Instructions:

•  By the due date, you should submit the following three files to Canvas:

–  1 PDF file

*  Your file should contain relevant explanations, mathematical notation, workings and SAS output.  Answers which do not include appropriate notation and/or workings, or conversely include too many irrelevant details will be penalised.

*  Units of measurement should be included with your answers where relevant (e.g. $, kg).

*  Where you are required to use SAS, a copy of the SAS output should be included within the PDF file (e.g. as an image). You do not need to includeSAS code within your PDF file.

*  Handwritten submissions are appropriate for some questions. These should be scanned for inclusion in the PDF file.

–  2 SAS files

*  Submit 1 SAS per question.

*  Use comments within your SAS code where appropriate.

*  All SAS files should run on any computer and should not require any additional files.

*  Files which contain unnecessary code or produce unnecessary output will be penalised.

*  SAS code will be assessed based on its accuracy and elegance.

–  Do not upload files as a zip.

•  Late Assignments:  Failure to submit the assignment on time will result in a penalty in accordance with the DCT late policy (i.e. 5% per day up to a maximum of 5 days).

•  Special Consideration:  If extenuating circumstances (e.g. illness) prevent the timely submission

of your assignment you can apply for special consideration. You may also apply for special consideration  if such circumstances result in your submission being incomplete. Applications for special consideration

should be submitted via Canvas.

•  Originality:  This assignment is an individual piece of work.  You are encouraged to discuss the

assignment with your lecturers and classmates, however, the work you submit must be your own.

Assignments that show similarities to work submitted by other students will be investigated for

plagiarism and treated very seriously. Plagiarism software, such as TurnItIn, maybe used to electronically compare submissions to those of other students and to documents on the internet.  Before you

submit this assignment, you should complete the Academic Integrity module on Canvas.

1.  The production manager at Stylise Hooper Fashions is planning their spring fashion line.  Stylise Hooper Fashions has designed and will manufacture two products for spring: jackets and dresses. Jackets sell for $200 and dresses sell for $150. Jackets require 4 metres of silk, 3 metres of lining and 2 hours of labour. Dresses require 2 metres of silk, 1 metre of lining and 2 hours of labour.

The production manager has ordered 300 metres of silk at a cost of $20 per metre and 200 metres of lining at a cost of $10 per metre. Any unused fabric can be returned, so they will only pay for what is actually used.

The spring fashion line must be ready in 4 weeks. Over the next four weeks there are 2 staff members available to sew the garments. Each staff member gets paid $40 per hour and can work at most 40 hours per week. Staff members are paid for the hours they work.

There is some concern about the number of dresses that can be sold, given the cooler weather expected this spring, so the maximum number of dresses that should be produced is 60.  Assume that fractional quantities of each garments can be produced.

How many jackets and dresses should Stylise Hooper Fashions manufacture for spring in order to maximise their profit (i.e. revenue - costs)?

Total for Question 1: 50 marks (a)  Formulate this problem as a linear programming problem by defining the decision variables,      (3 marks)

stating the objective function and stating the constraints.  Write the expressions out “in-full” rather than using vectors/matrices in your answer.

(b)  State the augmented form of this LP.                                                                                                                (2 marks)

(c)  Represent this linear programme (LP) graphically (using pen/paper or software of your choice).     (5 marks)

On your graph, indicate all corner points. For each corner point:

•  identify the values of the decision variables (not including the slack variables),

•  calculate the value of the objective function, and

•  identify the corner point that provides an optimal solution.

(d)  Solve the LP in part (a) using the Simplex Algorithm (by hand, not using SAS or other software).    (10 marks)

At each iteration of the algorithm clearly indicate the entering variable, the leaving variable, the variables in the basis, and the elementary row operations performed. After the final iteration, state the optimal solution and corresponding objective function value.

Hint: you should find an optimal solution after 2 iterations.

Note:  Typed or handwritten answers are acceptable for this question.   Handwritten answers should be scanned and included as an image in your PDF file.

(e)  Is the optimal solution unique? Justify your answer.                                                                                    (5 marks)

(f)  How much should Stylise Hooper Fashions be willing to pay for an additional metre of lining?      (5 marks)

As part of your answer, compute the range of feasibility for the relevant constraint.  Interpret your results in a way that is meaningful for the production manager at Stylise Hooper Fashions.

