38. Sales Forecast Modeling. Consulting Associates, Ltd., would like to generate a sales forecast based on the
assumption that next year sales are a function of current income, advertising, and advertising by a competing
retailer:
A.
Write an equation for predicting sales based on the assumption that the percentage change in sales is twice as large as the percentage
change in income and advertising; but only one-half as large as, and of the opposite sign of, the percentage change in competitor
advertising. Use the symbols S = sales, Y = income, A = advertising, and CA = competitor advertising.
B.
During the current year, sales total $500,000, income is $63,000 per capita, advertising is $50,000, and competitor advertising is
$100,000. Previous period levels were $60,000 (income), $40,000 (advertising), and $125,000 (competitor advertising). Forecast next
year sales.
39. Sales Forecast Modeling. Engineering Consultants, Inc., would like to generate a sales forecast based on
the assumption that next year sales are a function of current income, advertising, and advertising by a
competing manufacturer:
A.
Write an equation for predicting sales based on the assumption that the percentage change in sales is one-and-a-half as large as the
percentage change in income and advertising; but only one-half as large as, and of the opposite sign of, the percentage change in
competitor advertising. Use the symbols S = sales, Y = income, A = advertising, and CA = competitor advertising.
B.
During the current year, sales total $750,000, income is $71,750 per capita, advertising is $75,000, and competitor advertising is
$100,000. Previous period levels were $70,000 (income), $60,000 (advertising), and $80,000 (competitor advertising). Forecast next
year sales.
40. Sales Forecast Modeling. Kerry Weaver, office manager for Pediatric Medicine, Ltd., would like to
generate a sales forecast based on the assumption that next year sales are a function of current income,
advertising, and advertising by a competing local hospital:
A.
Write an equation for predicting sales based on the assumption that the percentage change in sales is three times as large as the
percentage change in income and advertising; but only one-fourth as large as, and of the opposite sign of, the percentage change in
competitor advertising. Use the symbols S = sales, Y = income, A = advertising, and CA = competitor advertising.
B.
During the current year, sales total $1,000,000, local disposable income is $78,750 per household, advertising is $90,000, and
competitor advertising is $100,000. Previous period levels were $75,000 (income), $80,000 (advertising), and $125,000 (competitor
advertising). Forecast next year sales.
41. Sales Forecast Modeling. Jeng-Mei Chen, Inc., an upscale Manhattan restaurant, would like to generate a
sales forecast based on the assumption that next year sales are a function of current income, advertising, and
advertising by a competing restaurant:
A.
Write an equation for predicting sales based on the assumption that the percentage change in sales is twice as large as the percentage
change in income and advertising; but only one-fourth as large as, and of the opposite sign of, the percentage change in competitor
advertising. Use the symbols S = sales, Y = income, A = advertising, and CA = competitor advertising.
B.
During the current year, sales total $2,000,000, income is $84,700 per capita, advertising is $500,000, and competitor advertising is
$300,000. Previous period levels were $77,000 (income), $400,000 (advertising), and $400,000 (competitor advertising). Forecast next
year sales.
42. Cost Forecast Modeling. Elliot Ness, manager of product packaging at Chicago Tool & Die, Inc., is
evaluating the cost effectiveness of the preventive maintenance program in his department. Ness believes that
the monthly downtime of the packaging line due to equipment breakdown is related to the hours spent each
month on preventive maintenance.
A.
Write an equation to predict next month’s downtime using the symbols D = downtime, M = preventive maintenance, t = time, a0 =
constant term, a1 = regression slope coefficient, and assuming that downtime in the forecast month decreases by the same percentage as
preventive maintenance increased during the preceding month.
B.
If 75 hours were spent last month on preventive maintenance and this month’s downtime was 50 hours, what should downtime be next
month if preventive maintenance this month is 90 hours? Use the equation developed in part A.
43. Cost Forecast Modeling. Robert Romano, CEO of Rocket Science, Inc., is evaluating the cost effectiveness
of the preventive maintenance program. Romano believes that the monthly downtime of the accelerator line due
to equipment breakdown is related to the hours spent each month on preventive maintenance.
A.
