LP Formulation
Page 1: Introduction to Business Decision Models
Course Title: OPM 3500
Course Focus: Business Decision Models
Key Topic: Linear Programming Formulation & Applications
Instructor: Yen-Ming Lee
Page 2: Product Mix Problem - Wyndor Glass Co.
New Products:
8-foot glass door with aluminum framing
4-foot by 6-foot double-hung, wood-framed window
Plants:
Plant 1: Produces aluminum frames and hardware.
Plant 2: Produces wood frames.
Plant 3: Produces glass and assembles the windows and doors.
Page 3: Step 1 - Define Decision Variables
Decision Variables arise from key questions:
What items can be chosen/controlled?
What decisions need to be made?
What factors affect costs, profits, etc.?
What information is needed for implementation?
Page 4: Step 2 - Define the Objective Function
Objective Function: A mathematical expression relating decision variables.
Common objectives include:
Minimizing costs
Maximizing profits
Page 5: Step 3 - Define the Constraints
Identifying Constraints involves asking:
Looking at the objective function, what ideal variable values are desired?
What limits or prevents achieving these ideal values?
Page 6: Algebraic Model for Wyndor Glass Co.
Decision Variables:
Let D = number of doors produced
Let W = number of windows produced
Objective Function:
Maximize P = $300D + $500W
Constraints:
D ≤ 4
2W ≤ 12
3D + 2W ≤ 18
D ≥ 0, W ≥ 0
Page 7: CompuTech Inc. Product Mix Problem
Objective: Decide quantities of PC1 and PC2 to maximize net profit.
Sales Limits:
Maximum of 600 PC1 and 1200 PC2 sales.
Pricing and Costs:
PC1: Sell for $300, cost $150
PC2: Sell for $450, cost $225
Available Hours:
10,000 assembly hours and 3000 testing hours.
Time Requirement:
PC1: 5 assembly hours, 1 testing hour.
PC2: 6 assembly hours, 2 testing hours.
Page 8: CompuTech Inc. Product Mix - Components
Decision Variables
Objective Function
Constraints
Page 9: The Diet Problem - Sing Sing Prison
Context: Kitchen manager designs a prisoner diet to minimize costs.
Food Options:
Milk, beans, oranges
Focus: Meet minimum nutritional requirements laid by law.
Page 10: Diet Problem - Framework
Decision Variables
Objective Function
Constraints
Page 11: The Diet Problem 2 - Summer Camp Lunch Planning
Context: Elizabeth Reed plans lunch for children while minimizing costs.
Food Choices: Peanut butter & jelly sandwiches with fruits/drinks.
Nutritional Goals:
300-500 calories, max 30% from fat
At least 60 mg of vitamin C, 10 g of fiber
Minimum: 2 slices of bread, 1 tbsp peanut butter/jelly, 1 cup liquid
Page 12: Diet Problem 2 - Framework
Decision Variables
Objective Function
Constraints
Page 13: Blending Problem - Agri-Pro Company
Objective: Mix four feeds to meet specified corn, grain, and mineral requirements.
Requirements:
8000 pounds mix, at least 20% corn, 15% grain, 15% minerals.
Feed Composition:
Costs and nutrient percentages outlined.
Page 14: Blending Problem - Components
Decision Variables
Objective Function
Constraints
Page 15: Electro-Poly Make vs. Buy Decisions
Context: Manufacturing slip rings for a $750,000 order.
Resources:
10,000 wiring hours available, 5,000 harnessing hours available.
Model Requirements: Determine how many to make vs. buy at minimum cost.
Page 16: Electro-Poly Decisions - Framework
Decision Variables
Objective Function
Constraints
Page 17: Nurse Staffing Problem - Daily Needs
Nurse Requirements:
Mon: 6, Tue: 5, Wed: 6, Thu: 5, Fri: 5, Sat: 5, Sun: 3
Work Schedule: Nurses work five 8-hour shifts with 2 consecutive days off.
Page 18: Nurse Staffing Problem - Framework
Decision Variables
Objective Function
Constraints
Page 19: Nurse Staffing Problem (cont.)
Constraints: Additional specifics on staffing requirements per day.
Page 20: Staffing Plan Extension - Maximizing Specific Shifts
Minimum Nurses Required: 8
Aiming to maximize those assigned to shifts 1, 2, and 7.
Focus: Reformulate decision variables, objective function, and constraints.
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Page 22: Staffing Plan Extension 2
Context: Part-time nurses available in addition to full-time nurses.
Staffing Cost:
Full-time: $100/day
Part-time: $80/day
Condition: No more part-time than full-time nurses on roster.
Objective: Minimize costs while meeting daily staffing requirements.
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Page 24: Multiperiod Planning Models
Importance: Decisions in current periods affect future outcomes.
Applications:
Production/inventory planning
Staffing requirements
Investment strategies
Capacity planning/location decisions.
Page 25: National Steel Co. - Production Planning Problem
Context: Orders for 2300, 2000, 3100, and 3000 tons of steel over four months.
Production and Inventory Options: Must meet demand with constraints on production and inventory.
January Production Costs:
Limit: 3000 tons monthly production.
Page 26: National Steel Co. - Components
Decision Variables
Objective Function
Page 27: National Steel Co. Problem (cont.)
Constraints: Requirements for production and inventory management.
Page 28: The Upton Co. - Production Planning Problem
Context: Planning production and inventory for heavy-duty air compressors over six months.
Challenges: Seasonal variations in utility costs, demand fluctuation, capacity variations.
Page 29: The Upton Corp - Constraints and Requirements
Maximum Warehouse Capacity: 6,000 units.
Safety Stock: Minimum of 1,500 units.
Production Options: Minimum production of half maximum capacity.
Page 30: Upton Corp - Details of Problem
Monthly Costs and Demands:
Detailed for six months including max/min production limits and carrying costs.
Page 31: Upton Corp Problem - Framework
Decision Variables
Objective Function
Page 32: Upton Corp Problem (cont.)
Constraints: Final formulation of constraints for decision-making.