You just won the lottery and you want to celebrate it with your friends. You decide to invest $1, 300,000 in a party: $1,000,000 for renting a fancy place and $300,000 for supplies. The objective of the event is to have fun! To measure this you will use a “happiness function” that depends on 7 variables (x): x_1 is the number of people in the party x_2 is the duration of the party in hours: x_3 the liters of beer (with unit cost of ca_3): x_4 the liters of other drinks (with unit cost of c_r5) x_5 the liters of soda (with unit cost of c_5) x_6 the number of confetti bags (with a cost per bag of c_6), and x_7 is the cost of the decoration. The happiness function is measured in “units of happiness: f’. You did some research and reached the following conclusions: For every 50 people invited 15 people are happy (i.e, 15f are produced). For every hour of the party 100 people are happy Each liter of beer makes 2 people happy Each liter of the other drinks makes 4 people happy Each liter of soda makes 1 person happy. Each bag of confetti makes 1 person happy For every $10,000 spent in decoration 15 people are happy In addition this information you know that you will have to spend more than $80,000 in decoration. The fixed cost of renting a fancy place is $100,000 and increases $150,000 for each hour of rent with a maximum of 8 hours. The place has a storage capacity for cold drinks for up to 800 liters. Moreover, for safety reasons the average alcohol percentage cannot be larger than 20 degree: beer has 5 and other drinks have 35 degree. Finally, the maximum number of people in the fancy place you are renting is 800. Model this problem using mathematical optimization solve this problem using MS Excel

## Expert Answer

a) Spreadsheet model of this problem is following

Max 15×1 + 100×2 + 2×3 4×4 + 1×5 + 1×6 + (15/10000)*x7

s.t.

x7 >= 80000

c3x3 + c4x4 + c5x5 + c6x6 + x7 <= 300000 *(note that values of c3, c4, c5, c6 is not given in the question, therefore while solving in MS Excel, i have considered these values as zero)*

150000×2 <= 1000000 – 100000

x2 <= 8

x3 + x4 + x5 <= 800

5×3 + 35×4 + 35×5 <= 20 (x3 + x4 + x5) or -15×3 + 15×4 + 15×5 <= 0

x1 <= 800

x1, x2, x3, x4, x5, x6, x7 >= 0

b) Solution of this problem using MS Excel is following

Note that value of coefficients c3, c4, c5, and c6 is not provided in the question. Therefore Solver will return that the cell values do not converge. This is because the feasible region is unbounded.

Formula: I2 =SUMPRODUCT(B2:H2,$B$11:$H$11) *copy to I2:I9*