Write an equation to predict next month’s downtime using the symbols D = downtime, M = preventive maintenance, t = time, a0 =
constant term, a1 = regression slope coefficient, and assuming that downtime in the forecast month decreases by half of the percentage
increase in preventive maintenance during the preceding month.
B.
If 60 hours were spent last month on preventive maintenance and this month’s downtime was 40 hours, what should downtime be next
month if preventive maintenance this month is 75 hours? Use the equation developed in part A.
44. Cost Forecast Modeling. John Carter, an intern at Medical Products, Inc., is evaluating the cost
effectiveness of a training program in his department. Carter believes that the monthly rejection rate is inversely
related to the hours spent each month on worker training.
A.
Write an equation to predict next month’s rejection rate using the symbols R = rejection rate, T = worker training, t = time, a0 = constant
term, a1 = regression slope coefficient, and assuming that the rejection rate in the forecast month decreases by twice the percentage
increase in worker training during the preceding month.
B.
If 40 hours were spent last month on worker training and this month’s rejection rate was 60, what should the rejection rate be next
month if worker training this month is 50 hours? Use the equation developed in part A.
45. Sales Forecasting. Samurai, Ltd., must forecast sales for a popular trivia game in order to avoid stockouts
or excessive inventory charges during the coming Christmas season. In percentage terms, the company
estimates that game sales fall at double the rate of price increases and grow at five times the rate of customer
traffic increases. Furthermore, these effects seem to be independent.
A.
Write an equation for estimating the Christmas season sales, using the symbols S = sales, P = price, T = traffic, and t = time.
B.
Forecast this season’s sales if Samurai sold 10,000 games last season at $20 each, this season’s price is anticipated to be $25, and
customer traffic is expected to rise by 10% over previous levels.
A.
= St + DS
= St – DSP + DST
= St – 2(Pt+1/Pt – 1)St + 5(Tt+1/Tt – 1)St
A.
= a0 + a1T
= Rt – DR
= 60 – 30
= 30 rejection rate
46. Sales Forecasting. The World Bazaar, Ltd., must forecast sales for indoor electric grills in order to avoid
stockouts or excessive inventory charges during the coming Christmas season. In percentage terms, the
company estimates that electric grill sales fall at triple the rate of price increases and grow at three times the rate
of customer traffic increases. Furthermore, these effects seem to be independent.
A.
Write an equation for estimating the Christmas season sales, using the symbols S = sales, P = price, T = traffic, and t = time.
B.
Forecast this season’s sales if the World Bazaar sold 500,000 indoor electric grills last season at $40 each, this season’s price is
anticipated to be $45, and customer traffic is expected to rise by 15% over previous levels.
47. Simultaneous Equations. The Metropolitan Symphony in Toledo, Ohio, has had great success with a
“Senior Citizens’ Night” promotion. By offering half off its regular $20 admission price, average nightly
attendance has risen from 4,000 to 6,000 persons. Beverage (B) and other concession (C) revenues tied to
attendance have also risen dramatically. Historically, Metropolitan Symphony has found that 75% of all concert
goers will buy a $3 beverage, whereas 10% of all concert goers plus 40% of those buying beverages will spend
$2 on hors d’oeuvres and other concessions.
A.
Write an expression describing total revenue from tickets plus beverage plus other concessions.
B.
Forecast total revenues for both regular and special senior citizens’ night pricing.
C.
If the profit contribution is 25% on concert ticket revenues, and 75% on beverages and other concession revenues, is the pricing
promotion profitable?
A.
If Q is the number of concert goers, then:
Ticket Revenue
= PQ
Beverage Revenue
= $3B
= 3(0.75Q)
= $2.25Q
Other Concession Revenue
= $2C
= $2(0.10Q + 0.40B)
= $2Q
A.
