Im a final year student needing a bit of help with the following lp assignment. Im a bit rusty at linear programming and am find it a bit confusing developing the layout and constraints for the following problem below. Any help would be most appreciated. Thanks!
Katherine Rally is meeting with Ted Lawson, the production manager, to decide upon next monthís production plan for the Autumn line of clothes. Specifically, she must determine the quantity of each clothing item to produce given the plantís capacity, limited resources, and demand forecasts. Accurate planning for next monthís production is critical to Autumn sales since the items produced next month will appear in stores during September and women generally buy the majority of the Autumn fashions when they appear in September.
She studies the clothing patterns and material requirements. Her Autumn lines consist of both professional and casual fashions. In her analysis of costs she has established the following table of information concerning the Autumn professional fashions
Clothing Item / Material Requirements / Price(£) / Labour and Machine Costs (£)
Tailored wool slacks / 3 yards of wool, 2 yards of acetate / 300 / 160
Cashmere sweater / 1.5 yards of cashmere / 450 / 150
Silk blouse / 1.5 yards of silk / 180 / 100
Silk camisole / 0.5 yards of silk / 120 / 60
Tailored skirt / 2 yards of rayon, 1.5 yards of acetate / 270 / 120
Wool blazer / 2.5 yards of wool, 1.5 yards of acetate / 320 / 140
The Autumn casual fashions include:
Velvet pants / 3 yards of velvet, 2 yards of acetate / 350 / 175
Cotton sweater / 1.5 yards of cotton / 130 / 60
Cotton mini-skirt / 0.5 yards of cotton / 75 / 40
Velvet shirt / 1.5 yards of velvet / 200 / 160
Button-down blouse / 1.5 yards of rayon / 120 / 90
She knows that for the next month she has ordered 45,000 yards of wool, 28,000 yards of acetate, 9,000 yards of cashmere, 18,000 yards of silk, 30,000 yards of rayon, 20,000 yards of velvet and 30,000 yards of cotton for production. The prices of the materials are listed below.
Material / Price per yard (£)
Wool / 9.00
Acetate / 1.50
Cashmere / 60.00
Silk / 13.00
Rayon / 2.25
Velvet / 35.00
Cotton / 2.50
Any material not used in production can be sent back to the textile wholesaler for a full refund, although scrap material cannot be sent back to the wholesaler.
She knows that the production of the silk blouse and cotton sweater results in scrap material. Specifically, for the production of one silk blouse or one cotton sweater, 2 yards of silk and cotton, respectively are needed. From these 2 yards, in both cases 1.5 yards is used in the clothe item and 0.5 yards is scrap material. She does not want to waste the material, so she plans to use the rectangular scrap of silk or cotton to produce a silk camisole or cotton mini-skirt respectively. Thus whenever a silk blouse is produced a silk camisole is produced and whenever a cotton sweater is produced, a cotton mini-skirt is produced. Note that it is possible to produce a silk camisole without producing a silk blouse and a cotton mini-skirt without producing a cotton sweater.
Some items have limited demand. A maximum of 5,500 pairs of velvet pants and 6,000 velvet shirts can be sold. The company does not want to produce more than the demand since the items go out of fashion and cannot be sold next year. The cashmere sweater also has limited demand and it is estimated that no more than 4,000 cashmere sweaters can be sold. Similarly, there is limited demand for the silk blouse and camisoles. Specifically, at most 12,000 silk blouses and 15,000 silk camisoles can be sold.
The demand forecasts also indicate that the wool slacks, tailored skirts and wool blazers have a great demand. There is demand for up to 7,000 pairs of wool slacks and 5,000 wool blazers. Katherine wants to meet at least 60% of demand for these two items to maintain her loyal customer base. The demand for tailored skirts is estimated to be 2,800.
Formulate and solve a linear programming problem maximizing profit given the production, resource and demand constraints.