Question

# Consider a capital budgeting problem with six projects represented by 0-1 text{variables} x1, x2, x3, x4, x5, text{and} x6. a. Write a constraint mode

Modeling

Consider a capital budgeting problem with six projects represented by $$0-1\ \text{variables}\ x_1,\ x_2,\ x_3,\ x_4,\ x_5,\ \text{and}\ x_6.$$
a. Write a constraint modeling a situation in which two of the projects 1, 3, and 6 must be undertaken.
b. In which situation the constraint "$$x_3\ -\ x_5 = 0$$" is used, explain clearly:
c. Write a constraint modeling a situation in which roject 2 or 4 must be undertaken, but not both.
d. Write constraints modeling a situation where project 1 cannot be undertaken IF projects 3. also is NOT undertaken.
e. Explain clearly the situation in which the following 3 constraints are used simulataneously (together):
$$\displaystyle{x}_{4}\le{x}_{1}$$
$$\displaystyle{x}_{4}\le{x}_{3}$$
$$\displaystyle{x}_{4}\ge{x}_{1}+{x}_{3}-{1}$$

2021-03-03

Since you have posted multiple sub parts questions but according to guidelines, we will solve first three sub parts for you. For rest of the sub parts resubmit the question again and specify the part you want us to solve.
Consider a capital budgeting problem with 6 projects represented by 0 or 1 and variables $$\displaystyle{x}_{{1}},{x}_{{2}},{x}_{{3}},{x}_{{4}},{x}_{{5}},{x}_{{6}}.$$
For part (a) it is required to find the constraint modelling a situation in which two of the project 1,3 and 6 must be undertaken.
$$\displaystyle{x}_{{1}}+{x}_{{3}}+{x}_{{6}}\ge{2}$$ For part (b) it is required to find the situation of the constraint $$\displaystyle{x}_{{3}}-{x}_{{5}}={0}.$$
constraint: $$\displaystyle{x}_{{3}}-{x}_{{5}}={0}$$
Project 3 and 5 must be selected together and none of them cannot be selected alone
$$\displaystyle\text{so}\ <\ {\left({x}_{{3}},{x}_{{5}}\right)}={\left({1},{1}\right)}{\quad\text{or}\quad}{\left({0},{0}\right)}$$
$$\displaystyle\Rightarrow{x}_{{3}}-{x}_{{5}}={0}$$
For part (c) it is required to write a constraint modelling a situation in which project 2 or 4 must be taken but not both.
$$\displaystyle{x}_{{2}}+{x}_{{4}}={1}$$
From the above constraint it is assure that one of the project is taken at a time not both as either
$$0\ +\ 1 = 1\ \text{or}\ 1\ +\ 0 = 1$$