ReCarry is a small manufacturing company which produces reusable shopping bags. It producestwo products: a standard bag and a designer bag. Both bags are the same size at 48.3 x 38.1 x 15.2cm, but differ in terms the fabric used and the market segment targeted.
For instance, the standardbag includes sponsor logos, whereas the designer bag comes with additional embellishments suchas gold plated sequins. A thorough investigation of the production process has revealed that thefollowing steps are involved in the manufacture of these bags.
• Cutting fabric
• Sewing
• Finishing (e.g. adding sponsor logos, embellishments etc.)
• Inspection and packagingIn order to manufacture a standard bag, 42 minutes are required to cut the fabric and 30minutes arerequired to sew the bag.
The inclusion of the sponsor logo occurs at the finishing stage and takes 1hour. The bag is then inspected and packed, which takes 6 minutes. In order to manufacture a designer bag, due to the delicate nature of the fabric, 1 hour is required to cut the fabric and 50 minutes are required to sew the bag. The inclusion of embellishments, such as gold plated sequins, takes 40 minutes. The bag is then inspected and packed using tissue paper andbubble wrap, which takes 25 minutes.Over the next financial quarter, 630 hours are available to cut the fabric, 600 hours are available forsewing, 708 hours are available for finishing, and 135 hours are available for inspection and packing. The profit contribution from standard bags is $10 per unit and for designer bags is $9 per unit. ReCarry want to know how many of each type of bag they should produce over the next financialquarter in order to maximise their profit.
(a) Formulate as a linear programming problem by defining the decision variables, stating the objective function and stating the constraints.
(b) Represent this LP graphically (using pen/paper or software of your choice) and fromyour graph. identify an optimal solution.
(c) Solve using the Simplex Method (by hand, not using SAS or other software). At each iteration of the algorithm clearly indicate the entering variable, the leaving variable, the variables in thebasis, and the elementary row operations performed. After the final iteration, state the optimalsolution and corresponding objective function value.
(d) Is the optimal solution unique?
(e) Compute the range of feasibility for each constraint. Interpret your results in a way that is meaningful for amanager at ReCarry.
(f ) Compute the range of optimality for each product. Interpret your results in away that ismeaningful for amanager at ReCarry.
(g) Suppose an additional 30 hourswere available for cutting the fabric. Howwould profit increase?
Howmuch should the firmbe willing to pay for an extra hour of cutting time?