Assumptions of the Transportation Model, Page 1 of 9
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Assumptions of the Transportation Model
Q:Are the assumptions of the Transportation model the same as the assumptions of payoff tables and decision trees?
A:No.
D:Let’s run through the assumptions for trees and tables and see which ones fit.
Q:What is the assumption of mutual exclusiveness?
A:Only one decision alternative can be chosen and only one state-of-nature will occur.
Q:Does this apply to the Transportation model?
A:No.
D:Think of all the different decisions you make in setting up a transportation solution. For each route, you must decide whether or not to use it, and if you use it, you have to decide how many units to place in it. Mutual exclusiveness implies a correct solution would use only one route to ship all the units. Since this isn’t true, the assumption is not relevant.
Q:What about states-of-nature?
A:There aren’t any.
D:More precisely, there is only one. The demand numbers you put into the transportation table could be considered a state-of-nature, but you put in only one set of demand numbers, so you are looking at only one possible future. This, by the way, is one of the disadvantages of the transportation model relative to tables and trees.
Q:What is the assumption of exhaustiveness?
A:We have a complete list of decision alternatives and states-of-nature.
Q:Does this assumption apply to the Transportation model?
A:No.
D:Let’s take a slightly different point-of-view and consider the solutions as decision alternatives. Exhaustiveness would say that you examine every feasible solution. We didn’t do that. You may remember that I talked about implicit enumeration earlier (if not, go back and find it). That told us two things: we don’t look at every solution and we don’t need to. This makes the assumption of exhaustiveness irrelevant.
Q:What about Exhaustiveness and states-of-nature?
A:You look at only one state-of-nature, so that cannot possibly be expected to be exhaustive.
Q:What is the assumption of determinism?
A:All data is 100% accurate.
Q:Does this assumption apply to the Transportation model?
A:Yes.
D:We have three sets of numbers that we use in the Transportation model: supply, demand and costs/unit for transportation. If we need these numbers to be precise, and they are not, then we need the assumption of determinism.
Q:Is the supply data accurate?
A:No.
D:Supply data comes from inventory counts, and we all know that inventory counts are never accurate. The wrong units are sent to us, or things get broken and not reported, or things get stolen. Even if you do regular inventory counts, when you get down to counting that last shelf (at 4 o’clock in the morning) the “counts” become estimates (“That looks like about … 20”).
Q:Is the demand data accurate?
A:No.
D:Demand data comes from customer orders (actual or forecast). If forecast, we know they are wrong, and even if actual, customers have (occasionally) been known to change their orders.
Q:Is the transportation cost/unit accurate?
A:No.
D:Transportation costs are largely based on fuel costs. Fuel costs tend to change a lot, and overnight.
Q:How do we allow for changes to our data?
A:By considering the affects of the changes.
Q:What part of the solution changes when the data changes?
A:That depends on which part of the data changes.
Q:What part of the solution changes when the supply or demand data changes?
A:For small changes, only the amounts shipped, not the routes that are used.
D:Remember that a solution to a transportation problem has two parts: which routes to use and how many units we ship on those routes. If a customer calls up and orders an extra unit, and you have it, you would simply toss it in with an existing delivery. If we are short a couple of units, we simply call another warehouse that is scheduled to deliver to the same customer and ask them to add in whatever we lack. There is no need to re-run the model; simply adjust your deliveries for the changes as they occur.
Q:What part of the solution changes when the cost/unit changes?
A:Only the total cost.
D:Changes to transportation costs tend to affect all routes equally. If all costs go up together, you would still use the same routes and ship the same amounts, you simply pay more.
Q:Is Determinism a problem for the Transportation model?
A:Not really.
