12.3-1.* The Research and Development Division of the Progres- sive Company has been developing four possible new product lines. Management must now make a decision as to which of these four products actually will be produced and at what levels. Therefore, an operations research study has been requested to find the most profitable product mix. A substantial cost is associated with beginning the production of any product, as given in the first row of the following table. Management's objective is to find the product mix that maximizes the total profit (total net revenue minus start-up costs). Product 1 2 3 4 $40,000 $70,000 $60,000 Start-up cost $50,000 Marginal revenue $ 70 $ 60 $ 90 $ 80 Let the continuous decision variables x₁, x2, x3, and x4 be the production levels of products 1, 2, 3, and 4, respectively. Manage- ment has imposed the following policy constraints on these vari- ables: 1. No more than two of the products can be produced. 2. Either product 3 or 4 can be produced only if either product 1 or 2 is produced. 3. Either 5x₁ + 3x₂ + 6x3 + 4x4 ≤ 6,000 or 4x1 + 6x₂ + 3x3 + 5x4≤ 6,000. (a) Introduce auxiliary binary variables to formulate a mixed BIP model for this problem. C (b) Use the computer to solve this model.
12.3-1.* The Research and Development Division of the Progres- sive Company has been developing four possible new product lines. Management must now make a decision as to which of these four products actually will be produced and at what levels. Therefore, an operations research study has been requested to find the most profitable product mix. A substantial cost is associated with beginning the production of any product, as given in the first row of the following table. Management's objective is to find the product mix that maximizes the total profit (total net revenue minus start-up costs). Product 1 2 3 4 $40,000 $70,000 $60,000 Start-up cost $50,000 Marginal revenue $ 70 $ 60 $ 90 $ 80 Let the continuous decision variables x₁, x2, x3, and x4 be the production levels of products 1, 2, 3, and 4, respectively. Manage- ment has imposed the following policy constraints on these vari- ables: 1. No more than two of the products can be produced. 2. Either product 3 or 4 can be produced only if either product 1 or 2 is produced. 3. Either 5x₁ + 3x₂ + 6x3 + 4x4 ≤ 6,000 or 4x1 + 6x₂ + 3x3 + 5x4≤ 6,000. (a) Introduce auxiliary binary variables to formulate a mixed BIP model for this problem. C (b) Use the computer to solve this model.
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
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A.?
B.?
Transcribed Image Text:12.3-1.* The Research and Development Division of the Progres-
sive Company has been developing four possible new product lines.
Management must now make a decision as to which of these four
products actually will be produced and at what levels. Therefore,
an operations research study has been requested to find the most
profitable product mix.
A substantial cost is associated with beginning the production
of any product, as given in the first row of the following table.
Management's objective is to find the product mix that maximizes
the total profit (total net revenue minus start-up costs).
Product
1
2
3
4
$50,000
$40,000
$70,000
$60,000
Start-up cost
Marginal revenue $ 70 $ 60 $ 90 $ 80
Let the continuous decision variables X₁, X2, X3, and x4 be the
production levels of products 1, 2, 3, and 4, respectively. Manage-
ment has imposed the following policy constraints on these vari-
ables:
1. No more than two of the products can be produced.
2. Either product 3 or 4 can be produced only if either product 1
or 2 is produced.
3. Either
5x₁ + 3x₂ + 6x3 + 4x4 ≤ 6,000
or
4x₁ + 6x₂ + 3x3 + 5x4≤ 6,000.
(a) Introduce auxiliary binary variables to formulate a mixed BIP
model for this problem.
c (b) Use the computer to solve this model.
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