1. A company produces two products, Product A and Product B. The company wants to maximize its profit while considering constraints on resources. Each unit of Product A requires 3 hours of labor and 2 units of raw material, while each unit of Product B requires 2 hours of labor and 1 unit of raw material. The company has 200 hours of labor and 150 units of raw material available. Product A generates a profit of Tsh. 1000 per unit sold, and Product B generates a profit of Tsh. 800 per unit sold.
2. A farmer has a total of 100 acres of land available for planting two crops: wheat and barley. The farmer wants to maximize the total profit earned from selling these crops while considering constraints on available labor and water. Each acre of wheat requires 2 hours of labor and 3 units of water, while each acre of barley requires 3 hours of labor and 2 units of water. The farmer has 200 hours of labor and 250 units of water available. Wheat sells for Tsh. 200 per acre, and barley sells for Tsh. 150 per acre.
3. A company produces two types of products, X and Y. The profit per unit for X is Tsh. 5, and for Y is Tsh. 7. Each unit of X requires 2 hours of labor and 3 units of raw material, while each unit of Y requires 4 hours of labor and 2 units of raw material. The company has 100 hours of labor and 80 units of raw material available. Determine the optimal production quantities to maximize profit.
4. An investor is considering investing in two types of assets, A and B. Asset A has an expected return of 8% and asset B has an expected return of 10%. The investor has a total of Tsh. 100,000 to invest. Asset A costs Tsh. 10,000 per unit, and asset B costs Tsh. 20,000 per unit. Determine the optimal investment strategy to maximize the expected return.
5. A person wants to plan a diet consisting of two types of food, X and Y. Each unit of X provides 10 grams of protein and 20 grams of carbohydrates, while each unit of Y provides 15 grams of protein and 10 grams of carbohydrates. The person needs to consume at least 50 grams of protein and 40 grams of carbohydrates daily. The cost per unit of X is Tsh. 2, and the cost per unit of Y is Tsh. 3. Determine the optimal quantities of X and Y to minimize the cost of the diet while meeting the nutritional requirements.
7. A company has two projects, P1 and P2, that require resources such as manpower and equipment. Each unit of P1 requires 3 hours of manpower and 2 units of equipment, while each unit of P2 requires 2 hours of manpower and 4 units of equipment. The company has 100 hours of manpower and 80 units of equipment available. Determine the optimal allocation of resources to maximize the total benefit from the projects.
8. A marketing manager is planning an advertising campaign using two types of media, TV and radio. Each unit of TV advertising costs Tsh. 1000 and reaches 5000 potential customers, while each unit of radio advertising costs Tsh. 500 and reaches 3000 potential customers. The total budget for the campaign is Tsh. 10,000. Determine the optimal allocation of budget between TV and radio advertising to maximize the total reach.
9. A manufacturing company produces two types of products, P1 and P2. Each unit of P1 requires 2 hours of machine time and 1 hour of labor, while each unit of P2 requires 1 hour of machine time and 2 hours of labor. The company has 80 hours of machine time and 60 hours of labor available. Determine the optimal production quantities to maximize profit, given that P1 sells for Tsh. 20 per unit and P2 sells for Tsh. 15 per unit.
10. A real estate developer is planning a housing development consisting of two types of houses, A and B. Each unit of house A requires 4 workers and 10 tons of building material, while each unit of house B requires 6 workers and 8 tons of building material. The developer has a total of 100 workers and 200 tons of building material available. Determine the optimal number of houses to build to maximize profit, given that house A sells for Tsh. 200,000 and house B sells for Tsh. 250,000.
11. An investor is considering investing in two financial assets, Stock X and Stock Y. Each unit of Stock X costs Tsh. 50 and has an expected return of 8%, while each unit of Stock Y costs Tsh. 80 and has an expected return of 12%. The investor has a total of Tsh. 10,000 to invest. Determine the optimal investment strategy to maximize the expected return on investment.
12. A manager needs to schedule two types of employees, full-time (F) and part-time (P), for a project. Each full-time employee costs Tsh. 100 per day and can work for 8 hours, while each part-time employee costs Tsh. 50 per day and can work for 4 hours. The manager has a total budget of Tsh. 1000 per day and needs to ensure that at least 10 hours of work are done each day. Determine the optimal number of full-time and part-time employees to hire to minimize costs while meeting the project requirements.
