# Case Problem 3

**Topics:**Limit of a function, Optimization, Limit of a sequence

**Pages:**5 (936 words)

**Published:**April 9, 2013

1. Let S = fraction of the Security Systems project funded by HVC M = fraction of the Market Analysis project funded by HVC

Max 1,800,000S + 1,600,000M

s.t.

600,000S + 500,000M ≤ 800,000 Year 1 600,000S + 350,000M ≤ 700,000 Year 2 250,000S + 400,000M ≤ 500,000 Year 3 S ≤ 1 Maximum for S M ≤ 1 Maximum for M S,M ≥ 0

The solution obtained using The Management Scientist software package is shown below:

OPTIMAL SOLUTION

Objective Function Value = 2486956.522

Variable Value Reduced Costs

-------------- --------------- ------------------

S 0.609 0.000

M 0.870 0.000

Constraint Slack/Surplus Dual Prices

-------------- --------------- ------------------

1 0.000 2.783

2 30434.783 0.000

3 0.000 0.522

4 0.391 0.000

5 0.130 0.000

OBJECTIVE COEFFICIENT RANGES

Variable Lower Limit Current Value Upper Limit

------------ --------------- --------------- --------------- S No Lower Limit 1800000.000 No Upper Limit M No Lower Limit 1600000.000 No Upper Limit

RIGHT HAND SIDE RANGES

Constraint Lower Limit Current Value Upper Limit ------------ --------------- --------------- --------------- 1 No Lower Limit 800000.000 822950.820

2 669565.217 700000.000 No Upper Limit

3 461111.111 500000.000 No Upper Limit

4 0.609 1.000 No Upper Limit

5 0.870 1.000 No Upper Limit

Solutions to Case Problems

Thus, the optimal solution is S = 0.609 and M = 0.870. In other words, approximately 61% of the Security Systems project should be funded by HVC and 87% of the Market Analysis project should be funded by HVC.

The net present value of the investment is approximately $2,486,957.

2.

Year 1 Year 2 Year 3

Security Systems $365,400 $365,400 $152,250 Market Analysis $435,000 $304,500 $348,000 Total $800,400 $669,900 $500,250

Note: The totals for Year 1 and Year 3 are greater than the amounts available. The reason for this is that rounded values for the decision variables were used to compute the amount required in each year. To see why this situation occurs here, first note that each of the problem coefficients is an integer value. Thus, by default, when The Management Scientist prints the optimal solution, values of the decision variables are rounded and printed with three decimal places. To increase the number of decimal places shown in the output, one or more of the problem coefficients can be entered with additional digits to the right of the decimal point. For instance, if we enter the coefficient of 1 for S in constraint 4 as 1.000000 and resolve the problem, the new optimal values for S and D will be rounded and printed with six decimal places. If we use the new values in the computation of the amount required in each year, the differences observed for year 1 and year 3 will be much smaller than we obtained using the values of S = 0.609 and M = 0.870.

3. If up to $900,000 is available in year 1 we obtain a new optimal solution with S = 0.689 and M =

0.820. In other words, approximately 69% of the Security Systems project should be funded by...

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