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Environmentally Conscious Design & Manufacturing. Class 24: Optimization - Operations Research. Prof. S. M. Pandit. Linear Programming. Problem statement Developing the objective function What are the constraints? Mathematical model Example 1 & 3 from Hamdy A. Taha, “Operations Research”
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Environmentally Conscious Design & Manufacturing Class24: Optimization - Operations Research Prof. S. M. Pandit
Linear Programming • Problem statement • Developing the objective function • What are the constraints? • Mathematical model • Example 1 & 3 from Hamdy A. Taha, “Operations Research” • and 2 from Hillier & Lieberman, “Introduction to • Operations Research”
Example 1 of LP • Given: • A small paint company produces both interior and • exterior house paints for wholesale distribution. • Two raw materials, A & B are used. • The price per ton is $3,000 for exterior paint and • $ 2,000 for interior paint. • Ask: • How much interior and exterior paints should the company produce daily to maximize gross income?
Construction of the Math. Model • What are the variables (unknowns ) of the problem? • What constraints must be imposed on the variables to satisfy the limitations of the modeled system? • What is the objective (goal) that need to be achieved to determine the optimum (best) solution from among all the feasible values of the variables?
Variables & Objective Function Variables: xE=tons produced daily of exterior paint xI =tons produced daily of interior paint Objective function: z=3 xE+2xI
Constraints Usage restriction: (raw material A) (raw material B) Demand restriction: (excess of interior over exterior paint) (maximum demand for interior paint) Nonnegativity restriction:
Model Maximize: (objective function) Subject to (constraints)
Example 2 Reclamation Center (spend $) 4 types of solid waste “Amalgamation” 3 types of products (earn $) Requires % of specific materials (Consume $)
Material pounds/week treatment cost available per pound 1 3,000 $ 3 2 2,000 $ 6 3 4,000 $ 4 4 1,000 $ 5
Formulation Q: What are the decision variables? - What information is needed? - What kind of decision variable(s) would best provide the information needed?
Decision Amount of each product grade yi=pounds of product grade “i” Zij=proportion of material “j” in product “i” In terms of weight, Quantity of material “j” used =ZAjyA+ZBjyB+ZCjyC (product of variables nonlinear function )
Replacing ‘product terms,’ so that xij=Zij yi {i=A,B,C j=1, 2, 3, 4} xij-decision variables Number of pounds of material j allocated to product grade i per week
We want: Total amount of product grade “i”: Proportion of material “j” in product grade “i”: Total profit:
Model Model: • Constraints: • Availability • Mixture specifications • Non-negativity
Example 3: Goal Programming • Given: • Two products are manufactured by passing sequentially • through two different machines • The time available for the two products on each machine is limited to 8 hours daily but may be exceeded by up to 4 hours on an overtime basis • Each overtime hour will cost an additional $ 5 • Require • To determine the production level for each product that • will maximize the net profit.
Goal Programming The constraints of the model without the overtime option (machine1) (machine2) The constraints of the model with the overtime option (machine1) (machine2)
Mathematical Representation To determine the number of units of each product (variables) that maximizes net profit (objective ) provided that the maximum allowable machine hours are exceeded only on an overtime basis (constraints)
Complete Model Maximize Subject to
Goal Programming To convert the model to a linear program, we use the substitution which is equivalent to
LP Model maximize Subject to
