# solution

The Institute for Operations Research and Management Science (INFORMS) has contracted with Sentinel Security to provide security service for its upcoming three-day, four-night national meeting in New York at a cost of \$B.Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â During the meeting hours from 8:00 A.M. to 6:00 P.M., Sentinel will pay its security guards \$D per hour. In the evening, security guards will be paid a lesser amount of \$E per hour. INFORMS requires that at least N guards be on duty during the evening. During the day there should be at least two guards as well as one additional guard for every 1 0 0 0 people attending the meeting. In addition to paying for its guards, Sentinel Security expects to have fixed expenses of \$F (for equipment, communications, etc.) during the meeting. Net profit to the firm is attained by subtracting its payments to its guards plus the fixed expenses from the contracted amount negotiated with INFORMS.

a. Formulate a model shell for this simplified problem. The objective of which is to maximize the net profit to Sentinel Security for supplying security service for T attendees to the national meeting in New York using the following decision variables:

X1 = number of guards needed each day

X2 = number of guards needed each evening

b. Explain why this problem of maximizing net profit is equivalent to minimizing the variable cost of payments to its security guards.

c. INFORMS expects 2000 attendees, and it wishes to have at least two guards on duty each evening. If the contracted price from INFORMS is \$10,000, fixed costs are expected to be \$1000, and Sentinel pays its guards \$15 per hour per day security and \$ 1 2 per hour for evening security, develop a complete mathematical model for security at the INFORMS meeting.

d. Suppose attendance is expected to be 2200. Now what would be the model? (Did you forget to include in your constraints that the number of guards must be an integer?)

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