This is Quantitative Decision Making for Business.

This assignment has seven problems. You are required to submit any 4 of them (you must include problem 1). Any other problems submitted will be for extra credit (1 point on your final grade); it is strongly recommended that you try them all.

Hand in both a summary sheet with your answers to the problems along with the needed computer backup work. Put your name on all of the work. Each person must submit his/her own answer and computer work. If anyone submits a copy of an assignment, everyone with that submission will get a zero on the assignment.

Problem 1 (TAY 44-45/161 adjusted) The Metro Food Services Company delivers fresh sandwiches each morning to vending machines throughout the city. The company makes three kinds of sandwiches – ham and cheese, bologna, and chicken salad. A ham and cheese sandwich requires a worker 0.90 minutes to assemble, a bologna sandwich requires a worker 0.80 minutes, and a chicken salad sandwich requires a worker 1.00 minutes to make. The company has 6 workers available each night to assemble sandwiches (i.e. 2,880 minutes). Vending machine capacity is available for 3,000 sandwiches each day. The profit for a ham and cheese sandwich is 70¢ (i.e. $0.70), the profit for a bologna sandwich is 85¢, and the profit for a chicken salad sandwich is 75¢. The company knows from past sales records that their customers buy as many or more of the ham and cheese sandwiches than the other two sandwiches combined. But customers need a variety of sandwiches available, so Metro stocks at least 300 of each sandwich type. Metro management wants to know how many of each sandwich it should stock to maximize profit.

a) Formulate this decision question as a Linear Programming Model. Define the variables, write down the constraints and the objective function in mathematical (algebraic) terms (variables, inequalities, etc.).

b) Find the optimal solution using SOLVER.

Answer as many parts below as possible without re-running Solver. (do each part independently from the others) Use the SENSITIVITY REPORT whenever possible.

c) If Metro Food Services could hire another worker and increase its available assembly time by 480 minutes, or increase its vending machine capacity by 200 sandwiches, which should it do? How much additional profit would your decision result in?

d) What would the effect be on the optimal solution if the profit for a ham and cheese sandwich were increased to 80¢? to 90¢?

Problem 2 (TAY 36/218 adjusted) Joe Henderson runs a small metal parts shop. The shop contains three machines – a drill press, a lathe, and a grinder. Joe has three operators, each certified to work on all three machines. However, each operator performs better on some machines than on others. The shop has contracted to do a big job that requires all three machines.

The times (in minutes) required by the various operators to perform the required operations on each machine are summarized in the table below. Joe wants to assign one operator to each machine so that the total operating time for all three operators is minimized.

a. Formulate a linear programming model for this problem

b. Solve the model using SOLVER

c. Joe’s brother, Fred, has asked him to hire his wife, Kathy, who is a machine operator. Kathy can perform each of the required machine operations in 25 minutes. Should Joe hire his sister-in-law?

Hint - define a variable for each machine-operator pair which is

= 1 if that operator is on that machine

= 0 if that operator is not on that machine

(This is a special case of the TRANSPORTATION PROBLEM (Class 4) with SUPPLY and DEMAND are all equal to 1. Do not have to put into SOLVER that the variables have to be integer; it always happens.)

(Hint: D1 + L1 + G1 ≤ 1 D1 + D2 + D3 ≥ 1 6 constraints in total)

Problem 3 (AWZ 73-923) A bus company believes that it will need the following numbers of bus drivers during each of the next 5 years

It costs $50,000 to hire (and train) a driver and $25,000 to fire a driver. A driver’s salary is $60,000 per year. At the beginning of Year 1 the company has 40 drivers. A driver hired at the beginning of a year can be used to meet the current year’s requirements and is paid full salary for the current year. How many drivers should the company hire and fire each year to minimize the company’s total (hiring, firing, and employing) drivers’ cost.

a. Formulate a linear programming model for this problem (A Spreadsheet approach is easier.)

b. Solve the model using SOLVER.

c. Does the optimal solution change if the hiring costs is cut in half (i.e. $25,000). If so, how?

Problem 4 EXERCISE 7.15 (BF 409) (Formulation only – no computer work is needed)

This requires the following steps:

1. Formulate the problem

a. Define the variables

b. Set up the constraints

c. Define the objective function

2. List and describe any assumptions that you made.

Problem 5 (TAY 56-226) The Douglas family raises cattle on their firm in Virginia. They also have a large garden in which they grow ingredients for making two types of relish – chow-chow and top-tomato. These they sell in 16-ounce jars at local sores and craft fairs in the fall. The profit for a jar of chow-chow is $3.00 and the profit for a jar of top-tomato relish is $2.50. The main ingredients in each relish are cabbage, tomatoes and onions. A jar of chow-chow must contain at least 65% cabbage, 5% onions, and 10% tomatoes, and a jar of top-tomato relish must contain at least 50% tomatoes, 5% onions, and 10% cabbage. Both relishes contain no more than 10% onions. The family has enough time to make no more than 800 jars of relish. In checking sales records for the past five years they know that they will sell at least 30% more chow-chow than top-tomato relish. They will have 400 pounds of cabbage, 450 pounds of tomatoes, and 50 pounds of onions available. The Douglas family wants to know how many jars of each relish to produce to maximize profits.

a. Formulate a linear programming model for this problem

b. Solve the model using SOLVER

Problem 6 (HH)

Problem 7 (HH)

TMESDrillLatheGrinder

Operator 1352218

Operator 2284130

Operator 3182536

Year 1Year 2Year 3Year 4Year 5

Drivers Needed for the year7050605070