1. A.
x1*p1 + x2*p2 --> max (where is p1 and p2 is unit profits)
x1*5 + x2*11 <= 550
x1*5 + x2*3 <= 300
x2 <= 40
x1 >= 0, x2 >= 0
B.
x1 - meters of light duty clothes
x2 - meters of heavy duty clothes
Each metre of light cloth requires five metres of raw cotton and each metre of heavy cloth
requires eleven metres of raw cotton. Due to we have only 550 metres of raw cotton comes this
equation: x1*5 + x2*11 <= 550
Same with processing time one metre of light cloth requires five hours, and each metre of heavy
cloth requires 3 hours. We have only 300 hours available, so we have one more equation:
x1*5 + x2*3 <= 300
Also we have demand for heavy duty cloth is at most 40 metres from which x2 <= 40
And also we can’t have negative meters of clothes, so x1 >= 0, x2 >= 0
C.
A linear programming problem or LP problem in two unknowns x1 and x2 is one in which we are
to find the maximum or minimum value of a linear expression
x1*p1 + x2*p2 (where p1 and p2 is constants)
called the objective function
The solution set of the collection of constraints is called the feasible region of the LP problem.
The largest or smallest value of the objective function is called the optimal value, and a pair of
values of x1 and x2 that gives the optimal value constitutes an optimal solution.
D.
For equation x1*5 + x2*11 <= 550 let’s draw line 5*x + 11*y = 550 (x1 = x, x2 = y)
Everything upon it will be red, and everything down it white.
For equation x1*5 + x2*3 <= 300 The line will be 5*x + 3*y = 300, everything upon it will be red,
and everything down it white.
And last one x2 <= 40, y = 40 with same principe as shown upon.
We will get such picture:
The feasible region is shown in white.
To find maxima we should draw family of lines p1*x = p2*y, p1 and p2 is constant, let’s suppose
p1 = p2 = 1, to make our life easier.
Only lines that has points from white area is needed. After that we much choose line that have
max distance to (0,0).
Not hard to understand that it’s this line
The answer is this x and y matched in circles bellow
X = 41.25, Y = 31.25
E. So company should make 41.25 metres of light duty clothes and 31.25 meters of heavy duty
clothes.
F.
model "Clothing"
uses "mmxprs"; !gain access to the Xpress-Optimizer solver
!sample declarations section
declarations
x1,x2: mpvar ! Decision variables: produced quantities
end-declarations
Profit:= x1 + x2 !objective function
rawCotton:= 5*x1 + 11*x2 <= 550 ! cotton in metres
processingTime:= 5*x1 +3*x2 <= 300 !hrs of processing time
demand:= x2 <= 40 !demand
maximize(Profit) !Solve the problem
writeln("Make ", getsol(x1), " light Duty Clothes")
writeln("Make ",getsol(x2), " heavy Duty Clothes")
writeln("Best profit is ", getobjval)
end-model
G. Graphically solution is good to understand LP problem but does not let you solve it 100%
correct. Firstly it works only when your LP problem has 2 unknown secondly, it’s very hard to get
accurately answer when. Mosel mode is hard to build but when you don’t know how to do it,
but after you find out LP problems is very easy to solve.
2. A.
x1 - number of cask golden glow
x2 - number of cask autumn fresh
x3 - number of cask sparkling light
7500 * x1 + 8200 * x2 + 9500 * x3 --> max
20*x1 + 24*x2 + 18*x3 <= 1700
x1 + x2 + x3 <= 80
11*x1 + 8*x2 + 16*x3
Mosel mode :
model "cider"
uses "mmxprs"; !gain access to the Xpress-Optimizer solver
!sample declarations section
declarations
x1,x2,x3: mpvar ! Decision variables: produced quantities
end-declarations
Profit:= 7500 * x1 + 8200 * x2 + 9500 * x3 !objective function
tonesApple:= 0.2*x1 + 0.24*x2 + 0.18*x3 <= 17 ! tones of apples
processingTime:= 11*x1 + 8*x2 + 16*x3 <= 2500 ! hrs of processing time
casks:= x1 + x2 + x3 <= 80 ! can store at most 80 casks
demand:= x3 <= (x1 + x2)/2 ! demand
maximize(Profit) !Solve the problem
writeln("Make ", getsol(x1), " cask golden glow")
writeln("Make ",getsol(x2), " cask autumn fresh")
writeln("Make ",getsol(x3), " cask sparkling light")
writeln("Best profit is ", getobjval)
end-model
Output
Make 15 cask golden glow
Make 38.3333 cask autumn fresh
Make 26.6667 cask sparkling light
Best profit is 680167