Formulate and solve this blending problem as a linear program.
Target Shirt Company makes three varieties of shirts: Collegiate, Traditional, and European. These shirts are made from different combinations of cotton and polyester.
The cost per yard of unblended cotton is $5 and for unblended polyester is $4. Target can receive up to 4000 yards of raw cotton and 3000 yards of raw polyester fabric weekly. The table below gives pertinent data concerning the manufacture of the shirts.
Formulation
Decision variables
Not only must we decide how many shirts to make and how much fabric to purchase, we also need to decide how much of each fabric is blended into each shirt.
Let sj = total number of shirt style j produced
fi = number of yards of material i purchased
xij = yards of fabric i blended into shirt style j
where i = 1 (cotton) or 2 (polyester) and
j = 1 (Collegiate), 2 (Traditional), or 3 (European)
Objective function
Maximize the overall profit.
To determine the profit function, subtract the cost of purchasing the fabric from the shirt sales revenue. Thus, the objective function is:
Max 14s1 + 15s2 + 18s3 ? 5f1 ? 4f2
Constraints
Definition of total number of shirts of each style
Total number of each style =
Total yardage used in making the style/Yardage per shirt)
(1) Collegiate: s1 = (x11 + x21)/1 >>> s1 ? x11 ? x21 = 0
(2) Traditional: s2 = (x12 + x22)/1.2 >>> 1.2s2 ? x12 ? x22 = 0
(3) European: s3 = (x13 + x23)/0.9 >>> 0.95s3 ? x13 ? x23 = 0
Definition of total yardage of materials
(4) Cotton: f1 = x11 + x12 + x13 >>> f1 ? x11 ? x12 ? x13 = 0
(5) Polyester: f2 = x21 + x22 + x23 >>> f2 ? x21 ? x22 ? x23 = 0
Weekly availability of the resources
(6) Cotton: f1 < 4000
(7) Polyester: f2 < 3000
Meet weekly contracts
(8) Collegiate: s1 > 500
(9) Traditional: s2 > 650
(10) European: s3 > 280
Do not exceed weekly demand
(11) Collegiate: s1 < 600
(12) Traditional: s2 < 850
(13) European: s3 < 675
Fabric requirements
Collegiate at least 50% cotton:
(Total yds. of cotton in Colleg. shirts) > [(0.5(1.00) yds./shirt)(number of Colleg. shirts)]
(14) x11 > 0.5s1 >>> x11 ? 0.5s1 > 0
Traditional at most 20% polyester:
(Total yds. polyester in Tradit. shirts) < [0.2(1.20) yds./shirt)(number of Tradit. shirts)]
(15) x22 < 0.24s2 >>> x22 ? 0.24s2 < 0
European at most 80% polyester:
(Total yds. polyester in Europ. shirts) < [0.8(0.90) yds./shirt)(number of Europ. shirts)]
(16) x23 < 0.72s3 >>> x23 ? 0.72s3 < 0
Nonnegativity of variables
sj, fi, xij > 0 for i = 1 or 2 and j = 1, 2, or 3
Total profit = $23,152.50
Cotton Collegiate yds. = 300.0, cotton Traditional yds. = 816.0, cotton European yds. = 121.5,
polyester Collegiate yds. = 300.0, polyester Traditional yds. = 204.0, polyester European yds. = 486.0, Collegiate shirts = 600, Traditional shirts = 850, European shirts = 675, cotton yards = 1237.5, polyester yards = 990.0
You might also like to view...
Which of the following procedures best accounts for the relative risk of a proposed project?
A. Adjusting the discount rate downward if the project is judged to have above-average risk. B. Reducing the NPV by 10% for risky projects. C. Picking a risk factor equal to the average discount rate. D. Ignoring risk because project risk cannot be measured accurately. E. Adjusting the discount rate upward if the project is judged to have above-average risk.
Productivity is measured as the ratio of inputs to outputs
Indicate whether the statement is true or false.
Credit terms 2/10, net 30 means ________
A) a discount of 10% is granted if payments are done within 30 days B) a discount of 10% is granted if payments are done within 2 days, net 30 days available C) a discount of 2% is granted if payments are done within 10 days, net 30 days available D) a discount of 2% is granted if payments are done within 30 days, beyond which a 10% interest is charged
While reviewing your current financial plan, you discover that you most likely won't achieve your long term financial goals. What should you do now?
A) Look at increasing your income. B) Look at cutting back on your expenses. C) Look at revising your goals. D) All of these would be realistic things to do.