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# 11-12-2Operations Research期末试题(应数)A

Operations Research（05005022）试卷 (A)

I. Judgement（２’×５=1０’） 1.A basic solution to the linear programming problem corresponds to a point of its feasible region．（ ） ) )

2.A basic solution is inevitably a feasible solution.(

3.The dual simplex method is devised for the dual problems.(

4.The big M method is able to find an initial feasible base for the linear programming problem．( )

5.The traveling salesman problem is related to the Hamiltonian graphs.(

)

II. Cloze（２’×５=1０’） 1.An optimal solution to the linear programming problem

?min z ? x1 ? x 2 ?s.t. x1 ? x 2 ? ?1 ? is_______________. ? x1 ? x 2 ? ?1 ? ? x1 , x 2 ? 0 ?

2.The

conventional

algorithm

for

the

assignment

problem

is_______________. 3.The content of the strong duality is that_______________. 4.Write out the LINGO program for the integer programming problem
?max z ? 2 x1 ? x 2 ?s.t. 5 x1 ? 4 x 2 ? 24 ? ：_______________. ? 2 x1 ? 5 x 2 ? 13 ? ? x1 , x 2 ? 0, int egers ?

5.A tree is_______________. III. Computation（60’） 1.（15 分）A plant plans to manufacture two kinds of products A and
B with 3500 labors，4000 kilograms materials and 2000 kilowatt-hours

electric power. The labors，materials and electric power consumed by per kilogram A and B ，as well as the profits of per kilogram A and
B ，are given as the table below：

products
A B

labors 7 5

materials electric power profit 5 8 2 5 6 7

Then how should a production plan be made to maximize the total profits？You are required to construct a linear programming model for this problem and solve it with the simplex method. 2.（15 分）Let the following data be for a certain transportation problem： m ? 3 ， n ? 4 ， d ? (10,7,5,5,6,3,8)T ， c ? (7,9,4,2,3,6,2,1,1,2,5,4)T . You are asked to formulate it as a mathematical model(concrete model needed) and find its initial basic feasible solution（two methods needed）. 3.（10 分）Consider the following linear programming problem

?max z ? x1 ? 2 x 2 ? 3 x3 ? 4 x 4 ?s.t. x1 ? 2 x 2 ? 3 x3 ?2 ? ? 2 x1 ? x 2 ? 2 x4 ? 2 ? ? x1 , x 2 , x3 , x 4 ? 0 ?

Given an optimal solution y ? (1,2)T to the dual problem for it，you are asked to find an optimal solution to it. 4.（10 分）A warehouse stores three kinds of commodities，the unit weight(ton per article) and unit value(dollar per article) of which are provided by the following table： unit weight 2 3 4

commodities 1 2 3

unit value 100 140 180

These commodities are to be transported to a market with a total transportation capacity of 10 tons. Then how should we make a transportation plan to maximize the total values? You are requested to formulate the problem as a mathematical programming model and write out the dynamic programming recursion formula for it. 5. 10 分） （ When we use the simplex method to solve the linear programming problem
?min z ? ?4 x1 ? 5 x 2 ? x3 ? 7 x 4 ?s.t. 8 x1 ? 4 x 2 ? 6 x3 ? x 4 ? 120 ? ? x1 ? 3 x 2 ? 2 x3 ? 2 x 4 ? 30 ? ? 3x1 ? 8 x 2 ? 2 x3 ? 2 x 4 ? 150 ? ? x1 , x 2 , x3 , x 4 ? 0 ?

we get the final simplex table

xB
x1 x4 x7

x1 x2 x3 x4 x5
1 0 0 0
1 3 4 3 13 3 17 ? 3 2 3 2 3 5 3 19 ? 3

x6
1 ? 15 8 15 13 ? 15 52 ? 15

x7
0 0 1 0

b
14 8 92 -112

0 1 0 0

z

2 15 1 ? 15 4 ? 15 1 ? 15

（1）Analyze the sensitivity for the coefficient c2 in the objective function ； （ 2 ） If the objective function changes into z ? ?4x1 ? 10x2 ? x3 ? 7 x4 ，then what about the optimal solution？ IV. Graphing（10’） Find the shortest paths from the vertex v1 to the rest in the following weighted digraphs:
v2
3 4 2 1 2 4

v5
2 1

v1

v3

5 1 2 4

v6 5
2

v8

v4

v7

V. Application（10 分） The hydrocarbon(烃)，the mother of organic compounds，is one kind of organic compound composed by carbon and hydrogen. The combination price of hydrogen tom in any type of hydrocarbon is always 1. The alkanes(烷烃) belong to the saturated group of hydrocarbons，the combination price of carbon atom in which is 4. Moreover，there is no cycle formed by any chemical bonds. According to the information above you are asked to prove that the molecular formula of alkanes is Cn H 2n?2 .