Mathematical Programming Homework 1: Workers at the Gotham City Police Department

Question 3.5.5

Problem:

Each day, workers at the Gotham City Police Department work two 6-hour shifts chosen from 12 am to 6 am, 6 am to 12 pm, 12 pm to 6 pm, 6 pm to 12 am.  The following number of workers are needed during each shift: 12 am to 6am-15 workers; 6 am to 12pm-5 workers; 12 pm to 6pm-12 workers;  6 pm to 12am-6 workers.  Workers whose two shifts are consecutive are paid $12 per hour; workers whose shifts are not consecutive are paid $18 per hour.  Formulate an LP that can be used to minimize the cost of meeting the daily workforce demands of the Gotham City Police Department.

Question 3.8.2

Problem: O.J. Juice Company sells bags of oranges and cartons of orange juice.  O.J. grades oranges on a scale of 1 (poor) to 10 (excellent).  O.J. now has on hand 100,000 lb of grade 9 oranges and 120,000 lb of grade 6 oranges.  The average quality of oranges sold in bags must be at least 7, and the average quality of oranges used to produce orange juice must be at least 8.  Each pound of oranges that is used for juice yields a revenue of $1.50 and incurs a variable cost of $1.05.  Each pound of oranges sold in bags yields a revenue of $.50 and incurs a variable cost of $.20.  Formulate an LP to help O.J. maximize profit.

Question 3.9.2

Problem:

Furnco manufactures tables and chairs. A table requires 40 board ft of wood, and a chair requires 30 board ft of wood. Wood may be purchased at a cost of $1 per board ft, and 40000 board ft of wood are available for purchase. It takes 2 hours of skilled labor to manufacture an unfinished table or an unfinished chair. Three more hours of skilled labor will turn an unfinished table into a finished table, and 2 more hours of skilled labor will turn an unfinished chair into a finished chair. A total of 6000 hours of skilled labor is available (and have already been paid for). All furniture produced can be sold at the following unite prices: unfinished table, $70; finished table, $140; unfinished chair, $60; finished chair, $110. Formulate LP that will maximize the contribution to profit from manufacturing tables and chairs.

Question 3.10.4

A manufacturing company produces two products, A and B. the company has agreed to deliver the product on the schedule shown in table 34. The company has two assembly lines, 1 and 2, with available production hours shown in table 35. The production rate for each assembly line and product combination, in term of hours per product, arc shown in table 36. It takes 0.15 hour to manufacture 1 unit of product A on line 1, and so on. It costs $5 per hours of line time to produce any product. The inventory carrying cost per month for each product is 200 per unit (charged on each month’s ending inventory). Currently, there are 500 units of A and 750 units of B in inventory management would like at least 1,000 units of each product in inventory at the end of April. Formulate an LP to determine the production schedule that minimizes the total cost incurred in meeting demands on time.

Question 3.12.2

An insurance company believes that it will require the following numbers of personal computers during the next six months: January, 9; February, 5; March, 7; April, 9; May, 10; June, 5. Computers can be rented for a period of one, two, or three months at the following unit rates: one- month rate, $200; two-month rate, $350; three-month rate, $450.

Formulate an LP that can be used to minimize the cost of renting the required computers. You may assume that if a machine is rented for a period of time extending beyond June, the cost of the rental should be prorated. For example, if a computer is rented for 3 months at the beginning of May, then a rental fee of 2/3 (450) = $300, not $450, should be assessed in the objective function

Question 4.14.3

During the next 3 months, Steelco faces the following demands per month for steel:  month 1, 100 tons; month 2, 200 tons; month 3, 50 tons.  During any month a worker can produce up to 15 tons of steel.  Each worker is paid $5000 per month.  Workers can be hired or fired at a cost of $3000 per worker fired and $4000 per worker hired.  The cost of holding one ton of inventory for a month is $100.  Demand can be backlogged at a cost of $70 per ton per month.  At the beginning of month 1, Steelco has 7 workers and no steel in inventory.  During any month, at most two workers can be hired and all demand must be satisfied at the end of month three.  It costs $300 in raw materials to produce a ton of steel.  Develop an aggregate plan that minimizes costs for Steelco