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MA569 linear program
创建日期：2024-08-10 16:55:48
所属分类：数学mathematics
标签： MA569
linear program

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MA569


1. Consider the following integer program:

objective Max 2x +5y

Subjective 2x +3y ≤ 15;

x +3y ≤ 10

x , y ≥ 0;

x , y ∈Z

(a) Sketch graph of the region. Determine the coordinates of the vertices for the correspond-

ing linear program (including vertices with non-integer coordinates).

(b) What is the optimal solution and optimal objective value to the linear program where the

variables are not required to be integers?

(c) List all of the integer-valued points that occur in the feasible region.

(d) What is the optimal solution and optimal objective value for the integer program?

1

2. A company manufactures three products whose daily labor and raw material requirements are

given in the following table:

Product Required daily labor (hr/unit) Required daily raw material (lb/unit)

1 4 3

2 3 4

3 5 6

The profits per unit of the three products are $25, $30, $22, respectively. The company has two

locations for locating its plant. The two locations differ primarily in their availability of labor

and raw material, as shown in the following table:

Location Available daily labor (hr) Available daily raw material (lb)

1 90 120

2 100 100

(a) Formulate an integer program to determine where they should locate the plant and how

much of each product they should produce in order to maximize their profit.

(b) Use Excel to solve the integer program. Where should they locate the plant? How much

of each product should they produce?

2

3. Consider the following BIP:

Max Z = 3x1 +3x2 +5x3?2x4? x5

Subject to x1 +2x2?3x4? x5 ≤ 03x1 +6x2?7x3 +9x4 +9x5 ≥ 10

x1, x2, x3, x4, x5 ∈{0, 1}.

Use the BIP branch-and-bound algorithm to solve this BIP. You can use Excel to solve the LP’s

that occur during the algorithm. Show that branching tree that results when you perform the

algorithm, and clearly label the vertices of the tree with the solution to the corresponding LP

and the resulting bound for Z .

3

4. Consider the following MIP:

Min Z = 5x1 +2x2 + x3 +2x4 +3x5

Subjective x2?5x3 + x4 +2x5 ≥?1

5x1? x2 + x5 ≥ 8

x1 + x2 +6x3 + x4 ≥ 5

x1, x2, x3, x4, x5 ∈Z+ ∪{0}.

Use the MIP branch-and-bound algorithm to solve this MIP. You can use Excel to solve the LP’s

that occur during the algorithm. Show the branching tree that results when you perform the

algorithm, and clearly label the vertices of the tree with the solution to the corresponding LP

and the resulting bound for Z .

4

5. Consider the following integer program:

Max Z = 100x1 +220x2 +50x3 +15x4 +17x5 +12x6 +4x7

Subjective 50x1 +30x2 +80x3 +9x4 +16x5 +5x6 +10x7 ≤ 800;

x1, x2, x3 ∈ {0, 1};

x4, x5, x6, x7 ∈ Z+ ∪{0}∩ [0, 200].

For each of the parts below, explain how to modify the integer program, so that the given con-

dition is also satisfied. Each of the parts in independent, so that no part depends on the parts

preceding it.

(a) Modify the integer program so that at least two of the following constraints are satisfied:

x4 ≥ 50; x5 ≤ 25; x6 + x7 ≤ 100.

(b) Modify the integer program so that x5 equals 9, 15, or 20.

(c) Modify the integer program so that 2x4 + x5 ≤ 50 or 4x4? x5 ≥ 20, but not both.

(d) Modify the integer program so that if x2 = 1, then x1 = 0.

(e) Modify the integer program so that x1, x2 and x3 cannot all equal 1.

(f) Modify the integer program so that x7 is divisible by 3 but not 6.

5

6. Consider a 3× 3 game board. You are required to field each square with a number between 1

and 9 such that the sum of the numbers in each row, each column, and each diagonal equals

15. Additionally, the numbers in all the squares must be distinct.

(a) Formulate an integer program to determine an assignment of numbers to squares. (Since

you are just trying to find a solution, your objective function can be anything.)

(b) Use Excel to solve the program. What is the solution?

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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 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