EXERCISE 1

• Individual homework for next class (Formulation) + Solution with graphical method

• In a radio chain there are two stations, one in FM and one in AM.

• The FM station daily broadcasts 6 hours of rock, 2 hours of classical music and 6 hours of news and discussions.
• The AM station broadcasts 2 hours of rock, 10 of classical music and 4 hours of news and discussions per day.

• Because of the audience requirements, the radio chain wants to broadcast at least 30 hours of rock, at least 40 of classical music and at least 48 hours of news and discussions per day.

• FM station daily costs are £240 and for the AM station are £120.

• How many days must the radio chain schedule each station to minimize the total cost and meet the minimum requirements from the audience?

EXERCISE 2
• Individual homework for next class (Formulation) + Solution with graphical method

• A business person, who intends to make a risky investment, needs a loan. She has a list of national and international banks to call and get an appointment to negotiate the conditions of the loan.

• Specifically, the list are 10 national banks and 8 international banks. His chances to get a successful appointment in national banks is 50%, while for international banks is only 65%. She does not have time to make more than 12 calls. The business person can only call once.

• How many national and international banks does she need to call in order to maximize her successful probability to get an appointment?

EXERCISE 3
• Team homework for next class (Formulation) + Solution with graphical method

• Boris Milkem, a cash manager, deals with French currency (francs) and US currency (US dollars). At midnight he buys francs and pays US$0.25 per franc and buy dollars at 3 francs per US dollar.

• Let x1 = Number of dollars purchased with francs and x2 = Number of francs purchased with dollars.

• Suppose both types of transactions are made simultaneously and the only limitation is that at 12:01 am Boris must have a nonnegative number of francs and dollars.

• Plan a linear programming model that allows Boris to maximize the number of US dollars he has after all transactions are completed.

EXERCISE 4

• Team homework for next class (Formulation) + Solution in solver

• Lizzie's Dairy produces cream cheese and cottage cheese. Milk and cream are blended to produce these two products. Both high-fat and low-fat milk can be used to produce cream cheese and cottage cheese. High-fat milk is 60% fat; low-fat milk is 30% fat. The milk used to produce cream cheese must average at least 50% fat and that for cottage cheese, at least 35% fat. At least 40% (by weight) of the inputs to cream cheese and at least 20% (by weight) of the inputs to cottage cheese must be cream. Assume that cream is 100% fat.

• Both cottage cheese and cream cheese are produced by putting milk and cream through the cheese machine. It costs €0.4 to process 1 lb of inputs into a pound of cream cheese. It costs
€0.4 to produce 1 lb of cottage cheese, but every pound of input for cottage cheese yields
0.9 lb of cottage cheese and 0.1 lb of waste.

• Cream can be produced by evaporating high-fat and low-fat milk. It costs €0.4 to evaporate 1 lb of high-fat milk. Each pound of high-fat milk that is evaporated yields 0.6 lb of cream. It costs €0.4 to evaporate 1 lb of low-fat milk. Each pound of low-fat milk that is evaporated yields 0.3 lb of cream. Each day, up to 3,000 lb of input may be sent through the cheese machine. Each day, at least 1,000 lb of cottage cheese and 1,000 lb of cream cheese must be produced. Up to 1,500 lb of cream cheese and 2,000 lb of cottage cheese can be sold each day. Cottage cheese is sold for €1.2/lb and cream cheese for €1.5/lb.

• High-fat milk is purchased for €0.8/lb and low-fat milk for €0.4/lb. The evaporator can process at most 2,000 lb of milk daily. Maximize Lizzie's daily profit using LP.

EXERCISE 5
• Formulation + Solution in Solver

• A pastry chef from Le Patissier bakery in Le Havre wishes to bake some cakes and cookies. Each cake requires 2.5 cups of flour and 2 cups of sugar, while one batch of cookies consumes 1 cup of flour and 0.5 cups of sugar.

• The chef can use at most 70 cups of flour and no more than 50 cups of sugar.

• Each cake can be sold in 10€ and each batch of cookies in 8€.

• How many cakes and batches of cookies must the chef sell to maximize the sales revenue?

EXERCISE 6
• Individual homework for next class (Formulation)

• A jewelry produces rings, bracelets and necklaces. The maximum availability of time is 8 hours per day.

• It takes 45 minutes to make a ring, 90 minutes to make a bracelet and a necklace 2 hours.

• The cost of a ring is 50 monetary units (m.u.), a bracelet 80 m.u. and a necklace 150 m.u.

• If the daily budget is 450 m.u., what is the maximum production that can be reached per day?

EXERCISE 7

• Team homework for next class (Formulation)

• James Beerd bakes cheesecakes and Black Forest cakes. During any month, he can bake at most 65 cakes. The costs per cake and the demands for cakes, which must be met on time, are listed below.

• It costs $0.50 to hold a cheesecake, and $0.40 to hold a Black Forest cake, in inventory for a month. Formulate an LP to minimize the total cost of meeting the next three months' demands.

EXERCISE 8
• Team homework for next class (Formulation) + Solution in Solver

• A bus company believes that it will need the following number of bus drivers during each of the next five years: year 1-60 drivers; year 2-70 drivers; year 3-50 drivers; year 4-65 drivers; year 5-75 drivers.

• At the beginning of each year, the bus company must decide how many drivers should be hired or fired. It costs $4,000 to hire a driver and $2,000 to fire a driver. A driver's salary is $10,000 per year.

• At the beginning of year 1, the company has 50 drivers. A driver hired at the beginning of a year may be used to meet the current year's requirements and is paid full salary for the current year.

• Formulate an LP to minimize the bus company's salary, hiring, and firing costs over the next five years.

EXERCISE 9
• Individual homework for next class (Formulation)

• A fleet of trucks belongs to CrabTruck. This fleet collects a series of goods in a 4 suppliers A to D for customers 1 to 5; however each truck can collect at most one of them. The travel time for each of the trucks to visit four possible suppliers is given below:

• Formulate the model that minimizes the time, knowing that all goods must be collected.

EXERCISE 10

• Individual homework for next class (Formulation)

• A set of police stations must be located to serve 7 districts of a city.

• Given the available properties, it is known that the possible locations are 6, each covering some of the districts, as indicated in the following table:

• Formulate the model to cover the seven districts at the minimum cost.

EXERCISE 11
• Individual homework for next class (Formulation)

• A executive has 7 days to prepare a strategic plan comprising 3 issues: Appraisal of assets (A), Budget (B) and cost reduction (C). Due to scarce resources:

• She does not want to exceed 16 hours of preparation.
• A requires at least twice the time of B.
• The number of hours to prepare C must be greater than or equal to those of A plus B.

• Formulate the model and maximize the expected balance scorecard of the company, given that it is proportional to the hours spent in preparing each of the statement multiplied by the prioritized factors 2, 3 and 1, respectively.

EXERCISE 12
• Formulation + Solution with graphical method

• A dairy company manufactures two types of milk: high-fat and lactose-free. The manufacturing process is carried out by a sterilization and a packaging machine. The summary of used hours to produce each product for each machine is given below :

• The maximum number of available hours for each of the machines is 20 hours. The firm must make at least 5 daily units of high-fat milk. The unit cost to process high-fat milk is $60MXP and the lactose-free for $90MXP.

Assuming that milk can be manufactured partially given that it will be stored in large tanks. Find the optimal quantity of each type of milk to be processed to minimize production cost.