Modeling following constraints in MILP. I want to know how I should formulate the following constraints in my MIP problem? x=x_1z_1+⋯+x_n z_n and y_1 <= y <= y_n and z_1+ cdots +z_n=1

caritatsjq 2022-10-24 Answered
Modeling following constraints in MILP
I want to know how I should formulate the following constraints in my MIP problem?
x = x 1 z 1 + + x n z n  and  y 1 y y n  and  z 1 + + z n = 1
OR
y = y 1 w 1 + + y n w n  and  x 1 x x n  and  w 1 + + w n = 1
x and y are continuous variables. z 1 , , z n and w 1 , , w n are binary decision variables. x 1 , , x n and y 1 , , y n are parameters.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

Szulikto
Answered 2022-10-25 Author has 22 answers
Step 1
Brace yourself, the notation is about to get a little ugly.
Let me list the variables first. In addition to x and y (continuous), we will have z 1 , , z n { 0 , 1 } and w 1 , , w n { 0 , 1 }, plus z ^ , z ~ , w ^ , w ~ [ 0 , 1 ] and one more binary variable t { 0 , 1 }. The constraints will be as follows:
x = i x i z i + x 1 z ^ + x n z ~ y = i y i w i + y 1 w ^ + y n w ~ i z i = 1 t z ^ + z ~ = t i w i = t w ^ + w ~ = 1 t .
Step 2
If t = 0, the fourth equation zeroes out the last two terms of the first equation and the third equation plus the first equation result in x being one of the x i . Meanwhile, the fifth equation zeroes out the summation in the second equation and the sixth equation and what's left of the second equation make y a convex combination of the endpoints y 1 , y n , so basically any value in the interval [ y 1 , y n ]. If t = 1, the reverse occurs.
Did you like this example?
Subscribe for all access
varsa1m
Answered 2022-10-26 Author has 4 answers
Step 1
(1) x min x x max (2) y min y y max (3) ( x min i x i ) t x i x i z i x max t (4) t i z i 1 ( n 1 ) t (5) ( y min i y i ) ( 1 t ) y i y i w i y max ( 1 t ) (6) ( 1 t ) i w i 1 ( n 1 ) ( 1 t )
Step 2
Constraints (1) and (2) are valid for both sides of the desired disjunction. Constraints (3) and (4) enforce
t = 0 ( x = i x i z i i z i = 1 ) .
Constraints (5) and (6) enforce
t = 1 ( y = i y i w i i w i = 1 ) .
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-09-17
The function y=3.5x+2.8 represents the cost y (in dollars) of a taxi ride of x miles.
a. Identify the independent and dependent variables.
b. You have enough money to travel at most 20 miles in the taxi. Find the domain and range of the function.
asked 2021-09-07
A city water department is proposing the construction of a new water pipe, as shown. The new pipe will be perpendicular to the old pipe. Write an equation that represents the new pipe.
asked 2021-09-25
You sell small and large candles at a craft fair. You collect $ 144 selling a total or 28 candles. How many of each type of candle did you sell?
Given:
Total selling price is $144.
Total number of candles is 28.
Cost of large candle is $6.
Cost of small candle is $4.
asked 2021-07-14
Use the fallowing terms to create a concept map that compares models of population growth: logistic model, exponential model, density dependent factors, density independent factors, limited growth, unlimited growth, disease, drought, food availability, and forest fires.
asked 2022-11-09
Exponential model for population
I am having trouble with a population modeling problem. The first part states:
Assume that Switzerland's population grows at a rate of 0.18 percent a year and that the 1988 population was 6.7 million.
I got the model to be:
y ( t ) = ( 6.7 × 10 6 ) e 0.0018 t
I am having trouble with the second part of the problem:
If there is a net immigration of 10,000 people a year into Switzerland, write an expression for the population in year t.
I have tried multiple variations of adding 10000 and 10000t to different parts of the equation with no luck.
How do I account for the 10000 immigrants?
asked 2021-03-07
The article “Modeling Arterial Signal Optimization with Enhanced Cell Transmission Formulations presents a new method for timing traffic signals in heavily traveled intersections. The effectiveness of the new method was evaluated in a simulation study. In 50 simulations, the mean improvement in traffic flow in a particular intersection was 654.1 vehicles per hour, with a standard deviation of 311.7 vehicles per hour.
a) Find a 95% confidence interval for the improvement in traffic flow due to the new system.
b) Find a 98% confidence interval for the improvement in traffic flow due to the new system.
c) A traffic engineer states that the mean improvement is between 581.6 and 726.6 vehicles per hour. With what level of confidence can this statement be made?
d) Approximately what sample size is needed so that a 95%
confidence interval will specify the mean to within ± 50 vehicles per hour?
e) Approximately what sample size is needed so that a 98% confidence
interval will specify the mean to within ± 50 vehicles per hour?
asked 2021-11-23
What interger operation is modeled below?