MATH39011Mathematical Programming I

Example Sheet 1

1. By graphical means

minimize z (x1; x2) = �x1 + x2subject to � x1 + 2x2 � 10

5x1 + 3x2 � 412x1 � 3x2 � 8

x1; x2 � 0

Identify all the corners (vertices) of the feasible region and the optimal vertex.

What is the minimum value of the LP?

By adding slack variables, write down the constraints in standard form

Ax = b

Identify the basic variables xB at the optimum and the optimal basis B. Show that

xB = B�1b

2. Let x� be any basic feasible solution (BFS) of the system

Ax = b

x � 0

corresponding to some basis B: Show that the LP minimization problem with cost z = cTxwhere c is de�ned by

cj =

�0 if Aj 2 B1 if Aj =2 B

has the unique minimum solution at x�:

[Hint: Observe that z (x) � 0 for any feasible x and that z (x�) = 0: To show uniqueness,suppose that y� is also optimal and deduce that y� = x�:]

3. Write the following LP problem in standard form:

min/max z = 2x1 + x2 � x3subject to x1 + x2 + 2x3 � 6

x1 + 4x2 � x3 � 4x1; x2; x3 � 0

and solve by evaluating the objective function at all basic feasible solutions. What bounds onx1; x2 and x3 are implied by constraint 1? If the �rst constraint is replaced by x1+x2�2x3 � 6show that x3 is no longer bounded and deduce that enumeration of vertices no longer resultsin a minimum point.

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4. [The re�nery problem]

An oil re�nery can buy two types of oil: light crude oil and heavy crude oil. The costs perbarrel of each type of crude are $11 (L) and $9 (H) respectively. The following quantities(mixes) of gasoline, kerosene and jet fuel are produced per barrel of each type of oil.

GASOLINE KEROSENE JET FUELLight crude oil 0.4 0.2 0.35Heavy crude oil 0.32 0.4 0.2

[Note that 5% and 8% of the crude are lost respectively during the re�ning process, thoughthis doesn�t a¤ect the formulation.] The re�nery has contracted to deliver 1 million barrelsof gasoline, 400,000 barrels of kerosene and 250,000 barrels of jet fuel. Find the number ofbarrels of each crude oil that should be re�ned in order to ful�l the order at minimum cost.

5. Consider the following problem in which x2 is a free variable

minimize z = x1 � 3x2 + 4x3subject to x1 � 2x2 + x3 = 5

2x1 � 3x2 + x3 = 6x1 � 0; x3 � 0

Show by substituting for x2 from the �rst constraint, that we can obtain a 2-variable LPproblem in standard form which can be solved graphically. (An alternative approach is tosubstitute x2 = u2 � v2 where u2; v2 � 0:)

6. Solve the following "continuous knapsack" problem by inspection, and justify your solutionin terms of the basic solutions of the problem in standard form.

Maximize 5x1 � 6x2 + 3x3 � 5x4 + 12x5subject to x1 + 3x2 + 5x3 + 6x4 + 3x5 � 90

x1; x2; x3; x4 x5 � 0

7. Show that the hyperplane X =�x 2 Rn : pTx = k

has no extreme points. [Hint: Consider

adding �y to x 2 Rn where pT y = 0:]

(a) Show that X = f(x1; x2) : x1 + x2 � 1g � R2 has no extreme points.[Hint: Consider the e¤ect of adding (";�") to any x 2 X].

(b) Consider the transformation to new non-negative variables u1; v1; u2; v2:

x1 = u1 � v1x2 = u2 � v2

and show that the new feasible region U does have extreme points e.g. (u1; v1; u2; v2) =(1; 0; 0; 0) :

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