Lecture 4- Outline
1. (0-1) IP problem: an example
2. LP solution using the DecisionPro software 3. IP solution using DecisionPro
4. (0-1) IP solution using DecisionPro
5. Assignment Problem solution using
DecisionPro
www.bradford.ac.uk/managem
(0-1) IP Problem: An
Recreation facilities selection to maximize daily usage
Example
by residents.
Resource constraints: £120,000 budget; 12 acres of land. Selection constraint: either swimming pool or tennis center (not both).
Data:
Recreation
Facility
Swimming pool
Tennis Center
Athletic field
Gymnasium
Expected Usage
(people/day)
Cost ($)
Land
Requirement
(acres)
300
90
400
150
35,000
10,000
25,000
90,000
4
2
7
3
2
The (0-1) Integer Formulation
For The Example x1 = construction of a swimming pool x2 = construction of a tennis center x3 = construction of an athletic field x4 = construction of a gymnasium
Maximize Z = 300x1 + 90x2 + 400x3 + 150x subject to:
£35,000x1 + 10,000x2 + 25,000x3 + 90,000x4 £120,000
4x1 + 2x2 + 7x3 + 3x3 12 acres x1 + x2 1 facility x1, x2, x3, x4 = 0 or 1
3
DecisionPro General
Features
• DecisionPro is a powerful application for decision-support modelling and analysis; • DecisionPro will help you make business decisions;
• DecisionPro combines basic quantitative methods in management;
• DecisionPro has analytic capabilities for business models.
4
DecisionPro’s capabilities
• General modelling and problem solving -for tackling complex problems; • Communicating with your ideas;
• Optimisation - for allocating; resources using linear programming;
• Sensitivity Analysis - for determining which parameters drive your decisions most.
5
How an LP model can be built • By using inaDecisionPro? hierarchical tree layout;
• The hierarchical layout allows you to work with meaningful node names;
• It automatically constructs a diagram that matches the logical structure of your model;
• This is a key benefit that makes it easier for you to tackle complex, unstructured problems.
6
DecisionPro's hierarchical layout • based on the divide and conquer concept;
• to solve a complex problem you simply divide it into two or more simpler components; • To solve these component problems, you apply the same process again, breaking them down into still finer elements;
• The result is a DecisionPro hierarchical tree.
7
An LP problem example to be transferred into
DecisionPro (1)
Department
Time to process on
Product A
(hrs)
Time to process on
Product B
(hrs)
Capacity
(hrs/week)
Machining
6
6
120
Assembly
8
4
120
8
An example of LP problem to be transferred into
DecisionPro (2)
Decision variables are:
XA= the number of product A made per week
XB= the number of product B made per week
Objective Function Coefficients:
Profit contribution of product A = £ 30
Profit contribution of product A = £ 35 9
An example of LP problem to be transferred into
DecisionPro (3)
Objective function: Max 30 XA + 35XB subject to constraints :
6 XA + 6 XB 120
8 XA + 4 XB 120
XA , X B 0 10
The LP problem built in
DecisionPro
11
The LP problem to be solved by
DecisionPro
12
The solution of DecisionPro for the LP problem
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DecisionPro's tree
• not only provides a great way to solve structure business problems, it also provides a
great way to present your analysis to others; • With DecisionPro, your models are inherently outlined;
• The root node represents the solution you seek and each branch provides increasingly more detailed information about how that solution is derived;
• This feature makes it easy for someone who is unfamiliar with your model to understand its function quickly.
14
Integer Programming Solution
Using DecisionPro Software
Primitive functions in DecisionPro such as simplex() can accept integer variables;
Just type simplex([Objective function,
Constraints],” Decision variables”, the number of integer variables);
Objective function and Constraints must be placed within the bracket,
Integer variables can be
