Question $4$

Bilmarket needs to determine a hiring and firing strategy for their next three months according

to their labor needs. At the beginning of the first month, Bilmarket has no workers. For the first

month, the market must have $20$ workers available, $16$ for second month and $25$ for third month.

Cost of hiring a worker is $$100$ and firing a worker is $$50.$ Each worker receives a monthly wage of

$$140.$ Bilmarket does not want to hire more workers than required in a month.

a$)$ Create an LP to determine a hiring and firing strategy that minimizes total cost for the next

three months according to those instructions:

$-$Let $x_{ij}$ be the number of workers hired at the beginning of month i and fired after working till the

end of month j$-1.$

$-$Create $4$ constraints respectively. $(1)$ For workers available in the first month, $(2)$ for workers

available in the second month, $(3)$ for workers available in the third month and $(4)$ for non$-$negativity.

b$)$ Replace the first three constraints in part $($a$)$ by the following constraints in order to obtain a

MCNFP$,$ and explain why the resulting LP is an instance of the MCNFP$.$

i$)$ Constraint $(1)$

ii$)$ Constraint$(2)-$Constraint$(1)$

iii$)$ Constraint$(3)-$Constraint$(2)$

iv$)-($Constraint$(3))$

c$)$ Draw the network corresponding to this MCNFP$,$ and provide all the parameter values. Question $3$

For the following requirements, provide a MCNFP instance with $4$ nodes. Provide both the network

and the related $b,c$ and $u$ data.

a$)$ Provide a feasible but not basic flow vector, using at most $5$ arcs.

b$)$ Provide a basic but not feasible flow vector.

c$)$ Provide a degenerate basic feasible solution.

d$)$ Provide a basic and feasible flow vector using $5$ arcs.Question $1$ Turkish Airlines is starting a new flight for a very competitive flight line. The flight line starts with $$stanbul$,$ and then it goes to Belgrade, Vienna and Hamburg, respectively. Turkish Airlines is planning to assign an aircraft to this line with a capacity of $200$ passengers. Definitely, Turkish Airlines cannot meet all the demand on this line, and it will lose some customers to its rivals. The operations management managers of the company now want to make a plan to figure out which flights of this line are financially more important. Formulate this problem as an MCNFP with the objective of minimizing the total opportunity cost based on the demands and the ticket prices of the flights of this line given in the table below. $($Note: When you are asked to model a problem as an MCNFP$,$ you are not expected to formulate the problem from scratch. You should only provide the parameters of the MCNFP formulation and explain what your decision variables mean.$)$