherwise| | | | X5| 1| If “Who Has Control? ” of Rita Losky is displayed| | 0| otherwise| | | | X6| 1| If “Domestication” of Rita Losky is displayed| | 0| otherwise| | | | X7| 1| If “Innocence” of Rita Losky is displayed| | 0| otherwise| | | | X8| 1| If “Aging Earth” of Norm Marson is displayed| | 0| otherwise| | | | X9| 1| If “Wasted Resources” of Norm Marson is displayed| | 0| otherwise| | | | X10| 1| If “Serenity” of Candy Tate is displayed| | 0| otherwise| | | | X11| 1| If “Calm Before the Storm” of Candy Tate is displayed| | 0| otherwise| | | X12| 1| If “Void” of Robert Bayer is displayed| | 0| otherwise| | | | X13| 1| If “Sun” of Robert Bayer is displayed| | 0| otherwise| | | | X14| 1| If “Storefront Window” of David Lyman is displayed| | 0| otherwise| | | | X15| 1| If “Harley” of David Lyman is displayed| | 0| otherwise| | | | X16| 1| If “Consumerism” of Angie Oldman is displayed| | 0| otherwise| | | | X17| 1| If “Reflection” of Angie Oldman is displayed| | 0| otherwise| | | | X18| 1| If “Trojan Victory” of Angie Oldman is displayed| | 0| otherwise| | | | X19| 1| If “Rick” of Rick Rawls is displayed| | 0| otherwise| | | X20| 1| If “Rick II” of Rick Rawls is displayed| | 0| otherwise| | | | X21| 1| If “Rick III” of Rick Rawls is displayed| | 0| otherwise| | | | X22| 1| If “Beyond” of Bill Reynolds is displayed| | 0| otherwise| | | | X23| 1| If “Pioneers” of Bill Reynolds is displayed| | 0| otherwise| | | | X24| 1| If “Wisdom” of Bear Canton is displayed| | 0| otherwise| | | | X25| 1| If “Superior Powers” of Bear Canton is displayed| | 0| otherwise| | | | X26| 1| If “Living Land” of Bear Canton is displayed| | 0| otherwise| | | | X27| 1| If “Study of a Violin” of Helen Row is displayed| | 0| otherwise| | | X28| 1| If “Study of a Fruit Bowl” of Helen Row is displayed| | 0| otherwise| | | | X29| 1| If “My Namesake” of Ziggy Lite is displayed| | 0| otherwise| | | | X30| 1| If “Narcissism” of Ziggy Lite is displayed| | 0| otherwise| | | | X31| 1| If “All That Glitters” of Ash Briggs is displayed| | 0| otherwise| | | | X32| 1| If “The Rock” of Ash Briggs is displayed| | 0| otherwise| | | | X33| 1| If “Winding Road” of Ash Briggs is displayed| | 0| otherwise| | | | X34| 1| If “Dreams Come True” of Ash Briggs is displayed| | 0| otherwise| Objective Function Max.

Total pieces displayed at the exhibit Z = X1 + X2 + X3 + X4 + X5 +…. +X34 Subject to Constraint 1 : Budget constraint not exceed $4,000,000 300,000X1 + 250,000X2 + 125,000X3 + 400,000X4 + 500,000X5 + 400,000X6 + 550,000X7 + 700,000X8 + 575,000X9 + 200,000X10 + 225,000X11 + 150,000X12 + 150,000X13 + 850,000X14 + 750,000X15 + 400,000X16 + 175,000X17 + 450,000X18 + 500,000X19 + 500,000X20 + 500,000X21 + 650,000X22 + 650,000X23 + 250,000X24 + 350,000X25 + 450,000X26 + 400,000X27 + 400,000X28 + 300,000X29 + 300,000X30 + 50,000X31 + 50,000X32 +50,000X33 +50,000X34 ? ,000,000 Constraint 2 : Only one collage to be displayed X9 + X16 + X29 + X30 = 1 Constraint 3 : At least one wire-mesh sculpture displayed if a computer-generated drawing is displayed (>1) X1 + X2 + X3 + X4 + X5 ? 2 Constraint 4 : At least one wire-mesh sculpture displayed if a computer-generated drawing is displayed (=0) X1 + X2 + X3 + X4 + X5 ? 1 Constraint 5 : At least one photo-realistic painting X14 + X15 + X19 ? 1 Constraint 6 : At least one cubist painting X20 + X27 + X28 ? 1 Constraint 7 : At least one expressionist painting X21 ? 1

