Question 1 Mitsumishi is a Japanese company whose number of vehicles (in thousands) sold
between January 2013 and December 2016 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. January
February
March
April
May
June
July
August
September
October
November
December Table 2. Number of Vehicles Sold (in
thousands)
Year
2013
2014
2015
2016
894.7 1165.4
1004.0
1177.7
1172.7 1214.7
1214.6
1283.2
1431.9 1475.5
1563.2
1578.1
1209.7 1311.9
1416.5
1484.8
1361.9 1468.9
1634.6
1664.9
1311.6 1429.4
1448.8
1513.4
1178.5 1339.4
1464.6
1545.0
1310.5 1526.4
1613.4
1609.4
1210.6 1162.2
1275.5
1473.3
1116.5 1234.4
1312.1
1485.1
1165.9 1265.9
1324.4
1347.1
1299.0 1389.2
1510.3
1653.7 January
February
March
April
May
June
July
August
September
October
November
December Table 3. Vehicle Sales (in $ million)
Year
2013
2014
2015
2016
14.343 15.702 15.645 17.064
14.937 15.875 15.884 16.751
14.573 15.662 16.839 17.508
14.831 15.746 16.622 17.121
14.416 15.730 17.028 18.092
14.482 16.040 17.133 17.429
14.467 16.050 16.867 17.943
14.502 16.030 17.655 18.206
15.153 15.750 16.856 18.516
14.790 15.733 16.889 18.551
15.411 16.479 17.439 18.551
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 2017. (Write down forecast model explicitly).
e) Compute an estimate of the average vehicle price for each year.
Question 2 Microbrewers Inc. makes three different types of beers: Ale, Dark and White. All the
products require water, malt, hops, and yeast. Since the supply of water is considered free, it is the amount of malt, hops and yeast resource that restricts capacity. They have only 180 gallons of malt, 270 gallons of hops, and 400 gallons of yeast available. Considering the aggregated weekly demand forecasts, they need to produce at least 60 gallons of beer per week. Ale is sold for $40 a gallon, dark is sold for $52 and white is sold for $58 a gallon. Brewing or production requirements are given in the following table:
Malt
Hops
Yeas
t Ale
1
2
5 Dark
2
4
2 White
3
5
1 a) Based on the production requirements, resource availability and revenue received for 1 gallon of
each product, develop a linear program to maximize the total revenue of the brewery. Provide
sensitivity reports. (Please write your notation, LP formulation explicitly, provide your computer models, solutions, answer and sensitivity reports.) b) What is the optimal production plan and the optimal revenue?
c) Are all the resources used up? How much of which resource is left unused?
d) By how much can the revenue (or price) of dark beer increase before the solution would change?
e) To what value can the price of ale beer decrease before the solution would change?
f) Which resource has the highest value for the brewery? Why?
g) If brewery introduces happy hour with prices $39 for Ale, $52 for White, would the current
solution remain optimal? Question 3 A hybrid electric plant uses three types of inputs to drive steam turbines in order to
produce electricity. Federal standards require that emissions from the furnace contain no more
than 2800 parts per million (ppm) of sulfur oxide and that no more than 40 kilograms per hour
(kg/hr) of particulate matter (smoke) be emitted from the stack. The following table gives the
amounts of both pollutants that result from burning the three types of energy resources.
Sulfur Oxide in
Input Particulates Emitted Stack Emissions per Ton of Input Burned Type (ppm) (kg/hr) A. Natural Gas 1200 1 B. Coal 3300 2 C. Lignite 2500 4 Burning one ton of natural gas (input A) results in 22,000 lb of steam, whereas burning one ton
of coal (B) or lignite (C), respectively, produces 25,600 or 33,000 lb of steam. The furnace has a
capacity for burning 26 tons per hour of any mixture of the three inputs. Also, the sulfur oxide
emissions that result from burning a mixture is equal to a weighted average of the parts­per­
million emissions of the individual inputs, where each weight is equal to the proportion of that input used in the mixture. Formulate and solve a linear programming model for operating the
electric plant so as to maximize the amount of steam generated per hour. Question 4 The Textile Mill produces five different fabrics. Each fabric can be woven on one or
more of the mill’s 38 looms. The sales department’s forecast of demand for the next month is
shown below, along with data on the selling price per yard, variable cost per yard, and purchase
price per yard. The mill operates 24 hours a day and is scheduled for 30 days during the coming
month.
Fabric Demand
(yards) Selling price
($/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
X3R
X4R
X5R
X1D
X2D
X3D
X4D
X5D
Y1
Y2
Y3
Y4
Y5
Constraint
1
2
3
4
5
6
7
Objective
Coefficient Value
Reduced Cost
27707.80815
0.00000
7500.00000
0.00000
62000.00000
0.00000
4668.80000
0.00000
22000.00000
0.00000
0.00000
-0.01394
0.00000
-0.01394
0.00000
-0.01748
11831.20000
0.00000
0.00000
-0.01000
34292.19185
0.00000
0.00000
-0.08000
0.00000
-0.06204
Slack/Surplus
Dual Value
0.00000
0.57530
0.00000
0.64820
0.00000
0.19000
0.00000
0.17000
0.00000
0.50000
0.00000
0.62000
0.00000
0.06204
Allowable
Increase Allowable
Decrease a)
b)
c) 0.61000
0.73000
0.20000
0.33000
0.31000
0.61000
0.73000
0.20000
0.19000
0.16000
0.50000
0.54000
0.00000 0.01394
Infinite
Infinite
0.01000
Infinite
0.01394
0.01394
0.01748
0.01575
0.01000
0.11000
0.08000
0.06204 0.11000
0.01394
0.01748
0.01575
0.01000
Infinite
Infinite
Infinite
0.01000
Infinite
0.01394
Infinite
Infinite RHS
Value
21600.00000
5760.00000
16500.00000
22000.00000
62000.00000
7500.00000
62000.00000 Allowable
Increase
6556.82444
2555.33477
Infinite
4668.80000
Infinite
27707.80815
22092.07648 Allowable
Decrease
5297.86007
1008.38013
11831.20000
11831.20000
34292.19185
7500.00000
27341.95794 What is the optimal production schedule and loom assignments for each fabric?
How many yards of each fabric must be purchased from another mill?
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 mill increased 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)
g)
h) How much is it worth for the company to have an extra regular hour available?
How much is it worth for the company to have an extra dobbie hour available?
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?

 

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