You must use a word processor and any computer program in preparing your answer.
Question 1. The OFLT Trucking Company needs to determine the number of trucks to satisfy the transportation demand between their two hub locations on a daily basis. The volume of the demand for the last 10 weeks is given in Table 1. (You can copy tables into Excel)
Table 1. Demand for trucks between Hub 1and Hub 2
|
Day
|
Week
|
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
|
Monday
|
67
|
68
|
76
|
75
|
75
|
82
|
77
|
88
|
87
|
84
|
|
Tuesday
|
54
|
57
|
59
|
57
|
58
|
69
|
65
|
57
|
72
|
56
|
|
Wednesday
|
47
|
49
|
49
|
52
|
57
|
59
|
52
|
57
|
68
|
59
|
|
Thursday
|
40
|
45
|
46
|
43
|
48
|
49
|
55
|
50
|
59
|
52
|
|
Friday
|
60
|
63
|
68
|
69
|
72
|
67
|
68
|
66
|
63
|
69
|
a) Plot the time series data and comment on the pattern you observe.
b) Use weighted moving average method to forecast the demand for 51 days.
c) Use exponential smoothing with optimal α to forecast the demand for 51 days.
d) Use multiple regression method (dummy variable approach) to forecast the demand for weeks 55 days.
e) Compute the following measures of forecast accuracy: mean absolute error, mean squared error, mean absolute percentage error for your forecasts in parts b, c and d.
f) Determine the most accurate forecast method and estimate the demand for week 11.
Question 2. Mitsumishi is a Japanese company whose number of vehicles (in thousands) sold between January 2012 and December 2015 is reported in Table 2. Company usually runs sales promotions in May and June. Their sales in millions dollars for the same time period are also reported in the Table 3.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Table 2.Vehicle Sales (in thousands)
|
|
|
Table 3.Vehicle Sales (in $ million)
|
|
|
Year
|
|
|
Year
|
|
|
2012
|
2013
|
2014
|
2015
|
|
|
2012
|
2013
|
2014
|
2015
|
|
January
|
934.7
|
1065.4
|
1034.0
|
1177.7
|
|
January
|
14.343
|
15.702
|
15.645
|
17.064
|
|
February
|
1172.7
|
1214.7
|
1214.6
|
1283.2
|
|
February
|
14.937
|
15.875
|
15.884
|
16.751
|
|
March
|
1431.9
|
1475.5
|
1563.2
|
1578.1
|
|
March
|
14.573
|
15.662
|
16.839
|
17.508
|
|
April
|
1209.7
|
1311.9
|
1416.5
|
1484.8
|
|
April
|
14.831
|
15.746
|
16.622
|
17.121
|
|
May
|
1361.9
|
1468.9
|
1634.6
|
1664.9
|
|
May
|
14.416
|
15.730
|
17.028
|
18.092
|
|
June
|
1311.6
|
1429.4
|
1448.8
|
1513.4
|
|
June
|
14.482
|
16.040
|
17.133
|
17.429
|
|
July
|
1178.5
|
1339.4
|
1464.6
|
1545.0
|
|
July
|
14.467
|
16.050
|
16.867
|
17.943
|
|
August
|
1310.5
|
1526.4
|
1613.4
|
1609.4
|
|
August
|
14.502
|
16.030
|
17.655
|
18.206
|
|
September
|
1210.6
|
1162.2
|
1275.5
|
1473.3
|
|
September
|
15.153
|
15.750
|
16.856
|
18.516
|
|
October
|
1116.5
|
1234.4
|
1312.1
|
1485.1
|
|
October
|
14.790
|
15.733
|
16.889
|
18.551
|
|
November
|
1165.9
|
1265.9
|
1324.4
|
1347.1
|
|
November
|
15.411
|
16.479
|
17.439
|
18.551
|
|
December
|
1384.0
|
1389.2
|
1540.3
|
1673.7
|
|
December
|
15.561
|
15.794
|
17.223
|
17.631
|
|
|
|
|
|
|
|
|
|
|
|
|
Perform an analysis of the sales data for the Mitsumishi by preparing report that summarizes your findings, forecasts, and recommendations based on your answers to the following questions:
a) Construct a time series plot for number of vehicles sold. What type of pattern exists in the data?
b) Construct a time series plot for vehicles sales. Comment on the underlying pattern in the time series.
c) How can you measure the effect of the promotions on sales performance? Please report annual sales improvements.
d) Using a dummy variable (multiple regression) approach, forecast the sales (as number of cars and amount of sales in dollars) for year 2016. (Write down forecast model explicitly).
e) Compute an estimate of the average vehicle price for each year.