13. Diet Problem: A bakery wants to mix two types of flour, A and B, to create a batch of bread. Flour A costs Tsh. 2 per kg and contains 3 grams of protein per kg, while flour B costs Tsh. 1 per kg and contains 2 grams of protein per kg. The bakery needs at least 100 grams of protein in the batch. How many kilograms of each flour should they use to minimize the cost while meeting the protein requirement?
14. Production Problem: A company manufactures two types of furniture, chairs and tables. Each chair requires 2 units of wood and 1 unit of labor, while each table requires 3 units of wood and 2 units of labor. The company has 120 units of wood and 80 units of labor available. They make a profit of Tsh. 10 per chair and Tsh. 15 per table. How many chairs and tables should they produce to maximize their profit while not exceeding their resource constraints?
- Investment Problem: An investor has Tsh. 10,000 to invest in two stocks, stock X and stock Y. Stock X has an expected return of 8% per year and a minimum investment of Tsh. 2,000, while stock Y has an expected return of 10% per year and no minimum investment requirement. The investor wants to maximize their expected return while investing at least Tsh. 2,000 in stock X. How much should they invest in each stock?
- Blending Problem: A juice manufacturer wants to create a new juice blend using three types of fruit juice: orange,apple, and mango. Orange juice costs Tsh. 0.50 per liter, apple juice costs Tsh. 0.60 per liter, and mango juice costs Tsh. 0.70 per liter. The new blend must contain at least 40% orange juice and at least 20% apple juice. How many liters of each juice should they use to minimize the cost per liter of the blend while meeting the composition requirements?
- Delivery Problem: A delivery company has three trucks with different capacities. They need to deliver packages to four locations. Each location requires a specific number of packages. The company wants to minimize the total number of trucks used while ensuring all packages are delivered.
- Scheduling Problem: A factory has three machines that can perform different tasks on a product. Each machine has a different processing time for each task. The factory needs to produce a certain number of products within a specific deadline. How should they schedule the tasks on the machines to minimize the total production time while meeting the deadline?
- Advertising Problem: A company wants to advertise its product on two different television channels. The reach of each channel and the cost per minute of advertising are different. The company wants to maximize the total number of people reached by their advertising campaign while staying within their budget.
- Crop Planting Problem: A farmer has a limited amount of land and wants to plant three different crops. Each crop has a different profit per hectare and requires different amounts of water and fertilizer. The farmer wants to maximize their total profit while not exceeding their water and fertilizer constraints.
- Knapsack Problem: A hiker has a limited weight capacity they can carry in their backpack. They have several items they want to bring, each with a different weight and value. The hiker wants to maximize the total value of the items they bring while not exceeding the weight limit.
- Assignment Problem: A company has five employees and five tasks that need to be completed. Each employee has different skills and experience levels, and their performance on each task varies. The company wants to assign each task to an employee who can complete it most efficiently.
23. Dietary Restrictions: A dietician wants to create a meal plan for a client with specific calorie and protein requirements. Two food options are available: vegetables (200 calories and 5 grams protein per serving) and meat (300 calories and 10 grams protein per serving). The client needs at least 1800 calories and 80 grams of protein daily. How many servings of each food should be included in the meal plan to meet the requirements while minimizing the total number of servings?
24. Travel Planning: You are planning a road trip and need to decide how many miles to drive each day. You have a fixed amount of time for the trip and want to minimize the total number of days spent traveling. Gas mileage for your car is constant. Formulate an LPP to determine the optimal daily driving distance.
25. Production Mix: A company produces two types of furniture, desks and chairs. Each desk requires 4 units of wood and 2 units of labor, while each chair requires 2 units of wood and 1 unit of labor. The company has 100 units of wood and 80 units of labor available. They make a profit of Tsh. 20 per desk and Tsh. 15 per chair. How many desks and chairs should they produce to maximize their profit while not exceeding their resource constraints?
26. Fabric Blending: A textile manufacturer wants to create a new fabric blend using two materials, cotton and polyester. Cotton costs Tsh. 2 per meter and is known for its breathability, while polyester costs Tsh. 1 per meter and is more durable. The new blend needs to be at least 60% cotton for comfort and at least 40% polyester for strength.How many meters of each material should be used to minimize the cost per meter of the blend while meeting the composition requirements?
27. Investment Strategy: An investor has Tsh. 5000 to invest in two stocks, A and B. Stock A has a higher expected return of 12% per year but requires a minimum investment of Tsh. 2000. Stock B has a lower expected return of 8% per year and has no minimum investment requirement. The investor wants to maximize their expected return while meeting the minimum investment requirement for stock A. How much should they invest in each stock?
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