D:The Transportation model has a quality called “robustness,” which simply means small changes to the data do not affect the solution. In affect, a robust model does not need a sensitivity analysis; it has already been tested and we know the solution is good. Part of the reason for this is that the Transportation model, by its nature, deals with short-term solutions. You don’t get a transportation plane for next year; you get one for next week, maybe two weeks, but rarely longer than that. With the short-term focus, there is less room for large changes. Certainly, catastrophic changes (a customer cancels an order) would require you to change your solution, but you know that already and would respond properly.
Q:Is Determinism the only assumption for the Transportation model?
A:No, we have two new ones.
Q:What is the first of the two new assumptions?
A:Proportionality.
Q:What is the assumption of proportionality?
A:For the Transportation model, the assumption of proportionality is that the cost per-unit of any route is constant, no matter how many units are shipped on that route.
D:The name comes from another way of saying the same thing – total costs are proportional to volume shipped. This definition is not as good as the other, because it uses the word you are defining (proportional) in the definition.
Q:Is proportionality a good assumption for the transportation model?
A:It is usually OK.
D:This assumption perfectly describes how we calculate the cost of using a route, but the question is whether or not we should be calculating costs this way. Let’s think about what the assumption means – if the cost per-unit on a route is, say, $6/unit, then one unit would cost us $6 to ship, two units would cost us $12, ten units would cost us $60, 1,000 units would cost us $6,000, etc.
Q:Does this seem reasonable?
A:Yes, unless we get volume discounts on our shipping costs.
D:You know how volume discounts work: the more you ship, the lower the cost per-unit. Proportionality says this never happens.
Q:If you get volume discounts, then is proportionality wrong as an assumption?
A:Yes.
Q:If an assumption is wrong, then you cannot use the model?
A:Right.
Q:So if you get volume discounts, then you can’t use the Transportation model?
A:Wrong.
D:You are not looking for reasons to avoid using the computer models; you are looking for reasons that it is OK to use the models. Very rarely is an assumption perfectly correct. That means that when the assumption is not perfectly correct, we have to see if there is some way we can still use the model.
Q:How can we use the Transportation model if we get volume discounts?
A:By applying the volume discounts manually.
D:let’s begin by thinking about what we know:
Q:When we set up a transportation problem, do we know whether or not (and on which routes) we will get volume discounts?
A:No.
D:When we set up the problem, even if we have an initial solution, we don’t have the optimal solution, so we don’t know how many units will be shipped on each route. Lacking that knowledge, we do not know whether or not we will qualify for volume discounts.
Q:If we don’t know whether or not we will qualify for volume discounts, which cost per-unit should we use in the transportation table – the regular cost or the discounted cost?
A:The regular (higher) cost.
D:This is the more conservative approach, since it tells us the highest optimal total cost.
Q:When will we know that we have qualified for volume discounts?
A:After we have the optimal solution.
D:Remember what “optimal” means. The solution is optimal for the data you type in. If you change the data, you get a different optimal solution. Both solutions are optimal, however they are simply optimal for different situations.
Q:How do you use the optimal solution to look for volume discounts?
A:you would need a volume discount chart, from your shipper, and would compare each shipping amount to that chart. Table 1 (below) shows a Volume Discount Chart and Table 2 (next page) shows the optimal solution from our example problem.
Units Shipped / Percent Discount1 – 99 / Full Price
100 – 149 / 5% off
150 – 199 / 10% off
200+ / 20% off
Table 1: Volume Discount Chart
Chicago / St. Louis / Cincinnati / SupplyKansas City / 6 / 8 / 10 / 150
25 / 125
Omaha / 7 / 11 / 10 / 175
175
Des Moines / 4 / 5 / 12 / 275
175 / 100
Demand / 200 / 100 / 300
Table 2: Optimal Solution
Q:Which routes qualify for a volume discount?
A:Kansas City – Cincinnati and Des Moines – St. Louis qualify for a 5% discount while Omaha – Cincinnati and Des Moines – Chicago qualify for a 10% discount.
Q:What’s the next step?
A:Calculate the new shipping costs on the qualifying routes and re-solve the problem.