= St + DS
= St – DSP + DST
= St – 3(Pt+1/Pt – 1)St + 3(Tt+1/Tt – 1)St
= St – 3(Pt+1/Pt)St + 3(Tt+1/Tt)St
= 500,000 – 3($45/$40)500,000 + 3(1.15)500,000
= 537,500
48. Simultaneous Equations. Buckeye Cinema, Inc., which runs a chain of movie theaters in the state of Ohio,
has had great success with a “Tuesday Night at the Movies” promotion. By offering half off its regular $8
admission price, average nightly attendance has risen from 400 to 800 persons. Popcorn (Pop) and other
concession (C) revenues tied to attendance have also risen dramatically. Historically, Buckeye Cinema has
found that 50% of all movie goers will buy a $4 cup of popcorn, whereas 25% of all movie goers plus 50% of
those buying popcorn will spend $3 on soda and other concessions.
A.
Write an expression describing total revenue from tickets plus popcorn plus other concessions.
B.
Forecast total revenues for both regular and special Tuesday-night pricing.
C.
If the profit contribution is 25% on movie ticket revenues, and 75% on popcorn and other concession revenues, is the pricing promotion
profitable?
Therefore,
Total Revenue
=
Ticket Revenue
+
Beverage Revenue
+
Other Concession Revenue
B.
Regular Price
= $97,000
Total Revenue
= $10(6,000) + $4.25(6,000)
C.
Yes. Note that:
Profit Contribution
= 0.25($20)(4,000) + (0.75)($4.25)(4,000)
= $32,750
Profit Contribution
= 0.25($10)(6,000) + 0.75($4.25)(6,000)
= $34,125
49. Simultaneous Equations. Macrosoft, Inc., based in Seattle, Washington, manufactures a wide range of
parts for the aircraft, automotive, and agricultural equipment industries. The company is currently evaluating
the merits of building a new plant in order to fulfill a new contract with the federal government. The alternative
to expansion is to use additional overtime, reduce other production, or a combination of both. The company will
add new capacity only if the economy appears to be expanding. Forecasting the general economic activity of the
United States is therefore an important input to the decision making process. The firm has collected data and
estimated the following relations for the U. S. economy:
Last year’s total profits (all corporations) Pt-1
= $1,200 billion
This year’s government expenditures G
= $3,000 billion
Annual consumption expenditures C
= $800 billion + 0.75Y + u
Annual investment expenditures I
= $1,520 billion + 0.9Pt-1
Annual tax receipts T
= 0.2 GDP
National income Y
= GDP – T
Gross domestic product GDP
= C + I + G
A.
Forecast each of the above variables through the simultaneous relations expressed in the multiple equation system. Assume that all
random disturbances average out to zero.
Special Price
Total Revenue
= $4(800) + $3.50(800)
= $6,000
Yes. Note that:
Regular Price
Profit Contribution
= 0.25($8)(400) + (0.75)($3.50)(400)
= $1,850
Special Price
Profit Contribution
= 0.25($4)(800) + 0.75($3.50)(800)
= $2,900
50. Simultaneous Equations. Gates Equipment, Inc., manufactures a wide range of parts for the agricultural
equipment industry. The company is currently evaluating the merits of building a new plant in order to fulfill a
new contract with a large export concern. The alternative to expansion is to use additional overtime, reduce
other production, or a combination of both. The company will add new capacity only if the regional economy
appears to be expanding. Forecasting the general economic activity of the United States is therefore an
important input to the decision making process. The firm has collected data and estimated the following
relations for the U. S. economy:
Last year’s total profits (all corporations) Pt-1
= $1,400 billion
This year’s government expenditures G
= $2,800 billion
Annual consumption expenditures C
= $800 billion + 0.8Y + u
Annual investment expenditures I
= $950 billion + 0.75Pt-1
Annual tax receipts T
= 0.25 GDP
National income Y
= GDP – T
Gross domestic product GDP
= C + I + G
A.
Forecast each of the above variables through the simultaneous relations expressed in the multiple equation system. Assume that all
random disturbances average out to zero.
Investment
I
= $950 + 0.75Pt-1
= $950 + 0.75(1,400)
= $2,000 billion
Gross Domestic Product
GDP
= C + I + G
= $800 + 0.8Y + $2,000 + $2,800
= $5,600 + 0.8(GDP – T)
= $5,600 + 0.8(GDP – 0.25GDP)
= $5,600 + 0.6GDP
0.4GDP
= $5,600
GDP
= $14,000 billion
