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. row1(; jackets, j = 1; dresses, j = 2; shirts, j=3; shorts, j = 4; pants, j = 5; camisoles, j = 6) row2(silk, i = 1; 4; 2; 0; 0; 0; 0.50) row3(lining, i = 2; 3; 1; 0; 0; 0; 0.00) row4(cotton, i = 3; 0; 0; 2; 2; 3; 0.00) row5(labour, i = 4; 2; 2; 2; 2; 2; 0.50) row6(price, pj; 200; 150; 150; 81; 90; 60.00) row7(maximum demand, uj; 20; 60; 140; 50; 80; 60.00) 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. row1(; Availability, si; UnitCost, ci) row2(silk, i = 1; 250; 20) row3(lining, i = 2; 150; 10) row4(cotton, i = 3; 600; 18) row5(labour, i = 4; 600; 40) 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. (b)Formulate this problem in SAS and solve using proc optmodel. Ensure that your formulation 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 unique? Justify your answer by referring to relevant SAS output. (d)In this problem, 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 be taken into consideration? Justify your answer with appropriate analysis/results.
(a) Linear Programming Formulation:
Decision Variables: Let ( x_j ) represent the quantity of garment type ( j ) produced, where ( j = 1, 2, 3, 4, 5, 6 ) for jackets, dresses, shirts, shorts, pants, and camisoles respectively.
Objective Function: The objective is to maximize the total profit, which can be expressed as: [ \text{Maximize} \quad Z = \sum_{j=1}^{6} p_jx_j - \sum_{i=1}^{4}c_ix_i ] where
Constraints:
Demand Constraints: [ x_j \leq u_j \quad \text{for} \quad j=1,2,3,4,5,6 ] These constraints ensure that the quantity of each garment type produced does not exceed the maximum demand.
Fabric Constraints: [ \sum_{j=1}^{6}a_{ij}x_j \leq s_i \quad \text{for} \quad i=1,2,3 ] where ( a_{ij} ) represents the amount of fabric or labor ( i ) used in producing one unit of garment ( j ) and ( s_i ) is the availability of fabric or labor ( i ).
Signature Product Satisfaction Constraint: [ \frac{x_2 + x_3}{u_2 + u_3} \geq 0.6 ] This constraint ensures that at least 60% of the demand for dresses and shirts is satisfied.
(b) Solution Using SAS:
I'm unable to execute SAS code directly as a text-based AI. However, I can guide you through the formulation process and explain how to input and solve the problem using proc optmodel in SAS. Here's a general structure for the SAS code:
proc optmodel; /* Decision Variables */ var x{j in 1..6} >= 0; /* Objective Function */ maximize Z = sum{j in 1..6} p[j]*x[j] - sum{i in 1..4} c[i]*x[i]; /* Constraints */ con demand_c{j in 1..6}: x[j] <= u[j]; con fabric_c{i in 1..3}: sum{j in 1..6} a[i,j]*x[j] <= s[i]; con signature_c: (x[2] + x[3]) / (u[2] + u[3]) >= 0.6; /* Resource Usage Table and Solution Output */ /* Include statements to print output */ solve; /* Print the required output */ create data output from [variables and constraints to be included in the output]; quit;
This structure outlines the basic formulation and solution process in SAS using proc optmodel. You can use appropriate statements within the proc optmodel block to print the required output as per the question.
(c) Uniqueness of Solution:
To determine if the solution obtained is unique, you can refer to the SAS output for the shadow prices (dual values). If the shadow prices are unique for each resource constraint, it indicates a unique solution. However, if the shadow prices are not unique, it suggests that there may be multiple optimal solutions.
(d) Switching Fabric Supplier Analysis:
When considering whether to switch fabric suppliers, factors to consider include the impact on profit, utilization of fabric resources, and any associated costs. To justify the recommendation, you should compare the total profit, fabric usage, and costs under the new supplier's terms. If the potential increase in profit due to better fabric utilization compensates for the higher fabric price, then switching suppliers may be recommended.
You can use the linear programming model to analyze the impact of the new fabric supplier's terms on profit and resources usage. By comparing the two scenarios (current and new supplier), you can determine the optimal decision and provide a justified recommendation based on the analysis/results.