Problems on Business Math including Forecasting, Exponential smoothing, decision analysis, Normal distribution are included in this posting.

1. A food distributor carries 64 varieties of salad dressing.  Appleton Markets stocks 48 of these flavors.  Beacon Stores carries 32 of them.  The probability that a flavor will be carried by Appleton or Beacon is 15/16.  Find the probability a flavor is carried by both Appleton and Beacon.

2. The time it takes to travel from home to the office is normally distributed with  = 25 minutes and  = 5 minutes.

a. What is the probability the trip takes more than 20 minutes?
b. What is the probability the trip takes less than 15 minutes?
c. What is the probability the trip takes between 30 and 35 minutes?
d. What is the probability the trip takes more than 40 minutes?

3. The table shows both prospective profits and losses for a company, depending on what decision is made and what state of nature occurs.  Use the information to determine what the company should do.

s1 s2 s3
d1 30 80 -30
d2 100 30 -40
d3 -80 -10 120
d4 20 20 20

a. if an optimistic strategy is used.
b. if a conservative strategy is used.
c. if minimax regret is the strategy.



4. Dollar Department Stores has the opportunity of acquiring either 3, 5, or 10 leases from the bankrupt Granite Variety Store chain.  Dollar estimates the profit potential of the leases depends on the state of the economy over the next five years.  There are four possible states of the economy as modeled by Dollar Department Stores, and its president estimates P(s1) = .4, P(s2) = .3, P(s3) = .1, and P(s4) = .2.  The utility has also been estimated.  Given the payoffs (in $1,000,000's) and utility values below, which decision should Dollar make using expected utility as its decision criterion?

Payoff Table

                               State Of The Economy
                                           Over The Next 5 Years
                Decision               s1     s2     s3     s4

       d1 -- buy 10 leases       10       5      0    -20
       d2 -- buy 5 leases             5       0     -1    -10
       d3 -- buy 3 leases             2       1      0     - 1
       d4 -- do not buy               0       0      0       0

Utility Table

Payoff (in $1,000,000's)     +10      +5      +2      +1      0      -1    -10    -20
Utility           +10      +5      +2      +1      0      -1    -20    -50


5. The number of girls who attend a summer basketball camp has been recorded for the seven years the camp has been offered.  Use exponential smoothing with a smoothing constant of .8 to forecast attendance for the eighth year.

47, 68, 65, 92, 98, 121, 146


6. Monthly sales at a coffee shop have been analyzed.  The seasonal index values are

Month Index
Jan 1.38
Feb 1.42
Mar 1.35
Apr 1.03
May .99
June .62
July .51
Aug .58
Sept .82
Oct .82
Nov .92
Dec 1.56

and the trend line is 74123 + 26.9(t).  

Assuming there is no cyclical component – but that there is a sesaonal component -- forecast sales for year 8 (months 97 - 108).

7. Find the complete optimal solution to this linear-programming problem:

Min   3X + 3Y
s.t. 12X + 4Y  >  48
10X + 5Y  >  50
  4X + 8Y  >  32
           X , Y  >  0


8. FarmFresh Foods manufactures a snack mix called TrailTime by blending three ingredients:  a dried fruit mixture, a nut mixture, and a cereal mixture.  Information about the three ingredients (per ounce) is shown below.

Ingredient Cost Volume Fat Grams Calories
Dried Fruit .35 .25 cup 0 150
Nut Mix .50 .375 cup 10 400
Cereal Mix .20    1 cup 1 50

The company needs to develop a linear programming model whose solution would tell them how many ounces of each mix to put into the TrailTime blend.  TrailTime is packaged in boxes that will hold between three and four cups.  The blend should contain no more than 1000 calories and no more than 25 grams of fat.  Dried fruit must be at least 20% of the volume of the mixture*, and nuts must be no more than 15% of the weight of the mixture**.  

Develop a model that meets these restrictions and minimizes the cost of the blend, using MS, Excel Solver, or an online LP solver. What is the optimal solution?

(*hint: develop an expression for the total volume of the mixture, and an expression for the volume of the dried fruit. Then create an inequality that states that the dried fruit volume must be at least 20% of the total volume; then re-arrange this inequality so that all expressions containing variables are on the left-hand side. Create constraints for calories, fat, and weight in similar fashion)

(** this is simpler, since each mixture has the same weight, one ounce)



9. true / false (two points each):


a. Critical activities are those that can be delayed without delaying the entire project.

b. PERT and CPM are applicable only when there is no dependence among activities.

c. A path through a project network must reach every node.

d A critical activity can be part of a noncritical path.

e. The earliest finish time for the final activity is the project duration.

10. Two airlines offer conveniently scheduled flights to the airport nearest your corporate headquarters.  Historically, flights have been scheduled as reflected in this transition matrix.

Current
Flight Next Flight
Airline A Airline B
Airline A .6 .4
Airline B .2 .8

If your last flight was on B, what is the probability your second next flight (that is, the flight after the next flight) will be on A?

Hints: (1) what are the possible 'paths' leading from "last flight was on B" to "flight after next is on A?" how many paths are there? what transitions occur along the way in each path? What is the probability for each path? (2) answering this question does not require developing Markov equations

See attached file for full problem description.

Attached file(s):

3mathfordecisionmaking1.doc  View File