Solved Problems

  1. Solved Problem 13-1 Higgins Plumbing and Heating maintains a stock of 30-gallon hot water heaters that it sells to homeowners and installs for them. Owner Jerry Higgins likes the idea of having a large supply on hand to meet customer demand, but he also recognizes that it is expensive to do so. He examines hot water heater sales over the past 50 weeks and notes the following:

    HOT WATER HEATER SALES PER WEEK NUMBER OF WEEKS THIS NUMBER WAS SOLD
    4 6
    5 5
    6 9
    7 12
    8 8
    9 7
    10 3
    Total 50

    1. If Higgins maintains a constant supply of 8 hot water heaters in any given week, how many times will he be out of stock during a 20-week simulation? We use random numbers from the seventh column of Table 13.4, beginning with the random number 10.

    2. What is the average number of sales per week (including stockouts) over the 20-week period?

    3. Using an analytic nonsimulation technique, what is the expected number of sales per week? How does this compare with the answer in part b?

    Solution

    The variable of interest is the number of sales per week.

    HEATER SALES PROBABILITY RANDOM NUMBER INTERVALS
    4 0.12 01 to 12
    5 0.10 13 to 22
    6 0.18 23 to 40
    7 0.24 41 to 64
    8 0.16 65 to 80
    9 0.14 81 to 94
    10 0.06 95 to 00
    1.00

    1. WEEK RANDOM NUMBER SIMULATED SALES WEEK RANDOM NUMBER SIMULATED SALES
      1 10 4 11 08 4
      2 24 6 12 48 7
      3 03 4 13 66 8
      4 32 6 14 97 10
      5 23 6 15 03 4
      6 59 7 16 96 10
      7 95 10 17 46 7
      8 34 6 18 74 8
      9 34 6 19 77 8
      10 51 7 20 44 7

      With a supply of 8 heaters, Higgins will be out of stock three times during the 20-week period (in weeks 7, 14, and 16).

    2. Average sales by simulation=Total sales20 weeks=13520=6.75 per week

    3. Using expected value,

      E(sales)=0.12(4 heaters)+0.10(5)+0.18(6)+0.24(7)+0.16(8)+0.14(9)+0.06(10)=6.88 heaters

      With a longer simulation, these two approaches will lead to even closer values.

  2. Solved Problem 13-2 The manager of a bank in Greensboro, North Carolina, is attempting to determine how many tellers are needed at the drive-in window during peak times. As a general policy, the manager wishes to offer service such that average customer waiting time does not exceed 2 minutes. Given the existing service level, as shown in the following data, does the drive-in window meet this criterion?

    DATA FOR SERVICE TIME
    SERVICE TIME (MINUTES) PROBABILITY (FREQUENCY) CUMULATIVE PROBABILITY RANDOM NUMBER INTERVAL
    0 0.00 0.00 (impossible)
    1.0 0.25 0.25 01 to 25
    2.0 0.20 0.45 26 to 45
    3.0 0.40 0.85 46 to 85
    4.0 0.15 1.00 86 to 00
    DATA FOR CUSTOMER ARRIVALS
    TIME BETWEEN SUCCESSIVE CUSTOMER ARRIVALS PROBABILITY (FREQUENCY) CUMULATIVE PROBABILITY RANDOM NUMBER INTERVAL
    0 0.10 0.10 01 to 10
    1.0 0.35 0.45 11 to 45
    2.0 0.25 0.70 46 to 70
    3.0 0.15 0.85 71 to 85
    4.0 0.10 0.95 86 to 95
    5.0 0.05 1.00 96 to 00

    Solution

    Average waiting time is a variable of concern.

    (1) CUSTOMER NUMBER (2) RANDOM NUMBER (3) INTERVAL TO ARRIVAL (4) TIME OF ARRIVAL (5) RANDOM NUMBER (6) SERVICE TIME (7) START SERVICE (8) END SERVICE (9) WAIT TIME (10) IDLE TIME
    1 50 2 9:02 52 3 9:02 9:05 0 2
    2 28 1 9:03 37 2 9:05 9:07 2 0
    3 68 2 9:05 82 3 9:07 9:10 2 0
    4 36 1 9:06 69 3 9:10 9:13 4 0
    5 90 4 9:10 98 4 9:13 9:17 3 0
    6 62 2 9:12 96 4 9:17 9:21 5 0
    7 27 1 9:13 33 2 9:21 9:23 8 0
    8 50 2 9:15 50 3 9:23 9:26 8 0
    9 18 1 9:16 88 4 9:26 9:30 10 0
    10 36 1 9:17 90 4 9:30 9:34 13 0
    11 61 2 9:19 50 3 9:34 9:37 15 0
    12 21 1 9:20 27 2 9:37 9:39 17 0
    13 46 2 9:22 45 2 9:39 9:41 17 0
    14 01 0 9:22 81 3 9:41 9:44 19 0
    15 14 1 9:23 66 3 9:44 9:47 21 0

    Read the data as in the following example for the first row:

    • Column 1: Number of customer.

    • Column 2: From third column of random number Table 13.4.

    • Column 3: Time interval corresponding to random number (random number of 50 implies a 2-minute interval).

    • Column 4: Starting at 9 a.m. the first arrival is at 9:02.

    • Column 5: From the first column of random number Table 13.4.

    • Column 6: Teller time corresponding to random number 52 is 3 minutes.

    • Column 7: Teller is available and can start at 9:02.

    • Column 8: Teller completes work at 9:05 (9:02+0:03).

    • Column 9: Wait time for customer is 0 as the teller was available.

    • Column 10: Idle time for the teller was 2 minutes (9:00 to 9:02).

    The drive-in window clearly does not meet the manager’s criteria for an average wait time of 2 minutes. As a matter of fact, we can observe an increasing queue buildup after only a few customer simulations. This observation can be confirmed by expected value calculations on both arrival and service rates.

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