Assignment -
Solve or prove that there is no solution to the following LP problems by verifying any graphical results using the Simplex method.
1) Minimize: z = x1 - x2
Subject to: x1 + x2 ≤ -1
With: x1 & x2 nonnegative
2) Minimize: z = x1 - x2
Subject to: x1 + x2 ≥ -1
With: x1 & x2 nonnegative
3) Maximize: z= x1 - x2
Subject to: x1 + x2 ≥ -1
With: x1 & x2 nonnegative
Redo problems 1-3 imposing the extra constraint that all basic variables are also integer.
Consider the following integer programs. Using Chvatal cuts formulate associated LP's and solve them graphically to obtain an optimal solution if it exists.
4) Maximize: z = 10x1 + x2
Subject to: 2x1 + 5x2 ≤ 11
With: x1 & x2 nonnegative & integer
5) Maximize: z = 3x1 + 4x2
Subject to: 2x1 + x2 ≤ 6
2x1 + 3x2 ≤ 9
With: x1 & x2 nonnegative & integer
6) Maximize: z = x(5π-x)on [0, 20]
7) Maximize: z =|x2 - 8|on [-4, 4]
8) Maximize: z = x1 (x2 - 1) + x3 (x3^2 - 3) over the entire real line
9) Consider the following problem:
minx1,x2 f, f(x1, x2) = (x1 - 4)2+ (x2 - 4)2
Subjected to:
x1 + x2 ≤ 4
x1 +3x2 ≤ 9
Derive algebraically the Kuhn-tucker conditions of the problem. Then, plot these conditions graphically as well as level contours of the objective function. Finally, using the plot obtain a solution to the optimization problem considered.
10) In general, there are two sizes of oil tankers: those that can go through the Suez canal and those that cannot and in effect have to travel around Africa to get from the Persian Gulf region to a NW European oil terminal. In this problem we are interested in calculating the minimum capacity (in weight/mass of cargo carried) of Suez incompatible tankers so that they are competitive to Suez compatible ones. The following values are of interest.
L, B, T: Principal vessel dimensions
Δ: Vessel mass displacement
F: Fare charged to customer per unit mass of cargo for single transit
C: Cargo single transit cost per unit mass of cargo
R: Empty ship seagoing cost per nm traveled
WS: Average waiting time per Suez Canal transit
XS: Fee per single Suez Canal transit
P: Port time (for loading or unloading) per unit mass of cargo
V: Vessel's service speed in open seas
Z: Distance in nm between Gulf and European port of call
For simplicity assume that cargo capacity is equal to mass displacement Δ and that Δ = ρgLBT.
Denote variables related to the Suez compatible vessel by an index of S and those related to the Suez incompatible one by I.
Suez Canal constraints: LS <= Lmax, BS <= Bmax, TS <= Tmax. Also: Z1 = 5ZS.
Determine ΔI for economic feasibility if the following are known: ρ, g, Lmax, Bmax, Tmax, ZS, WS, XS, FS, CS/I, RS/I, PS/I, VS/I.