Constraint 8 : At least one watercolor painting X10 + X31 + X32 + X33 + X34 ? 1 Constraint 9 : At least one oil painting X12 + X13 + X22 + X23 + X26 ? 1 Constraint 10 : the number of paintings to be no greater than twice the number of other art forms X10 + X11 + X12 + X13 + X14 + X15 + X20 + X21 + X22 + X23 + X26 + X27 + X28 + X31 + X32 +X33 + X34 ? 2*(34 – sum of total number of painting) Constraint 11 : All Ash’s painting X31 + X32 + X33 + X34 = 4 Constraint 12 : All Candy’s painting X10 + X11 = 2 Constraint 13 : At least one David’s painting X14 + X15 ? 1

Constraint 14 : At least one Rick’s painting X19 + X20 + X21 ? 1 Constraint 15 : David’s painting is equal to Rick’s painting X14 + X15 – X19 – X20 – X21 ? 0 Constraint 16 : Only one Ziggy’s painting X29 + X30 ? 1 Constraint 17 : At least one piece from a female artist for every two pieces included from a male artist 2(X4 + X5 + X6 + X7 + X10 + X11 + X16 + X17 + X18 + X27 + X28) – (X1 + X2 +X3 +X8 +X9 +X12 +X13+X14 +X15+X19+X20 +X21+X22+X23 +X24+X25+X26 +X29 + X30+ X31+X32+X33 +X34 ) ? 0 Constraint 18 : Include either one or both of the pieces “Aging Earth” and Wasted Resources” X8 + X9 ? Constraint 19 : At least one Bear Canton’s paintings X24 + X25 + X26 ? 1 Constraint 20 : At least one of the following pieces: Chaos Resigns, Who has Control, Beyond and Pioneer X4 + X5 + X22 + X23 ? 1 Constraint 21 : Floor space for four sculptures X1 + X2 + X3 + X8 +X17 +X18 ? 4 Constraint 22 : Space for 20 paintings, collages, and drawings X4+X5 +X6+X7+X9 +X10+X11+X12 +X13+X14+X15 +X16 + X19+ X20+X21+X22+X23 +X24+X25+X26 +X27+X28+X29 +X30+X31+X32 +X33 + X34 ? 20 Constraint 23 : Narcissism is displayed, Reflection should also be displayed X17 + X30 ? Optimal Solution No feasible area. The budget ($4,000,000) does not cover all the requirements b) To ensure that the exhibit draws the attention of the public, Celeste decides that it must include at least 20 pieces. Formulate and solve a BIP model to minimize the cost of the exhibit while displaying at least 20 pieces and meeting the requirements set by Ash and Celeste. How much does the exhibit cost? Which pieces are displayed? New Objective Function : Min. Total Cost Z = 300,000X1 + 250,000X2 + 125,000X3 + 400,000X4 + 500,000X5 + 400,000X6 + 50,000X7 + 700,000X8 + 575,000X9 + 200,000X10 + 225,000X11 + 150,000X12 + 150,000X13 + 850,000X14 + 750,000X15 + 400,000X16 + 175,000X17 + 450,000X18 + 500,000X19 + 500,000X20 + 500,000X21 + 650,000X22 + 650,000X23 + 250,000X24 + 350,000X25 + 450,000X26 + 400,000X27 + 400,000X28 + 300,000X29 + 300,000X30 + 50,000X31 + 50,000X32 +50,000X33 +50,000X34 Revised Constraint Constraint 1 : Total pieces to be displayed are at least 20 pieces X1 + X2 + X3 + X4 + X5 +…. +X34 ? 20 Solution At optimal point, the exhibit cost is $5,575,000. Total pieces are 20 pieces.

