运筹学英文版北信科作业
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CHAPTER 3
3.1-5、Use the graphical method to solve the problem：
Maximize Z=2 x1+ x2
Subject to
X2<=10
2 x1+5 x2<=60
X1+x2<=18
3 x1 +x2<=44
And
X1>=0,x2>=0
Solution:
X2
22
18
(11.3 , 10)
12
10
Feasible region
14.61830X1
As the figure above shows, the corner-point is the optimal solution, x1=11.3 and x2=10.
3.1-13、consider the following problem, where the value of k has not yet been ascertained.
Maximize Z=x1+2 x2,
Subject to
-x1+x2<=2
X2<=3
K x1+x2<=2k+3, where k>=0
And
X1>=0, x2>=0
The solution currently being used is x1=2,x2=3. Use graphical analysis to determine the values of k such that this solution actually is optimal.
Solution:
X2
Corner-point
3
2
2X1
As the picture above shows, the value of k is boundless，any value is ok.
3.1-14、consider the following problem, where the value of c1 and c2 has not yet been ascertained.
Maximize Z=c1 x1+c2 x2
Subject to
2x1+ x2<=11
-x1+2x2<=2
And
X1>=0, x2>=0
Use graphical analysis to determine the optimal solution for(x1, x2) for the various possible values of c1 and c2.(hint: separate the cases where c2=0.c2>0,and
c2<0.for the latter two case，
Focus on the ratio of c1 to c2.)
X2
11
(4, 3)
1
5.5X1
If the value of c2 is 0, the optimal solution is x1=5, x2=0;
If the value of c2 is greater than zero and c1/c2 great than 2, the optimal solution is still x1=5, x2=0;
If the value of c2 is greater than zero and c1/c2 equal 2, the optimal solution has
multiple optimal solutions;
If the value of c2 is greater than zero and c1/c2 less than 2, the optimal solution is x1=4, x2=3;
If the value of c2 is less than zero and c1/c2 less than 1/2, the optimal solution is x1=4, x2=3;
If the value of c2 is less than zero and c1/c2 equal 1/2, the optimal solution has multiple optimal solutions;
If the value of c2 is less than zero and c1/c2 greater than 1/2, the optimal solution is x1=0, x2=1;
3.4-10. Larry Edison is the director of the Computer Center for Buckly College. He now needs to schedule the staffing of the center. It is open from 8 A.M. until midnight. Larry has monitored the usage of the center at various times of the day, and determined that the following number of computer consultants are required.
Time of Day Minimum Number of Consultants Required to Be Duty
8 A.M.-noon Noon-4 P.M.
4 P.M.-8 P.M.
8 P.M.-midnight
4 8 10 6
Two types of computer consultants can be hired: full-time and part-time. The full-time consultants work for 8 consecutive hours in any of the following shifts: morning (8 A.M.-4 P.M.), afternoon (noon-8 P.M.), and evening (4 P.M.-midnight).Full-time consultants are paid $40 per hour.
Part-time consultants can be hired to work any of the four shifts listed in the above table. Part-time consultants are paid $30 per hour.
An additional requirement is that during every time period, there must be at least 2 full-time consultants on duty.
Larry would like to determine how many full-time and how many part-time workers should work each shift to meet the above requirements at the minimum possible cost.
(a)Formulate a liner programming model for this problem.
C(b) Solve this model by the simplex method.(不会…希望老师多讲讲)
Solution：let the full-time consultants be x1、x2、x3, let the part-time consultants be