x7 x6 2x6 x4 1 1 2 0
12.7-9. Use the MIP branch-and-bound algorithm presented in Sec. 12.7 to solve the following MIP problem interactively. MaximizeZ 3x1 4x2 2x3 x4 2x5,
subject to 2x1 x1 2x1 and xj 0, for j 1, 2, 3, 4, 5 xj is binary, for j 1, 2, 3.
D,I
x2 3x2 x2
x3 x3 x3
x4 x4 x4...
Company’s product mix problem as follows, using linear
programming:
Maximize profit = $7X1 + $5X2
subject to LPconstraints:
2 X1 + 1X2 ≤ 100
4 X1 + 3 X2 ≤ 240
where X1 equals the number of Walkmans produced and X2 equals the number of Watch-TVs produced.
To convert these inequality constraints to...
tables and no chairs to get the max profit.
3.4 Special Cases in Graphical Method
3.4.1 Multiple Optimal Solution Example 1 Solve by using graphical method Max Z = 4x1 + 3x2Subject to 4x1+ 3x2 ≤ 24 x1 ≤ 4.5 x2 ≤ 6 x1 ≥ 0 , x2 ≥ 0 Solution
The first constraint 4x1+ 3x2 ≤ 24, written in a...
assignment problem can be stated as:
Minimize (maximize): Z=
Subject to:
ij
Xij
for j= 1,2,………, m
for i= 1,2,………, n
Xij = 0 or 1 for al i and j
The following is a step by step algorithm that uses the Hungarian method to solve
the general assignment problem.
Step 1: for the original...
ﬁrst LP problem is
max z = 4x1 + 5x2
subject to
x1 + x2 ≤5
6x1 + 10x2 ≤45
x2 ≤3.
17 / 80
Choice of way to split
The second LP problem is given by
max z = 4x1 + 5x2
subject to
x1 + x2 ≤5
6x1 + 10x2 ≤45
x2 ≥4.
18 / 80
Bounds on the optimal value and adding branches
1. If the...
represent the amount of an unused resource. We formulate the Shader Electronics Company’s product mix problem as follows, using linear programming: Maximize profit Subject to LPconstraints: 2X1 4X1 1X2 3X2 100 240 $7X1 $5X2
where X1 equals the number of Walkmans produced and X2 equals the number of...
(100, 0). Therefore, the values of x1 and x2 are 100 and 0 respectively.
Maximum Profit, Zmax = 4x1 + 2x2
= 4(100) + 2(0)
= Rs. 400.00
Example 10:
Solve the following LPP by graphical method.
Minimize Z = 18x1+ 12x2
Subject to...
recorded, it is optimum, or else no integer valued feasible solution exists.
Solved Problem 2 Use branch and bound technique to solve the following IPP Maximise z = 7x1 + 9x2 ---------------- (1) Subject to the constraints – x 1 + 3x2 < 6 -–------------ (2) 7x1 + x2 < 35 0 < x1, x2 < 7...
, equivalently,
z , x1 , x2 = 0:
Putting this equation together with the constraints, we get the following system of linear equations. = 0 z ,x1 ,x2 2x1 +x2 +x3 = 4 x1 +2x2 +x4 = 3 Row 0 Row 1 Row 2 7.1
7.2. SOLUTION OF LINEAR PROGRAMS BY THE SIMPLEX METHOD
89
Our goal is to maximizez...
optimality d. Range for multiple changes
PROBLEMS
1. Solve these problems using graphical method and answer the questions that follow. Use simultaneous equations to determine the optimal values of the decision variables. a. MaximizeZ 4x1 3x2Subject to 4x2 48 kg Material 6x1 8x2 80 hr Labour 4x1 x1...
based on the simplex method to
solve the problem.
4.6-15.* Consider the following problem.
MaximizeZx1 4x2,
subject to
x2
6
3x1
x1 2x2
4
x1 2x2
3
(no lower boundconstraint for x1).
(a) Solve this problem graphically.
(b) Reformulate this problem so that it has only two functional...
following in equations:
4 x1 + 2 x 2 ≤ 10 8 2 x1 + x 2 ≤ 8 3 x2 ≤ 6 x1, x 2 ≥ 0
Formulate & solve the LP problem by using graphical method so as to optimize both P1 & P2. Solution Objective: Maximise Z = 4 x1 + 3 x2
Since the origin (0,0) satisfies each and every constraint, all points below the...
M7-1
Convert the followingconstraints and objective function into the proper form for use in the simplex method: Minimize cost = 4X1 + 1X2 subject to 3X1 + X2 = 3 4X1 + 3X2 Ú 6 X1 + 2X2 … 3
SOLVED PROBLEMS
M7-41
Solution
Minimize cost = 4X1 + 1X2 + 0S1 + 0S2 + MA 1 + MA 2 subject to...
artiﬁcial variables) and 3 constraints.
b. The dual would have 2 constraints and 5 variables
(3 decision variables and 2 slack variables).
c. The dual problem would be smaller and easier to solve.
maximize proﬁt ϭ 0.5X1 ϩ 0.4X2
primal constraints: 2X1 ϩ 1X2 р 120
2X1 ϩ 3X2 р 240
X1, X2 у 0...
values of (x1, x2) are estimated to be those given in the following table:
3.4-3. Use the graphical method to solve this problem: MaximizeZ 15x1 20x2,
subject to x1 2x1 x1 and x1
D
2x2 3x2 x2
10 6 6
0,
x2
0.
3.4-4. Use the graphical method to solve this problem: Minimize Z...
be sold for a $15 profit. Formulate and solve this LP production mix situation to find the best combination of air conditioners and fans that yields the highest profit. Use the corner point graphical approach.
Let X1 = the number of air conditioners scheduled to be produced
X2 = the number...
5 = 20 Marks
Answer all Questions :
1. Use Branch and Boundmethod to solve the following LPP.
Maximizez = gx1 + gx2
Subject to the constraints
- x1 + 3x2 ≤ 6
7x1 + 40 x2 ≤ 35
x2 ≤ 7
x1
, x2 ≥ 0 and integers
2. A manufacturing company produces two products A and B. The time requirement...