Question 1

Giant Manufacturing is a Taiwanese bicycle manufacturer, which makes premium bicycles . An important decision facing the firm is whether to manufacture wheels itself or purchase wheels from an external supplier. If Giant manufactures wheels itself, it has to set up a factory for the production of wheels, which incurs a high cost and thus affects the firm’s profitability . Since the demand for bicycles is uncertain, Giant needs to examine the historic sales data and, based on this,  makes  a  reasonable  estimation  about  possible  demand  scenarios .  The  decision  also requires  Giant  to  work  out  the  estimate  of  profit  for  each  demand  scenario  and  decision alternative.  The  following  payoff  table  shows  the  projected  profits  (in  millions  of  dollars) together with the probability of each demand scenario:

a.   Use a decision tree to recommend a decision for Giant. [6 marks]

b.   Compute EVPI (Expected Value of Perfect Information) to determine whether Giant should attempt to obtain a better estimate of demand. [4 marks]

A market study of the potential demand for bicycles is expected to report either a favourable (F) or Unfavourable (U) condition. The relevant conditional probabilities are as follows:

c.   Suppose Giant will conduct the above market study. Use a decision tree to recommend a decision for Giant. [14 marks]

d.   Determine whether Giant should conduct the market study if it costs 3.5 million dollars.      [4 marks]

Question 2

Pear Corp. is planning to launch a new version of iPear, iPear Max, to compete with Apple’s new phone. The future demand for iPear Max is uncertain. Suppose that the demand distribution and the relevant financial information for iPear Max are given as follows .

Demand Probability

600                 0.35

800                 0.45

1000                 0.20

Fixed cost

Variable cost

Revenue

$8,000

$6/unit

$22/unit

Use the random numbers 0.51, 0.97, and 0.16 (as cumulative probabilities) to generate three         demand scenarios. Then simulate three iterations and calculate the net profit for each iteration. It

is assumed that the number of units sold equals the demand.                                               [12 marks]

Question 3

Consider the following linear program:

Max 3x1 + 4x2 ($ Profit)

s.t. x1 + 3x2 ≤ 12

2x1 + x2 ≤ 8

x1 ≤ 3

x1, x2 ≥ 0

Use the graphical solution approach to solve the above linear programming model, and answer the following questions (please use the graph paper at the end of this booklet to draw graphs):

a.   What are the optimal solutions (including the optimal values for decision variables and the optimal objective function value)?   [8 marks]

b.   Suppose the unit profit on x1 is increased from $3 to $7. Are the values for decision              variables in (a) still optimal? What is the optimal value of the objective function when this  unit profit is increased to $7?     [8 marks]

Question 4

The specialty retailer Sky-Umbrella wants to determine how many of four different styles of          umbrellas to stock in order to maximize its profit. It is assumed that every umbrella stocked will   be sold. The variables measure the numbers of women’s, golf, men’s, and folding umbrellas,         respectively. The constraints measure storage space in units, special display racks, demand, and a marketing restriction, respectively.

Let x1 = The number of women’s umbrellas

X2 = The number of golf umbrellas

X3 = The number of men’s umbrellas

X4 = The number of folding umbrellas

The linear programming model is formulated as follows:

Max 4 x1 + 6 x2 + 5 x3 + 3.5 x4

s. t.

2 x1 + 3 x2 + 3 x3 + x4 <= 120 (storage space)

1.5 x1 + 2 x2 <= 54 (special display racks)

2 x2 + x3 + x4 <= 72 (demand)

X2 + x3 >= 12  (marketing restriction)

x1, x2, x3, x4 >= 0 (non-negativity)

The partial Excel Solver output is given below:

Answer Report

Objective Cell (Max)

Cell

Name

Original

Value

Final Value Integer

number of umbrellas

12    Contin

$D$11    number of umbrellas golf                               0                            0    Contin

$E$11    number of umbrellas men                             0                          12    Contin

$F$11    number of umbrellas folding                         0                          60    Contin

Constraints

Cell Name Cell Value Formula Status

$C$17    storage space LHS                                        120    $C$17<=$E$17 Binding

$C$18    special display racks LHS                              18    $C$18<=$E$18 xxx

72    $C$19<=$E$19 Binding

$C$20    marketing restriction LHS                             12    $C$20>=$E$20    Binding

Sensitivity Report

Variable Cells

Use the above output to answer the following questions.

a.   Fill out the above tables. Note that the unfilled cells are marked by “xxx” .               [3 marks]

b.   How much storage space is left unused? How many display racks are left unused?[2 marks]

c.    By how much can the total profit on women’s umbrellas increase before the optimal solution would change?

d.   Interpret the shadow price for the storage space constraint.

[4 marks]

[2 marks]

e.   Sky-Umbrella considers stocking a new arrival umbrella, Travel Umbrella. It uses 3 units of  storage space and 2 special display racks. Sky-Umbrella makes a profit of $7 per unit of        Travel Umbrella. What is your recommendation on whether or not to stock this new arrival umbrella?                                                                                                                                [5 marks]

Question 5

The Tots Toys Company is trying to schedule production of two very popular toys for the next three months: a rocking horse and a scooter. Information about both toys is given below.

Toy

Beginning Inventory on June 1

Required Plastic

Required Time

Production Cost

Inventory

Cost

Rocking Horse Scooter