LP-assignments
--------------

Solve the following LP-problems using CPLEX:

1)

      max  4 x1 + 2 x2 + x3

s.t.         x1             <= 1
           4 x1 +   x2      <= 4
           8 x1 + 4 x2 + x3 <= 16

             x1, x2, x3 >= 0


2)

      max   2 x1 +   x2 

s.t.          x1 +   x2 <= 5
             -x1 +   x2 <= 0
            6 x1 + 2 x2 <= 21
 
              x1, x2 >= 0

3)    

      max     x1 +   x2 +   x3

s.t.                 x2 + 2 x3 <= 1
              x1 + 2 x2 +   x3 <= 1
            2 x1               <= 1

              x1, x2, x3 >= 0

4)

      min     x1 +   x2 +   x3

s.t.                 x2 + 2 x3 >= 1
              x1 + 2 x2        >= 1
            2 x1 +   x2        >= 1

              x1, x2, x3 >= 0


5) Formulation of an IP-problem
-------------------------------

In the strategic manpower planning problem we consider a strategic
assessment of the demand of manpower for an organisation. Consider an
organisation that is open 7 days a week with 1 shift a day.

- The number of employees needed varies from day to day but is
  constant on a weekly basis (we will call those numbers b1 for
  monday, b2 for tuesday etc.)

- All employees must work 5 consecutive days and have two days off.

- The objective is to minimize the number of employees and find out
  when they have to work.

a) Formulate a mathematical model that solves the strategic manpower
   planning problem as described above.

b) Solve the problem for b1=10, b2=5, b3=10, b4=5, b5=10, b6=5, b7=10.

c) Solve the problem for b1=8, b2=8, b3=8, b4=8, b5=8, b6=8, b7=7. 

d) How can you compute the unused number of man days? Calculate the
   number of unused days for the two examples above.

e) The labour costs vary depending upon the days off. How can we
   change the objective function to incorporate that?


6) Formulation of a "prototype" LP-problem
------------------------------------------

The food production company NiceMeat produces hot dogs and
bratwursts. Hot dog bundles provide a profit of 1000 Danish kroner a
piece, while bratwursts provide a profit of 1500 Danish kroner (each
bundle contains 1000 items). 

There are only three factors that limit the production of hot dogs and
bratwursts. First, the preprocessing machine runs a total of 150
machine-hours per week available. A hot dog bundle requires 1 hour of
preprocessing, while a bratwurst bundle requires 2 hours. Second, the
mixing facility can only handle 5000 pounds per week. Each hot dog and
bratwurst bundle weighs 45 pounds. Finally, the packing department
only has 400 boxes per week. Each hot dog bundle requires 3 boxes,
and each bratwurst bundle requires 4 boxes.

Formulate an LP-model for the problem, and solve it using CPLEX.


7) Formulation of a "prototype" LP-problem
------------------------------------------

The oil company Dat Oil buys crude oil, refines it and sells the
refined products. Prices, amounts etc. are given below:


Raw oil type   Octane   Barrels/day   Price/barrel

     1           68         4000         1.02 $
     2           86         5050         1.15 $
     3           91         7100         1.35 $
     4           99         4300         2.75 $

Refined oil   Min. Oct.   Demand      Sales Price/Barrel

     1           95        <= 10000        5.15 $
     2           90        infinity        3.95 $
     3           85        >= 15000        2.99 $

Raw oil not used in the production is sold at he prices 1.85 $ a
barrel for types 1 and 2, and 2.95 $ a barrel for types 3 and 4.

Dat Oil wants to identify a production plan to maximize its
profit. Formulate an LP-model for the problem and solve it using
CPLEX.