Let's Go for a Hike: 7 Exercises

The diagram shows 4 hikers heading up the hill. The distance traveled is measured in yards. The gaps between the hikers are also measured in yards. Hiker #1 brings up the rear. She is the strongest hiker, and her"natural pace" is 30 yards/minute which amounts to 1,800 yards/hour. There are 1,760 yards in a mile, so her pace is 1.02 miles/hour. It's a 5 mile hike, so her natural pace would allow the hike to be completed in less than 5 hours. The 1st hiker will travel at her natural pace if she is happy with the gap between herself and the 2nd hiker. If the gap becomes too large, she will increase her pace in an attempt to close the gap. If the gap becomes too small, she will slow her pace to allow the gap to grow. The 2nd and 3rd hikers follow a similar approach. They speed up when the gaps get too big; they slow down when the gaps become too small. The 4th hiker leads the way. There is no gap to worry about, so she hikes at her natural pace. The group has decided that the best approach is to place the slowest hiker in the front, the fastest hiker in the rear. They reason that the group will remain in closer contact during the long hike, so they can call out to each other if there is a problem.The hike begins at noon, and the entire team must complete the 5 mile hike before nightfall (6 pm). The following exercises challenge you to build and test a model of the hikers. The exercises conclude with interesting reading from The Goal, Goldratt's (1986) "novel" on manufacturing.

1. Model #1: Build and Simulate. Do the Hikers Complete the Hike in Six Hours?

The flow diagram shows a simple model to simulate the distance traveled by each hiker. The stocks accumulate the distance traveled in yards; the flows give the pace of travel in yards per minute. The natural paces are listed below:    1st hiker - rear  30 yards/min 2nd hiker 27 yards/min 3rd hiker 26 yards/min 4th hiker- front 22 yards/min The 4th hiker leads the way traveling at her natural pace. (The "Pace 4" flow is identical to the "Natural Pace 4.") The 3rd hiker adjusts her pace based on "Gap 3," the gap in front of her. The "Pace Adjustment 3" is a nonlinear graph (~) whose value is 1 when the 3rd hiker can travel at her natural pace. Values below 1 mean she is slowing her pace because the gap is too small. Values above 1 mean she is increasing her pace because the gap is too large. The 1st and 2nd hikers follow a similar approach and make the same adjustments:  gap = 20 yards Pace Adjustment = 1, i.e. 20 yards is considered an ideal gap.  gap between 0 and 20 yards Pace Adjustment falls below 1. Draw the graph to represent how you would slow your own pace if the gap gets too small.  gap between 20 and 40 yards Pace Adjusted upward. Over the long run, the hikers can not exceed their natural pace by more than 5%. Set the adjustment to 1.05 if the gap reaches 40 yards.

Build the model with time in minutes. Set DT to 0.5 minutes and let the simulation run for 360 minutes (to simulate the 6 hours from noon to 6 pm). You may initialize the four stocks with the hikers only 5 yards apart (distance1=0, distance2=5, distance3=10 and distance4=15). Document your simulation with a time graph of the distances traveled by all 4 hikers. Include a second time graph showing the three gaps between the hikers. A gap beyond 200 yards is considered unsafe (because the hikers could not communicate with each other.) Does the hike proceed in a safe manner? Do all four hikers complete the 5 mile hike within 6 hours?

2. Model #2: Do You Get the Same Results?

This flow diagram shows a more compact form of the same model. This approach was suggested by Professor K. K. Fung (1999). He uses stocks to simulate the gaps between the hikers. For example, Gap 1, the gap in front of the 1st hiker, decreases due to the pace of the 1st hiker and increases due to the pace of the 2nd hiker. The gap will remain constant if the two hikers travel at the same pace. Assume the natural paces are the same as the previous exercise. Also, assume the pace adjustments are the same as the previous exercise.

Build this new model with time measured in minutes. Set DT to 0.5 minutes and simulate the model for 360 minutes. You may initialize each of the gaps at 5 yards. Simulate the model and document your results with a time graph of the three gaps. Do you get the same results as in the previous exercise? Does the hike proceed in a safe manner? Do all four hikers complete the 5 mile hike within 6 hours?

3. Let the Fast Hikers Go to the Front
Let's use the 1st model to simulate the hike with the fastest hiker in front and the slower hikers behind. (Every hike from my youth turned out this way -- the fastest hikers made their way to the front. My friends and I brought up the rear.) This approach can be simulated by changing the natural paces to:

	1st hiker, at the rear	22 yards/min
	2nd hiker	26 yards/min
	3rd hiker	27 yards/min
	4th hiker, in front	30 yards/min

Document your results with a time graph of the 3 gaps. Does the hike proceed in a safe manner (ie, do the gaps remain less than 200 yards?).

4. Mr. Rogo's Solution:

Eliyahu Goldratt describes this hike in The Goal. The hikers are Boy Scouts, and Mr. Rogo is the scout master. (Mr. Rogo is also the head of a troubled manufacturing facility. He is the lead character in Goldratt's novel about manufacturing.)

Mr. Rogo puts the slowest hiker at the front of the pack; the faster hikers are at the rear. The hike proceeds in a safe manner, but the fast hikers are constantly grumbling about the restrictions on their pace. They direct a lot of abusive comments at Herbie, the slow hiker who is leading the way. Mr. Rogo is constantly forced to restrain the fast hikers from bolting to the front. He is also constantly admonishing the fast hikers to limit their abusive remarks about Herbie (while wishing to himself that Herbie would walk a little faster).

