Question #1 (25 points)
Manufacturers in Dalton, GA, produce more than 70% of the total output of the $9 billion worldwide carpet industry. Competition in this industry is intense and forces producers to strive for maximum efficiency and economies of scale. It also forces producers to continuously evaluate investments in new technology.
Kamm Industries is one of the leading carpet producers in the Dalton area. Its owner, Geoff Kamm, asked for your assistance in planning the production schedule for the next quarter (13 weeks). The company has orders for 15 different types of carpet that can be produced on two types of looms: Dobbie looms and Pantera looms. Pantera looms produce standard tufted carpeting. Dobbie looms can also produce standard tufted carpeting but also allow the incorporation of designs (such as flowers or corporate logos) into the carpeting. The following table summarizes the orders for each type of carpet that must be produced in the coming quarter along with their production rates and costs on each type of loom, and the cost of subcontracting each order. Note that the first 4 orders involved special production requirements that can only be achieved on a Dobbie loom or via subcontracting. Assume that any portion of an order may be subcontracted.
Demand Dobbie Pantera Subcontract
Carpet (Yds) Yd/Hr Cost/Yd Yd/Hr Cost/Yd Cost/Yd
1 14,000 4.510 $2.66 na na $2.77
2 52,000 4.796 $2.55 na na $2.73
3 44,000 4.629 $2.64 na na $2.85
4 20,000 4.256 $2.56 na na $2.73
5 77,500 5.145 $1.61 5.428 $1.60 $1.76
6 109,500 3.806 $1.62 3.935 $1.61 $1.76
7 120,000 4.168 $1.64 4.316 $1.61 $1.76
8 60,000 5.251 $1.48 5.356 $1.47 $1.59
9 7,500 5.223 $1.50 5.277 $1.50 $1.71
10 69,500 5.216 $1.44 5.419 $1.42 $1.63
11 68,500 3.744 $1.64 3.835 $1.64 $1.80
12 83,000 4.157 $1.57 4.291 $1.56 $1.78
13 10,000 4.422 $1.49 4.558 $1.48 $1.63
14 381,000 5.281 $1.31 5.353 $1.30 $1.44
15 64,000 4.222 $1.51 4.288 $1.50 $1.69
Kamm currently owns and operates 15 Dobbie looms and 80 Pantera looms. To maximize efficiency and keep pace with demand, the company operates 24/7. Each machine is down for routine maintenance for approximately 2 hr/week.
a) Formulate a linear programming model for this problem that can be used to determine the optimal production/subcontracting plan. Make sure to include the managerial problem definition. Assume that produced and purchased quantities can take continuous values (e.g., you could produce 9562.34 yards of a carpet type). Identify:
i. Decision variables
ii. Objective function
iii. All relevant constraints.
Note: There is an Excel template available for this problem.
b) Use Excel Solver to determine the optimal solution. Include a copy of the Solver Answer Report as part of the main file.
c) By editing the model in (a) and explaining those edits, answer the questions below:
1) What would happen to the total cost if one of the Dobbie machines broke and could not be used at all during the quarter?
2) What would happen to the total cost if an additional Dobbie machine was purchased and available for the quarter?
3) What would happen to the total cost if one of the Pantera machines broke and could not be used at all during the quarter?
4) What would happen to the total cost if an additional Pantera machine was purchased and available for the quarter?
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