Pieces to be displayed at the exhibit are listed as follows ; Colin Zweibell| “Burden”| A wire mesh sculpture of a mule| 250,000| sculpture| Male| Colin Zweibell| “The Great Equalizer”| A wire mesh sculpture of a gun| 125,000| sculpture| Male| Rita Losky| “Chaos Reigns”| A series of computer-generated drawings| 400,000| drawing| Female| Rita Losky| “Domestication”| A pen-and-ink drawing of a house| 400,000| drawing| Female| Norm Marson| “Aging Earth”| A sculpture of trash covering a $700,000 larger globe| 700,000| sculpture| Male|

Candy Tate| “Serenity”| An all blue watercolor painting| 200,000| painting| Female| Candy Tate| “Calm Before the Storm”| A painting with an all blue watercolor background and a black watercolor center| 225,000| painting| Female| Robert Bayer| “Void”| An all black oil painting| 150,000| painting| Male| Robert Bayer| “Sun”| An all yellow oil painting| 150,000| painting| Male| David Lyman| “Harley”| A photo-realistic painting of a Harley-Davidson motorcycle| 750,000| painting| Male|

Angie Oldman| “Reflection”| A mirror (considered a sculpture)| 175,000| sculpture| Female| Rick Rawls| “Rick III”| An expressionist self-portrait(painting)| 500,000| painting| Male| Bear Canton| “Wisdom”| A pen-and-ink drawing of an Apache chieftain| 250,000| drawing| Male|

Helen Row| “Study of a Violin”| A cubist painting of a violin| 400,000| painting| Female| Helen Row| “Study of a Fruit Bowl”| A cubist painting of a bowl of fruit| 400,000| painting| Female| Ziggy Lite| “Narcissism”| A collage of photographs of Ziggy Lite| 300,000| collage| Male| Ash Briggs| “All That Glitters”| A watercolor painting of the Golden Gate Bridge| 50,000| painting| Male| Ash Briggs| “The Rock”| A watercolor painting of Alcatraz| 50,000| painting| Male| Ash Briggs| “Winding Road”| A watercolor painting of Lombard Street| 50,000| painting| Male|

Ash Briggs| “Dreams Come True”| A watercolor painting of the San Francisco Museum of Modern Art| 50,000| painting| Male| Total cost = $5,575,000 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 exhibit and that all of Rita’s pieces are displayed. How much does the patron have to pay? Which pieces are displayed? Additional Constraint

Constraint 24 : All of Rita’s pieces are displayed X4 + X5 + X6 + X7 ? 4 Total cost is $5,975,000 which is covered 20 pieces of art and also included all Rita’s drawings. The additional cost (add on from $4,000,000) is = $1,975,000. Pieces to be displayed at the exhibit are listed as follows ; List of pieces chosen in C. Colin Zweibell| “The Great Equalizer”| A wire mesh sculpture of a gun| 125,000| sculpture| Male| Rita Losky| “Chaos Reigns”| A series of computer-generated drawings| 400,000| drawing| Female| Rita Losky| “Who Has Control? | A computer-generated drawing intermeshed with lines of computer code| 500,000| drawing| Female| Rita Losky| “Domestication”| A pen-and-ink drawing of a house| 400,000| drawing| Female| Rita Losky| “Innocence”| A pen-and-ink drawing of a child| 550,000| drawing| Female| Norm Marson| “Aging Earth”| A sculpture of trash covering a $700,000 larger globe| 700,000| sculpture| Male| Candy Tate| “Serenity”| An all blue watercolor painting| 200,000| painting| Female| Candy Tate| “Calm Before the Storm”| A painting with an all blue watercolor background and a black atercolor center| 225,000| painting| Female| Robert Bayer| “Void”| An all black oil painting| 150,000| painting| Male| Robert Bayer| “Sun”| An all yellow oil painting| 150,000| painting| Male| David Lyman| “Harley”| A photo-realistic painting of a Harley-Davidson motorcycle| 750,000| painting| Male| Angie Oldman| “Reflection”| A mirror (considered a sculpture)| 175,000| sculpture| Female| Rick Rawls| “Rick III”| An expressionist self-portrait(painting)| 500,000| painting| Male| Bear Canton| “Wisdom”| A pen-and-ink drawing of an Apache chieftain| 250,000| drawing| Male|

Helen Row| “Study of a Fruit Bowl”| A cubist painting of a bowl of fruit| 400,000| painting| Female| Ziggy Lite| “Narcissism”| A collage of photographs of Ziggy Lite| 300,000| collage| Male| Ash Briggs| “All That Glitters”| A watercolor painting of the Golden Gate Bridge| 50,000| painting| Male| Ash Briggs| “The Rock”| A watercolor painting of Alcatraz| 50,000| painting| Male| Ash Briggs| “Winding Road”| A watercolor painting of Lombard Street| 50,000| painting| Male| Ash Briggs| “Dreams Come True”| A watercolor painting of the San Francisco Museum of Modern Art| 50,000| painting| Male| Total cost = $5,975,000