AMS 540 / MBA 540 (Fall, 2010)
Linear Programming
Estie Arkin
Instructor: Estie Arkin, Math tower 1106, phone: 6328363, email: [email protected] course web site: http:/www.ams.sunysb.edu/estie/estie.html/courses/540/ams540; Oce hours: TBA. You may al
AMS 540 / MBA 540 (Fall, 2010)
Estie Arkin
Homework Set # 4
Due in class on Tuesday, October 12, 2010. 1). Solve using the big M method: (Note: you can check your answer using Lindo or ampl.) max z = 2x1 5x2 + x3 s.t. x1 + x2 + x3 x1 x2 + x3 5x1 + 3x2 x3
AMS 540 (Fall, 2014)
Estie Arkin
Homework Set # 1: Solution notes
1). B3 , B5 the number of 3 and 5 speed bicycle made, F r the number of frames bought, F i, As the additional
hours of nishing and assembly obtained.
max
s. t.
z = 12B3 + 15B5 4F r 6F i 4.5
AMS 540 / MBA 540 (Fall, 2011)
Estie Arkin
Linear Programming
Homework Set # 9
Due in class on Tuesday, December 6, 2011.
1). Consider the following minimum cost netwrok ow problem: (I am using the notation from
class bi < 0 is a supply.)
3
8
5
2
4
4
2

AMS 540 / MBA 540 (Fall, 2011)
Estie Arkin
Linear Programming  Midterm solution sketch
Mean 67.9, median 68, high 97, low 32.
1). (12 points) (a). State a non degenerate BFS for the following LP: (make sure to say which variables are
basic, which are non
AMS 540 / MBA 540 (Fall, 2010)
Estie Arkin
Homework Set # 7: Solution notes
1). (a). Graphically, nd the range of values of b2 for which the current basis remains optimal. The optimal solution is when both constraints are binding. As b2 increases, the opt
AMS 540 / MBA 540 (Fall, 2011)
Estie Arkin
Homework Set # 4: Solution notes
1). (a). First solution: Let a1 and a2 be the articial variables in constraints 1,2. Cleaning up the
objective function, we get:
z
1
0
0
z
a1
a2
x1
1
1
1
x2
1
1
1
x3
M
1
2
x4

AMS 540 / MBA 540 (Fall, 2010)
Estie Arkin
Linear Programming
Homework Set # 8
Due in class on Thursday, November 18, 2010. 1). Consider the following resource allocation problem and the accompanying optimal tableau (x5 , x6 , x7 are the respective slack
AMS 540 / MBA 540 (Fall, 2009)
Estie Arkin
Linear Programming  Final solution sketch
Mean 79, high 98, low 32.
1). (10 points) Consider a transshipment problem dened in class, in which bi is the demand/supply of node
i, n bi = 0, and all cij and bi are i
AMS 540 / MBA 540 (Fall, 2010)
Estie Arkin
Homework Set # 1: Solution notes
1). B3 , B5 the number of 3 and 5 speed bicycle made, F r the number of frames bought, F i, As the additional hours of nishing and assembly obtained. max s. t. z = 12B3 + 15B5 4F
AMS 540 / MBA 540 (Fall, 2009)
Estie Arkin
Linear Programming  Midterm solution sketch
Mean 76.14, median 77, high 95. 1). (20 points) Consider the following LP, in which c1 , c2 are some constants: max z = c1 x1 + c2 x2 s.t. x1 2x2 x1 2x1 + 3x2 x1 , x2
AMS 540 / MBA 540 (Fall, 2011)
Estie Arkin
Linear Programming
Homework Set # 6
Due in class on Thursday, November 3, 2011.
1). Consider the following Primal Linear Programming problem: (Here x, y, a, b, c are all vectors in
and y are the variables, a, b,
AMS 540 / MBA 540 (Fall, 2008)
Estie Arkin
Linear Programming  Midterm solution sketch
Mean 58.3, high 91, low 28.
1). (30 points) Consider the standard form polyhedron P = cfw_x  Ax = b, x 0. (As usual, assume that A is
an m by n matrix whose rows are
AMS 540 / MBA 540 (Fall, 2011)
Estie Arkin
Homework Set # 3: Solution notes
1). (a). The basic variables are x5 and x6 , and the initial B = I .
(b). Either x1 or x2 enters, and the unique winner of the min ratio test x6 leaves.
Now, consider one of the l
AMS 540 / MBA 540 (Fall, 2010)
Estie Arkin
Homework Set # 2
Due in class on Thursday, September 23, 2010. 1). (a) Show that the set of all feasible solutions to the following linear program forms a convex set: mincfw_cx  Ax = b, x 0 (b) Prove that the se
AMS 540 / MBA 540 (Fall, 2010)
Estie Arkin
Homework Set # 3
Due in class on Thursday, September 30, 2010. 1). (a). State a non degenerate BFS for the following LP: (make sure to say which variables are basic, which are non basic and what each variable is
AMS 540 / MBA 540 (Fall, 2010)
Estie Arkin
Linear Programming
Solutions Homework Set # 9
1). (a). Use the arcs (4,1) (4,3) (3,2) (5,4) as your starting tree solution. The edge (1,2) is non basic at its upper bound, and all other non basic edges are at the
AMS 540 / MBA 540 (Fall, 2010)
Estie Arkin
Linear Programming  Homework Set # 1
Due in class on Tuesday, September 14, 2010. Formulate the following problems. I suggest you solve your formulations by some LP package, to check whether the formulation make
AMS 540 (Fall, 2014)
Estie Arkin
Homework Set # 2
Due in class on Tuesday, September 16, 2014.
1). (a) Show that the set of all feasible solutions to the following linear program forms a convex set:
mincfw_cx  Ax = b, x 0
(b) Prove that the set of optima
AMS 540 / MGT 540 (Fall, 2010)
Estie Arkin
Homework Set # 4: Solution notes
1). Solve using the big M method: Set up a standard form, with a1 and a2 the articial variables: max z = 2x1 5x2 + x3 M a1 M a2 s.t. x1 + x2 + x3 x4 + a1 x1 + x2 x3 + a2 5x1 + 3x2
AMS 540 (Fall, 2016)
Estie Arkin
Homework Set # 3
Due in class on Thursday, September 29, 2016.
1). (a). State a non degenerate BFS for the following LP: (make sure to say which variables are basic, which
are non basic and what each variable is equal to.)
AMS 540 (Fall, 2016)
Estie Arkin
Homework Set # 1: Solution notes
1). Variables: wi are the number of workers that have quarter i = 1, 2, 3, 4 off. xi number of mixers produced
in quarter i, and Ii inventory at the end of quarter i.
min
z = 30, 000(w1 + w
AMS 540 (Fall, 2016)
Estie Arkin
Homework Set # 2
Due in class on Tuesday, September 20, 2016.
1). (a) Show that the set of all feasible solutions to the following linear program forms a convex set:
mincfw_cx  Ax = b, x 0
(b) Prove that the set of optima
AMS 540 (Fall, 2014)
Estie Arkin
Homework Set # 3: Solution notes
1). (a). State a non degenerate BFS for the following LP: (make sure to say which variables are
basic, which are non basic and what is variable is equal to.)
max z = 6x1 + 4x2 + x3
s. t.
x1
AMS 540 (Fall, 2017)
Estie Arkin
Linear Programming  Homework Set # 1
Due in class on Thursday, September 7, 2017. Formulate the following problems. I suggest you solve your formulations
by some LP package, to check whether the formulation makes sense! Y
AMS 540 (Fall 2017)
Estie Arkin
Cover Sheet: Homework number
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