Do not resolve the LP.

(g)  Suppose that if Stylise Hooper Fashions discounts dresses by $p, they will be able to sell additional   (5 marks)

dresses (above the existing limit of 60) at their end-of-season sale.  They will still sell up to 60 dresses at the original price, but any additional dresses will sell for the discounted price. What is the maximum discount that should be provided per dress? As part of your answer, compute the range of feasibility for the relevant constraint.

The production manager has suggested making and then selling an extra 10 dresses with a discount of $12 at the end-of-season sale. Would you recommend that they do so? Justify your answer using appropriate calculations and interpret your results in a way that is meaningful for the production manager at Stylise Hooper Fashions.

Do not resolve the LP.

(h)  Suppose that Stylise Hooper Fashions believes that the price of jackets could be increased from  $200 to $220. How would this impact the production plan? Justify your answer using appropriate calculations and interpret your results in a way that is meaningful for the production manager at Stylise Hooper.  As part of your answer, compute the range of optimality for the relevant objective function coefficient.

Do not resolve the LP.          (5 marks)

(i)  Formulate the LP from part (a) in SAS using proc  optmodel. Verify that the solution is the same as part (d). Use SAS to print the following output:

•  the profit (formatted as a $XXX.XX),

•  the number of each garment produced,

•  a resource usage table showing for each resource: the amount used, the amount available, the percentage used (formatted as a percentage), and the shadow price. (10 marks)

2.  The production manager at Stylise Hooper Fashions also needs to plan for their summer fashion line Over summer, they have a much bigger product range and there are different operating requirements.

The design team has provided an estimate about the fabric and labour requirements for this year’s product line.  Demand forecasting has also been undertaken to estimate the maximum number of each garment that can be sold over this year’s summer season.  The number of each garment produced cannot exceed this number.  The required fabric, required labour, price and maximum demand for each garment are provided below.

Furthermore, in order to maintain their loyal customer base, they must ensure that at least 60% of demand is satisfied for their signature products, dresses and shirts.

The amount of the fabric ordered and the cost per metre, and the availability of labour and the cost per hour are shown in the table below.

In contrast to the spring order, all fabrics ordered for summer must be paid for in full. Due to nature of the summer season, any fabric that is not used can be returned for a 50% refund. Staff members have a fixed contract so must be paid the same amount, regardless of how much they work.

Stylise Hooper Fashions is an environmentally-conscious organisation and has a waste-minimisation policy which requires efficient use of off-cuts, where possible.  Off-cuts cannot be returned, so will  be wasted if not used. Whenever a jacket is produced, there are sufficient off-cuts to also produce a  camisole. Therefore, whenever a jacket is produced a camisole must also be produced.  Camisoles produced with a jacket do not require any fabric, only 0.5 hours of labour.  Camisoles produced without a jacket require fabric and labour, as outlined in the table above.

How many of each garment should Stylise Hooper Fashions manufacture for summer in order to maximise their profit (i.e. revenue - costs)?

Assume that fractional quantities of each product can be produced.

(a)  Formulate this problem as a linear programming problem by defining the decision variables, stating the objective function and stating the constraints. A brief explanation of the objective function and each constraint should be provided.

Note: you can use appropriately defined vectors and matrices in your formulation.       (20 marks)

(b)  Formulate this problem in SAS and solve using proc  optmodel. Ensure that your formulation          (10 marks)

is a linear programming problem.

Use SAS to print the following output:

•  the profit (formatted as $XXX.XX),

•  the number of each garment produced and the percentage of demand met per product (formatted as a percentage),

•  a resource usage table showing for each resource: the amount used, the amount available, the percentage used (formatted as a percentage), and the shadow price.

(c)  Is the solution obtained in part (b) unique?  Justify your answer by referring to relevant SAS output.            (5 marks)

(d)  The production manager is considering switching to a different fabric supplier for next year. The new supplier provides a refund of 80% on unused fabric, but has a higher price for silk at $25 per metre. Would you recommend they switch supplier? What factors should betaken into consideration? Justify your answer with appropriate analysis/results. (5 marks)