Question 3.
The AMD Inc. made a contract with the government to supply 1200 microcomputers this year and 2500 next year. The company has the production capacity to make 1400 microcomputers each year, and it has already committed its production line for this level. Based on managerial decisions, the production line can be used for at most 80 overtime shifts each year, each shift costing the company an additional $20,000. In each overtime shift, 50 microcomputers can be manufactured. Units produced this year to meet next year's demand must be stored at a cost of $100 per unit. Formulate and solve a linear programming model to find the best production schedule that minimizes total costs. (Please write your notation, LP formulation explicitly, submit your managerial reports with computer models, solutions, answer and sensitivity reports attached.)
Question 4.
A hybrid electric plant uses three types of inputs to drive steam turbines in order to produce electricity. Federal standards require that emissions from the furnace contain no more than 2500 parts per million (ppm) of sulfur oxide and that no more than 40 kilograms per hour (kg/hr) of particulate matter (smoke) be emitted from the stack. The following table gives the amounts of both pollutants that result from burning the three types of energy resources.
|
|
Sulfur Oxide in
|
Particulates Emitted
|
|
Input
|
Stack Emissions
|
per Ton of Input Burned
|
|
Type
|
(ppm)
|
(kg/hr)
|
|
A. Natural Gas
|
1200
|
1
|
|
B. Coal
|
3300
|
2
|
|
C. Lignite
|
2100
|
5
|
Burning one ton of natural gas (input A) results in 22,000 lb of steam, whereas burning one ton of coal (B) or lignite (C), respectively, produces 26,500 or 33,500 lb of steam. The furnace has a capacity for burning 26 tons per hour of any mixture of the three inputs. Also, the sulfur oxide emissions that result from burning a mixture is equal to a weighted average of the parts-per-million emissions of the individual inputs, where each weight is equal to the proportion of that input used in the mixture. Formulate and solve a linear programming model for operating the electric plant so as to maximize the amount of steam generated per hour.
Question 5. The Textile Millproduces five different fabrics. Each fabric can be woven on one or more of the mill's 38 looms. The sales department's forecast of demand for the next month is shown below, along with data on the selling price per yard, variable cost per yard, and purchase price per yard. The mill operates 24 hours a day and is scheduled for 30 days during the coming month.
|
Fabric
|
Demand (yards)
|
Selling price ($/yard)
|
Variable Cost ($/yard)
|
Purchase Price ($/yard)
|
|
1
|
16,500
|
0.99
|
0.66
|
0.80
|
|
2
|
22,000
|
0.86
|
0.55
|
0.70
|
|
3
|
62,000
|
1.10
|
0.49
|
0.60
|
|
4
|
7,500
|
1.24
|
0.51
|
0.70
|
|
5
|
62,000
|
0.70
|
0.50
|
0.70
|
The mill has two types of looms: dobbie and regular. The dobbie looms are more versatile and can be used for all five fabrics. The regular looms can produce only three of the fabrics. The mill has a total of 38 looms: 8 are dobbie and 30 are regular. The rate of production for each fabric on each type of loom is given in the following table. The time required to change over from producing one fabric to another is negligible and does not have to be considered.
|
Loom Rate (yards/hour)
|
|
Fabric
|
Dobbie
|
Regular
|
|
1
|
4.63
|
-
|
|
2
|
4.63
|
-
|
|
3
|
5.23
|
5.23
|
|
4
|
5.23
|
5.23
|
|
5
|
4.17
|
4.17
|
The Textile Mill satisfies all demand with either its own fabric or fabric purchased from another mill. Fabrics that cannot be woven at the Scottsville Mill because of limited loom capacity will be purchased from another mill. We use following linear programming model to maximize the profit of the Textile Mill and to answer the management's questions:
Let X3R = Yards of fabric 3 on regular looms
X4R = Yards of fabric 4 on regular looms
