Problem 1 - Flextrola and Solectrics are two manufacturers that produce electronics products. Both companies serve global markets. Their market sizes are as shown in Table 1. All demand is in millions of units.
Table 1: Market Demands and Duties
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Markets
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N. America
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S. America
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Europe (EU)
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Europe (Non EU)
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Japan
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Rest of Asia/Australia
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Africa
|
|
Demand
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Flextrola
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10
|
4
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20
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3
|
2
|
2
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1
|
|
Solectrics
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12
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1
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4
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8
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7
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3
|
1
|
|
Total
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22
|
5
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24
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11
|
9
|
5
|
2
|
|
Import Duties
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3%
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20%
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4%
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15%
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4%
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22%
|
25%
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Both companies have three production facilities. Flextrola's plants are in Europe (EU), N. America, and S. America. Solectrics' plants are in Europe (EU), N. America, and Rest of Asia/Australia. The capacity (in millions of units), annual fixed cost (in millions of $), and variable production costs ($ per unit) for each plant are as shown in Table 2.
Table 2: Plant Capacities and Costs
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Plants
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Capacity (in millions of units)
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Fixed Cost / year (in millions of $)
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Variable Cost / Unit
|
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Flextrola
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Europe (EU)
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20
|
100
|
$6.00
|
|
N. America
|
20
|
100
|
$5.80
|
|
S. America
|
10
|
60
|
$5.30
|
|
Solectrics
|
Europe (EU)
|
20
|
100
|
$6.00
|
|
N. America
|
20
|
100
|
$5.80
|
|
Rest of Asia
|
10
|
50
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$5.00
|
Transportation costs from plant regions and market regions are as shown in Table 3. All transportation costs are shown in dollars per unit.
Table 3: Transportation Costs
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N. America
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S. America
|
Europe (EU)
|
Europe (Non EU)
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Japan
|
Rest of Asia /Australia
|
Africa
|
|
N. America
|
1
|
1.5
|
1.5
|
1.8
|
1.7
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2
|
2.2
|
|
S. America
|
1.5
|
1
|
1.7
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2
|
1.9
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2.2
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2.2
|
|
Europe (EU)
|
1.5
|
1.7
|
1
|
1.2
|
1.8
|
1.7
|
1.4
|
|
Rest of Asia
|
2
|
2.2
|
1.7
|
1.6
|
1.2
|
1
|
1.8
|
When products are shipped from a plant region to a market region that is different from the plant region, then duties are applied on each unit based on the sum of fixed cost per unit capacity, variable cost per unit, and transportation cost per unit. For example, a unit produced in Flextrola's N. America plant and shipped to Europe (EU) market has a fixed cost per unit of capacity of $5.00 ($100/20), a variable production cost of $5.80, and a transportation cost of $1.50. The 4% import duty is thus applied on $12.30 (5.00 + 5.80 + 1.50), i.e., import duty = 4%*$12.30 = $0.492. Thus, the total cost (variable production costs, transportation costs and duties) on importing a unit from N. America plant to Europe (EU) market is $5.80 + $1.50 +$0.492 = $7.792. Note that if products are shipped from a plant region to a market region that is same as the plant region, then no duties will be applied. For example, the total cost (variable production costs, transportation costs and duties) on importing a unit from N. America plant to N. America market is $5.80 + $1.00 = $6.80.
Questions:
a) Calculate the total cost per unit (variable production costs, transportation costs and duties) from plants to markets, i.e. complete the following table.
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N. America
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S. America
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Europe (EU)
|
Europe (Non EU)
|
Japan
|
Rest of Asia /Australia
|
Africa
|
|
Flextrola
|
Europe (EU)
|
|
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$7.792
|
|
|
|
|
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N. America
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6.80
|
|
|
|
|
|
|
|
S. America
|
|
|
|
|
|
|
|
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Solectrics
|
Europe (EU)
|
|
|
|
|
|
|
|
|
N. America
|
|
|
|
|
|
|
|
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Rest of Asia
|
|
|
|
|
|
|
|
b) Formulate an Integer Linear Programming model and solve it using ASPE to find out Flextrola's optimal production and distribution network that minimizes the total cost incurred by the company. Check Figure: ~$560-$570 (in millions of $). Describe Flextrola's optimal production and distribution network. Which plants serve which markets? What plants have excess capacities? The ILP model and write-ups inserted as word objects.
c) Formulate an Integer Linear Programming model and solve it using ASPE to find out Solectrics' optimal production and distribution network that minimizes the total cost incurred by the company. Check Figure: ~$510-$520 (in millions of $). Describe Solectrics' optimal production and distribution network. Which plants serve which markets? What plants have excess capacities? The ILP model and write-ups inserted as word objects.
