Review of the course will midsem:
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1) (a) Show that if a variable x_j leaves the basis, it cannot return to it in the next iteration of the simplex
algorithm (regardless of the pivot rule). Deduce that there are no cycles of size two in the algorithm.
                                                                                                                            
(b) Give an example showing that a variable x_j can join the basis and leave it in the next iteration.
                                                                                                                            
2)  Solve Linear Prpgramming Problem , identifying the Basic variables, Basis, B Inverse.
	Maximize
					z = x_1 + 2x_1 - x_3
	Subject to
					2x_1 +  x_2  +   x_3 <= 14
					4x_1 + 2x_2 + 3x_3 <= 28
					2x_1 + 5x_2 + 5x_3 <= 30
					x_1,x_2,x_3 >= 0
                                                                                                                            
3) The fundamental theorem of linear algebra is often stated in the form of 'Fredholm alternatives'. For any A and b,
one and only one of the following systems has a solution:
                                                                                                                            
         (1) Ax = b
         (2) Transpose(A)*y = 0, Transpose(y)* b not= 0.
                                                                                                                            
Show that, it is contradictory for (1) and (2) both to have solutions.
(In other words, either b is in column space of A or there is a y in the null space of A such that  Transpose(y)* b not=
0).
                                                                                                                            
4) The Two Mines Company own two different mines that produce an ore which, after being crushed, is graded into three
classes: high, medium and low-grade. The company has contracted to provide a smelting plant with 12 tons of high-grade,
8 tons of medium-grade and 24 tons of low-grade ore per week. The two mines have different operating characteristics as
detailed below.
                                                                                                                            
Mine    Cost per day (£'000)    Production (tons/day)
                                High    Medium    Low
X       180                     6       3         4
Y       160                     1       1         6
                                                                                                                            
How many days per week should each mine be operated to fulfil the smelting plant contract?