The hike does proceed in a safe manner, but Mr. Rogo becomes concerned that they will not reach their destination in a timely manner. He sees that Herbie's slow pace is the key constraint in the system, so he decides to take a closer look. He notices that Herbie's backpack is larger than the other scouts, so he calls the troop to a halt. They examine Herbie's backpack and are surprised by what they see. It is filled with canned soda pop, jars of pickles, candy bars, frying pans, etc. (Herbie is following the scout's motto to "Be Prepared.")

Mr. Rogo then seizes on a new solution. Herbie's supplies are split up among the remaining scouts. With their heavier backpacks, the fast hikers find that their natural pace is reduced. Herbie's backpack is much lighter, and he can now travel at 25 yards/min (instead of 22 yards/min). He is still the slowest hiker, so Mr. Rogo puts him at the head of the pack.

Use the 1st model to simulate Mr. Rogo's solution based on the natural paces shown below:

	1st hiker at the rear, still the fastest, but not as fast as before	28 yards/min
	2nd hiker	26 yards/min
	3rd hiker	25.5 yards/min
	4th - Herbie still in front, but with a lighter backpack.	25 yards/min

Assume that these paces apply from the start of the simulation. Document your results with a time graph of the distances traveled and the gaps between the hikers. Does the hike proceed in a safe manner? Do all four hikers complete the 5 mile hike within 6 hours?

5. What About the Slope?

The previous sketch shows the hikers heading up a hill with constant slope. Each hiker has a natural pace that does not change during the hike. This assumption may be reasonable if the hikers face the same slope for the entire 5 miles. But suppose the hikers are heading up the slope shown in this chart. The hike begins at an elevation of 6,000 feet. Five miles later, they make camp at 6,554 feet. They climb 554 feet after 5 miles. With 5,280 feet/mile, the average slope is 2.1%.

the elevations every half mile: 6,000 feet 6,053 feet 6,317 feet 6,317 feet 6,581 feet 6,581 feet 6,449 feet 6,449 feet 6,502 feet 6,554 feet 6,554 feet

Let's assume that each hiker's natural pace listed in exercise #4 applies when the slope is 2%. But their paces will be lower when they are hiking on a steeper slope; they will be 20% higher when hiking on a flat slope; and they will be even higher when hiking downhill. Use the chart shown here to represent the change in each hiker's natural pace with changes in the slope. The horizontal axis shows the slope (in %) ranging from a downhill slope of 10% to an uphill slope of 10%. The multiplicative effect on natural pace is shown on the vertical axis. The neutral point is 2% where the "multiplier" is 1.0. If the slope falls to zero, the multiplier rises to 1.2 which means that each hiker's natural pace will be 20% faster. Each hiker's natural pace can be up to 80% faster (when the slope reaches -4% or lower). On the other hand, each hiker's natural pace can fall to 10% of normal values if the slope reaches 10%.

Expand your model from the 4th exercise to represent the change in each hiker's natural pace due to changes in the slope. The slope at the start of the hike is around 2%, so the natural paces should be the same as in the previous exercise at the beginning of the simulation. Simulate the new model and document your results with two graphs. The first is a time graph of the gaps between the hikers. (Can the hikers maintain proper gaps in the new situation?) The second graph is an X-Y graph with slope encountered by the first hiker on the X axis and the multiplicative effect on the first hiker's natural pace on the Y axis. (Does the X-Y graph match the expected pattern?) If your new model seems to working properly, use it to learn if Mr. Rogo's solution will bring all the hikers to camp within 6 hours.

6. Is DT = 0.5 min Short Enough?

Repeat the simulation from the 5th exercise with DT cut in half. Do you get essentially the same result?
Now repeat the simulation from the 1st exercise with DT cut in half. Do you get essentially the same result?

Based on first principles (see Chapter 11 in the book), which model do you think will require the smaller DT

	the 1st model for hikers on a flat slope, or
	the 5th model for hikers on a variable slope?

7. The Goal

If you are interested in manufacturing, read Goldratt's (1986) The Goal. You will learn his analogy between a scout hike and a manufacturing line.

	The scout master is concerned with yards/minute of hiking pace; The operations manager is concerned with parts/minute of processing speed.
	The scout master is concerned with gaps accumulating between hikers. The operations manager is concerned with inventories accumulating between machines.

To reinforce the analogy, draw a stock and flow diagram of a manufacturing line comprised of four machines. The first machine is fed by a steady flow of parts. (The flow is timed to match the 1st machine's natural pace of operation.) The 1st machine alters the part and deposits the altered part in an inventory. This inventory provides the buffer for the inflow of parts to the 2nd machine. The 2nd machine has its own natural pace of operation, but that pace will be adjusted downward if the inventory falls too low. On the other hand, if the inventory climbs to excessive levels, the 2nd machine can be operated slightly faster than normal. The output from the 2nd machine is deposited into inventory to buffer the flow to the 3rd machine. The 3rd machine has variable operation and deposits its altered parts into an inventory to feed the 4th machine. The 4th machine is the final machine in the manufacturing line. It has variable operation based on the inventory produced by the 3rd machine. The final product are the parts that emerge from the 4th machine.

Does your model resemble one of the models shown in exercise #1 or exercise #2?
Could your model be used to simulate the final parts produced in a 6 hour run of the assembly line?