Question

The New England Cycle Company is planning next month’s production of its two bicycles: its B2 tandem cycle built for two and its S10 single-rider 10-speed model. Both models use the same seats and tires. Management has been informed that 2000 seats, 1000 gear assemblies (only the S10 requires a gear assembly), and 2400 tires are available next month for production. On the average, it takes two hours to produce a B2 and three hours to produce a S10 bicycle. The accounting department estimates that each B2 yields a profit of $40 and each S10 yields a profit of $100.

Management has set the following prioritized goals:

Priority 1: Fulfill a contract for 400 B2 bicycles that it promised to deliver to vendors in Ocean City, MD next month.

Priority 2: Produce at least 1000 total bicycles during the month

Priority 3: (1) Achieve at least $100,000 profit for the month, (2) use no more than 1600 labor-hours during a month.

Each $1000 short of $100,000 goal is considered 30 times as important as utilizing an extra labor-hour.

Priority 4: The company is developing a new cycle, so it would like to have: (1) at least 200 tires left over at the end of the month, (2) at least 100 gear assemblies left over at the end of the month.

Each leftover gear assembly is deemed twice as important as a leftover tire.

Management wants to determine the production schedule that best meets its prioritized objectives.

Formulate this goal programming problem. Solve with the software of your choice

Answer 1

Let X = no of B2 cycles which are going to manufacture in next month

Y = no of S10 cycles which are going to manufacture in next month

Using the constraints in the question, the equations are:

As we know that total no of seats are 2000, 2X + Y ≤ 2400

and Gear Assemblies should be 1000 which implies that Y ≤ 1000

Total tires are 2400 which implies that 2X + 2Y ≤ 2400

Also, X,Y ≥ 0

Variables for the priority constraints,

Priority 1 = On achieving a contract of 400 B2 cycles (U1),

Our required equation will be X + U1 - E1 = 400

Min V1 = U1

Priority 2 = On achieving atleast 1000 production level (U2),

Our required equation will be X + Y + U2 - E2 = 1000

Min V2 = U2

Priority 3 = On acheiving $100000 profit (U3),

Our required equation will be 0.4X + 0.1Y +U3 - E3 = 100

When we will use more than 1600 labor hrs (E4)

then our required equation will be 2X + 3Y + U4 - E4 = 1600

Min V3 = 30U3 + E4

Priority 4 = When we use more than 2200 tires (E5),

then our required equation will be 2X + 2Y + U5 - E5 = 2200

when we use more than 900 gear assemblies (E6)

then our required equation will be Y + U6 - E6 = 900

Min V6 = E5 + 2E6

Solving it using the excel, we will get the answer as B2 = 400 cycles and S10 = 700 cycle.