Assignment - Introduction to Math Programming
Directions -
Formulate a linear programming model for the following description.
Include definitions of decision variables, Objective function, and constraints.
Augment your linear programming model into standard form (Ax = b) using slack/surplus variables, and conversions as necessary.
Program and solve model in LINGO, email me LINGO code, and attach LINGO output.
Wood Built Bookshelves (WBB) is a small wood shop that produces three types of bookshelves: models A, B, and C. Each bookshelf requires a certain amount of time for cutting each component, assembling them, and then staining them. WBB also sells unstained versions of each model. The times, in hours, for each phase of construction and the profit margins for each mode, as well as the amount of time available in each department over the next 2 weeks, are given below.
|
Model
|
Labor (h)
|
Profit Margin
|
|
Cutting
|
Assembling
|
Staining
|
Stained
|
Unstained
|
|
A
|
1
|
4
|
7
|
$60
|
$30
|
|
B
|
0.5
|
3
|
5
|
$40
|
$20
|
|
C
|
2
|
6
|
8
|
$75
|
$40
|
|
Labor Available
|
200
|
700
|
550
|
|
|
Since this is the holiday season, WBB can sell every unit that it makes. Also, due to the previous year's demand, at least 20 model B bookshelves (stained or unstained) must be produced. Finally, to increase sales of the stained bookshelves, at most 5o unstained models are to be produced.
Homework - Introduction to Math Programming
Directions - The focus of this homework is on linear programming models and methods. You may discuss problems with others as necessary. Turn in your own unique assignment. Please print out and email me any LINGO utilized to complete the homework. This homework will be graded for completion and a subset of problems will be graded for correctness.
1. Consider the following LP:
maximize Z = x1 + 2x2
subject to: x1 + 2x2 ≤ 10
x1 + x2 ≥ 1
x2 ≤ 4
x1, x2 ≥ 0
(a) Apply the graphical method (or a technique based on similar logic) to solve this model.
(b) What do you observe about the solution?
2. Consider the following LP:
minimize Z = 20x1 + 15x2
subject to: 0.3x1 + 0.4x2 ≥ 2
0.4x1 + 0.2x2 ≥ 1.5
0.2x1 + 0.3x2 ≥ 0.5
x1 ≤ 2
x2 ≤ 2
x1, x2 ≥ 0
(a) Apply the graphical method (or a technique based on similar logic) to solve this model.
(b) What do you observe about the solution?
3. Consider an optimization model with decision variables x1, x2 and the following constraints:
5x1 + 2x2 ≤ 10
x2 ≥ 0
(a) Graph the feasible region for these constraints.
(b) Devise a maximizing linear objective which results in an unbounded optimal solution. Graph the iso-profit line for your objective demonstrating the unboundedness.
4. Consider the following linear program:
maximize Z = 10x1 + 18x2 + 14x3
subject to: x1 + 3x2 + 2x3 ≤ 10
6x1 + 8x2 + 4x3 ≤ 24
4x1 + 2x2 + 4x3 ≤ 16
x1, x2, x3 ≥ 0
(a) Work through the simplex method step by step in tabular form. You must provide the tabular form for each basic solution visited and be sure to identify your logic in identifying entering/leaving variables.
(b) Implement and solve the problem with LINGO.
5. Consider the following linear program:
minimize Z = 2x1 + x2 + 3x3
subject to: 10x1 + 4x2 + 14x3 = 840
3x1 + 2x2 + 5x3 ≥ 280
x1, x2, x3 ≥ 0
(a) Formulate this in standard form.
(b) Using Big M generate an initial tableau.
(c) Formulate and solve part (a) problem with LINGO.
Attachment:- Assignment File.rar