A manufacturer produces three products, X, Y and Z with two machines A and B. The time taken to
produce each unit of X is 20 minutes for machine A and 26 minutes for machine B. Each unit produced
of Y takes 24 minutes using machine A and 28 minutes using machine B. Lastly, the time taken to
produce each unit of Z is 20 minutes for machine A and 16 minutes for machine B. In a particular week,
the manufacturer has a limited time of working hours. In particular, 48 hours of work are available on
machine A and 52 hours are available on machine B. Moreover, each week starts with a stock of 60
units of X, 50 units of Y and 40 units of Z.
Weekly demand for the products are stochastic but the manufacture estimates an average of 70 units
for each product X, Y and Z being demanded on a weekly basis.
(i) Using the information above, formulate the linear programming (LP) model for the
manufacturer, if they wish to plan production in order to end the week with the maximum
possible stock. (6 marks)
(ii) This question requires you to write out the Excel formula for specific cells from the Excel
sheet provided below. Write out the Excel formulas to cells B13, B17 and B19 as if you are
typing out the formulae in Excel. Please clearly label your answers and write out your
answers in your script booklet and NOT on the exam paper itself. (4 marks)
The LP has been solved and the sensitivity analysis report generated. For the following questions,
please refer to this sensitivity analysis output to answer the questions.
(iii) For the optimal solution, how much time in hours is spent on using Machine A? (2 marks)
(iv) For the optimal solution, how much stock of Product X is left? How much stock of Product Y is left?
How much stock of Product Z is left? (3 marks)
(v) Would the solution change if the number of hours available on machine A was 50 hours instead of
48 hours? If yes, then what is the value of the objective function? (4 marks)
(vi) What would the optimal solution be if the machines can now produce an extra 0.5 units of Product
Y? To answer this question, think about what it means in terms of the objective coefficient of Product
Y in the objective function. (4 marks)
(vii) Define reduced cost. Explain why the reduced costs for products X, Y and Z are equal to 0? (2
(i) 15 marks
The Grand Strand Oil Company produces regular and premium gasoline for independent service
stations. The Grand Strand refinery manufactures the gasoline products by blending 3 petroleum
components. The gasolines are sold at different prices, and the petroleum components have different
costs. Data available show that regular gasoline can be sold for $2.9 per gallon and premium gasoline
for $3 per gallon. For the current production planning period, Grand Strand can obtain the 3 petroleum
components at the cost per gallon and in the quantities as follows:
Product specifications, as shown in the following table, for the regular and premium gasolines restrict
the amounts of each component that can be used in each gasoline product. Current commitments to
distributors require Grand Strand to product at least 10,000 gallons of regular gasoline.
Formulate the LP model to determine the amounts of gallons of components 1, 2, and 3 should Grand
Strand mix or blend into regular and premium gasoline to reach the maximum profits.