X5R = Yards of fabric 5 on regular looms
X1D = Yards of fabric 1 on dobbie looms
X2D = Yards of fabric 2 on dobbie looms
X3D = Yards of fabric 3 on dobbie looms
X4D = Yards of fabric 4 on dobbie looms
X5D = Yards of fabric 5 on dobbie looms
Y1 = Yards of fabric 1 purchased
Y2 = Yards of fabric 2 purchased
Y3 = Yards of fabric 3 purchased
Y4 = Yards of fabric 4 purchased
Y5 = Yards of fabric 5 purchased
Max 0.61X3R + 0.73X4R + 0.20X5R + 0.33X1D + 0.31X2D + 0.61X3D + 0.73X4D + 0.20X5D + 0.19Y1 + 0.16Y2 + 0.50Y3 + 0.54Y4
Subject to:
0.1912X3R + 0.1912X4R + 0.2398X5R ≤21600 (Regular Hours Available)
0.21598X1D + 0.21598X2D + 0.1912X3D + 0.1912X4D + 0.2398X5D ≤ 5760 (Dobbie Hrs Available)
X1D + Y1 = 16500
X2D + Y2 = 22000(Demand Constraints)
X3R + X3D + Y3 = 62000
X4R + X4D + Y4 = 7500
X5R + X5D + Y5 = 62000
ALL variables >=0
OPTIMAL SOLUTION OBTAINED WITH LINGO:
|
Optimal Objective Value
|
|
|
62531.49090
|
|
|
|
|
|
|
|
Variable
|
Value
|
Reduced Cost
|
|
X3R
|
27707.80815
|
0.00000
|
|
X4R
|
7500.00000
|
0.00000
|
|
X5R
|
62000.00000
|
0.00000
|
|
X1D
|
4668.80000
|
0.00000
|
|
X2D
|
22000.00000
|
0.00000
|
|
X3D
|
0.00000
|
-0.01394
|
|
X4D
|
0.00000
|
-0.01394
|
|
X5D
|
0.00000
|
-0.01748
|
|
Y1
|
11831.20000
|
0.00000
|
|
Y2
|
0.00000
|
-0.01000
|
|
Y3
|
34292.19185
|
0.00000
|
|
Y4
|
0.00000
|
-0.08000
|
|
Y5
|
0.00000
|
-0.06204
|
|
|
|
|
|
Constraint
|
Slack/Surplus
|
Dual Value
|
|
1
|
0.00000
|
0.57530
|
|
2
|
0.00000
|
0.64820
|
|
3
|
0.00000
|
0.19000
|
|
4
|
0.00000
|
0.17000
|
|
5
|
0.00000
|
0.50000
|
|
6
|
0.00000
|
0.62000
|
|
7
|
0.00000
|
0.06204
|
|
|
|
|
|
|
|
|
|
Objective
|
Allowable
|
Allowable
|
|
Coefficient
|
Increase
|
Decrease
|
|
0.61000
|
0.01394
|
0.11000
|
|
0.73000
|
Infinite
|
0.01394
|
|
0.20000
|
Infinite
|
0.01748
|
|
0.33000
|
0.01000
|
0.01575
|
|
0.31000
|
Infinite
|
0.01000
|
|
0.61000
|
0.01394
|
Infinite
|
|
0.73000
|
0.01394
|
Infinite
|
|
0.20000
|
0.01748
|
Infinite
|
|
0.19000
|
0.01575
|
0.01000
|
|
0.16000
|
0.01000
|
Infinite
|
|
0.50000
|
0.11000
|
0.01394
|
|
0.54000
|
0.08000
|
Infinite
|
|
0.00000
|
0.06204
|
Infinite
|
|
|
|
|
|
RHS
|
Allowable
|
Allowable
|
|
Value
|
Increase
|
Decrease
|
|
21600.00000
|
6556.82444
|
5297.86007
|
|
5760.00000
|
2555.33477
|
1008.38013
|
|
16500.00000
|
Infinite
|
11831.20000
|
|
22000.00000
|
4668.80000
|
11831.20000
|
|
62000.00000
|
Infinite
|
34292.19185
|
|
7500.00000
|
27707.80815
|
7500.00000
|
|
62000.00000
|
22092.07648
|
27341.95794
|
a) What is the optimal production schedule and loom assignments for each fabric?
b) How many yards of each fabric must be purchased from another mill?
c) What is the maximum profit attainable with the suggested production schedule?
d) If the purchase price of fabric 3 is decreased by $0.10, would the optimal solution change?
e) If the millincreased the selling price of fabric 2 on dobbie looms to $1.00, would the production schedule change? How much profit change would you expect?
f) How much is it worth for the company to have an extra regular hour available?
g) How much is it worth for the company to have an extra dobbie hour available?
h) What is the maximum value of the 9th Dobbie Loom; i.e., how much they should be willing to pay for the additional dobbie loom?
i) Management would like to understand the effects of different demand levels for different fabrics on the optimal solution and the total profit. Discuss the range of feasibility and the value of extra demand for each fabric.
j) If the company has to choose only one fabric to promote by additional advertisement, which fabric they should choose and why?
k) If they increase the selling price for fabric 1 and 4 by $0.10 simultaneously, would the optimal solution change? What would be the optimal total cost?
l) After implementing lean strategies, they plan to increase available regular hours to 25000 and available dobbie hours to 4000. Will there be any savings or total cost increase?