d) Suppose that Flextrola and Solectrics have recently merged. Formulate an Integer Linear Programming model and solve it using ASPE to find out the merged company's optimal production and distribution network that minimizes the total cost incurred assuming no plants will be shut down. Check Figure: ~$1070-$1075 (in millions of $). Describe the merged company's optimal production and distribution network. Which plants serve which markets? What plants have excess capacities? The ILP model and write-ups inserted as word objects.
e) Suppose that the merged company wants to shut down some plants to reduce the cost. Shutting down a plant (either 10 million or 20 million units) saves 80 percent in fixed costs. Fixed costs are only partially recovered because of costs associated with a shutdown. For example, if Flextrola's plant in Europe (EU) is shut down, it will still incur $20 millions shutdown costs. Formulate an Integer Linear Programming model and solve it using ASPE to find out the merged company's optimal production and distribution network that minimizes the total cost incurred if plants can be shut down. Check Figure: ~$990-$995 (in millions of $). Describe the merged company's optimal production and distribution network. Which plants will be shut down? Which plants serve which markets? What plants have excess capacities? The ILP model and write-ups inserted as word objects.
Problem 2 - A computer manufacturer is developing a production schedule for the next five months (say April -- August). Demands for this manufacturer's laptop computer in the next five months are forecasted to be 1200, 2100, 1500, 1000, and 800, respectively. Assume that it costs this manufacturer $1200 to produce each laptop computer. At the end of each month, a holding cost of $300 per computer left in inventory is incurred. Increasing production from one month to the next incurs costs for hiring and training new employees. It is estimated that a cost of $2000 per computer is incurred if production is increased from one month to the next. Decreasing production from one month to the next incurs costs for laying off employees, loss of morale, and so forth. It is estimated that a cost of $1800 per computer is incurred if production is decreased from one month to the next. All demands must be met on time, and the units produced in one month can be used to meet demand for the current month as well as for future months. In the current month (March), 1500 laptop computers were produced. Assume that at the beginning of April, there are 100 computers in inventory.
Questions:
a) Formulate an Integer Linear Programming model for this problem and solve it via ASPE. Check Figure: ~$9.50million-9.55million. Please do not use nonlinear functions such as "If", "ABS", "Min" and "Max".
b) What is the optimal solution? Briefly explain it.
The ILP model and write-ups inserted as word objects.
Problem 3 - Appalachian State University operates its own power-generating plant. The electricity generated by this plant supplies power to the university and to local businesses and residences. The plant burns three types of coal, which produce steam that drives the turbines to generate the electricity. The Environmental Protection Agency (EPA) requires that for each ton of coal burned, the emissions from the coal furnace smoke stacks contain no more than 2500 parts per million (ppm) of sulfur and no more than 2.8 kilograms (kg) of coal dust. However, the managers of the plant are concerned about the environment and want to keep these emissions to a minimum. The following table summarizes the amounts of sulfur, coal dust, and steam that result from burning a ton of each type of coal.
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Coal
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Sulfur (in ppm)
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Coal Dust (in kg)
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Pounds of Steam Produced
|
|
1
|
1,100
|
1.7
|
24,000
|
|
2
|
3,500
|
3.2
|
36,000
|
|
3
|
1,300
|
2.4
|
28,000
|
The three types of coal can be mixed and burned in any combination. The resulting emission of sulfur or coal dust and the pounds of steam produced by any mixture are given as the weighted average of the values shown in the table for each type of coal. For example, if the coals are mixed to produce a blend that consisted of 35% of coal 1, 40% of coal 2, and 25% of coal 3, the sulfur emission (in ppm) resulting from burning one tone of this blend is: 0.35*1,100 + 0.4*3,500 + 0.25*1,300 = 2,110. The manager of this facility wants to select a blend of coal to burn while considering the following objectives:
Objective 1: Maximize the pounds of steam produced.
Objective 2: Minimize sulfur emissions.
Objective 3: Minimize coal dust emissions.
Questions:
a) Formulate an MOLP model for this problem, and implement the model in a spreadsheet "Prob3a". Determine the best possible value for each objective in the problem. Record all three solutions.
b) Determine the solution that minimizes the maximum percentage deviation from the optimal objective function values. Suppose management considers maximizing the amount of steam produced five times as important as achieving the best possible values for the other objectives. Write the LP formulation. Describe your solution including the actual percentage deviation for each objective.