The sequential construction of a structure involves two phases. Initially, the f

Question

The sequential construction of a structure involves two phases. Initially, the foundation is built, then work commences on the superstructure. Assume that both the foundation and superstructure can be completed at three different rates, say a, b, or c. These rates modify the probability of completion of each phase of construction as shown in the following table. Monthly costs vary for the different rates. In addition, if the construction is not completed in 11 months, the contractor must pay a penalty of $300,000 per month. 2. probability of time of completion phase rate cost/month at 4 months 5 months 6 months 7 months rate, S 30,000 36,000 42,000 25,000 40,000 50,000 foundation foundation foundation 0.3 0.5 0.3 0.7 0.5 0.7 0.1 0.3 0.5 superstructure a superstructure b superstructure C 0.4 0.3 0.3 0.5 0.4 0.2 (a) Compute the expected cost of the foundation performed at rate a where E[S foundation at rate al summation of the total cost for each time multiplied by each probability. Note that the total cost for four months at rate a would be $30,000/mo times 4 months. (b) Compute all expected costs (ie foundation & structure at rate a, b, c) (c) Compute the total expected penalty for each possible strategy of completion of the whole structure. HINT: compute the Expected time to complete the foundation and structure using each rate, and if the expected time to complete F+S> 11 months, the penalty is $300,000. E[time to complete foundation, rate a] = 4 months"(0.3) + 5 months"(0.7)-4.7 months (d) Determine the best strategy by minimizing the sum of total expected cost and penalty

Explanation / Answer

a)The expected cost of foundation at rate a is the sum of total cost for each month multiplied by with its probability. Thus Expected cost=30000*(4*0.3+5*0.7+6*0+7*0)=30000*(1.2+3.5)=30000*4.7=141000

b) For foundation

At rate a..Expected cost =141000(shown in part a))

At rate b Expected Cost=36000*(4*0.5+5*0.5+6*0+7*0)=36000*(2+2.5)=36000*4.5=162000

At rate c, Expected Cost=42000*(4*0.3+5*0.7+6*0+7*0)=42000*(1.2+3.5)=42000*4.7=197400

For superstructure

At rate a..Expected cost =25000*(0*4+0.1*5+0.4*6+0.5*7)=25000*(0.5+2.4+3.5)=25000*6.4=160000

At rate b Expected Cost=40000*(0*4+0.3*5+0.3*6+0.4*7)=40000*(1.5+1.8+2.8)=40000*6.1=244000

At rate c, Expected Cost=50000*(0*4+0.5*5+0.3*6+0.2*7)=50000*(2.5+1.8+1.4)=50000*5.7=285000

c) The expected time for each strategy is summation of time multiplied by the respective probability.

For foundation

At rate a Expected Time= 4*0.3+5*0.7+6*0+7*0=1.2+3.5=4.7,

At rate b, Expected Time=4*0.5+5*0.5+6*0+7*0=4.5

At rate c, Expected Time=4*0.3+5*0.7+6*0+7*0=4.7

For superstructure,

At rate a Expected Time= 0*4+0.1*5+0.4*6+0.5*7=6.4

At rate b, Expected Time=0*4+0.3*5+0.3*6+0.4*7=6.1

At rate c, Expected Time=0*4+0.5*5+0.3*6+0.2*7=5.7

Thus total expected time for different rates are:

Rate a=4.7+6.4=11.1

Rate b=4.5+6.1=10.6

Rate c=4.7+5.7=10.4

Thus as the Time is exceeding 11 months only in rate a, there the extra penalty of 300,000 per month is levied. Thus the total costs including the penalty for each rate is:

Penaty:

Rate a...(11.1-11)*300000=30000

Rate b....0(No penalty as total time is less than 11 months)

Rate c...0(No penalty as total time is less than 11 months)

Rate a....141000+160000+(11.1-11)*300000=331000

Rate b....162000+244000=406000

Rate c....197400+285000=482400

d) As computed above, sum of total cost and penalty for strategy with rate a is the least among all rates, thus that is the strategy which should be followed.

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