Q:Why do we have to re-solve the problem? Couldn’t we simply calculate the change in total cost by hand?
A:You could calculate the change in total cost by hand, but you do not know whether the solution will be the same after using the volume discounts. You must re-solve the problem to learn that. Table 3 show the new optimal solution with the discounted rates highlighted.
Chicago / St. Louis / Cincinnati / SupplyKansas City / 6 / 8 / 9.5 / 150
25 / 125
Omaha / 7 / 11 / 9 / 175
175
Des Moines / 3.6 / 4.75 / 12 / 275
175 / 100
Demand / 200 / 100 / 300
Table 3: Optimal Solution
Q:Did the solution change?
A:No.
Q:Did anything change?
A:The total cost decreased from $4,380 to $4,017.50, a savings of $362.50.
D:For this problem, in part because it is very small, the solution did not change, so the only affect of the volume discounts was to lower total costs. In more complex problems, it is common for the algorithm to change the solution when volume discounts are entered into the table.
Q:What changes might occur to the solution?
A:Which routes are used and the number of units shipped on each route.
D:When you lower the cost on a route, that route becomes more attractive to the algorithm. The algorithm might choose to put additional units on that route, to take advantage of the lower rate. This could simply mean decreasing the amount shipped on another route, or it could mean no longer using another route, as all the units are shifted to the route with the volume discount.
Q:Do changes to the solution create any problems for us?
A:Potentially.
D:As you increase shipping on a route with a volume discount, you might qualify for a further volume discount. Conversely, as the algorithm steals units from one route to ad them to the route with a volume discount, you might lose the volume discount on a route that previously had one.
Q:What do you do if you gain a further discount or lose a discount you had qualified for?
A:Calculate the new costs per-unit, enter them into the table and solve the problem again.
D:It is not unusual for this to take several passes before you find the best solution using the volume discounts.
Q:Is there software that will do this for us?
A:Yes, there is.
D:Companies that do a lot of shipping have more sophisticated software that can check for volume discounts automatically and re-solve the problem for you. Smaller companies probably wouldn’t find such software worthwhile.
Q:So, is proportionality a problem?
A:No.
D:Proportionality simply warns you that the algorithm does not allow for volume discounts. If you do qualify for volume discounts, you must allow for that, either manually, or by having extra software that re-runs the algorithm for you. Whether it is you doing the work or the software, the changes to the rates have to be calculated outside of the algorithm and then entered if you want to find the true lowest cost shipping solution.
Q:What is the other new assumption?
A:Additivity.
Q:What is the assumption of Additivity?
A:For the Transportation model, the assumption of Additivity is that the volume shipped on one route does not affect the cost of shipping on any other route.
D:The name comes from the ability to calculate total cost by adding the cost of the individual routes.
Q:Is Additivity a good assumption for the transportation model?
A:It is usually OK.
D:This assumption perfectly describes how we calculate the total cost of a solution, but the question is whether or not we should be calculating total costs this way. Let’s think about what the assumption means, using Figure 1 (next page) as an example:
Figure 1: Warehouse W1 with Customers C1 and C2
Q:Is there any reason to expect that using the route between W1 and C1 would affect the cost of shipping from W1 to C2?
A:No, they are in opposite directions.
D:Whether you are shipping to C1 or not, whether you are shipping 1 unit or 1,000,000, the cost of shipping to C2 is not affected. This is a perfect example of Additivity. Suppose you were to ship 100 units to C1 and 50 units to C2. Then you get total cost by simply adding the cost of shipping to C1 (100 units X $6/unit = $600) to the cost of shipping to C2 (50 units X $8/unit = $400) to get the total cost of $1,000.
Consider, though, a different arrangement, as shown in Figure 2:
Figure 2: Alternative Arrangement
Q:What has changed?
A:Now, C1 is on the way to C2.
Q:Will this arrangement necessarily affect the costs?
A:No.
D:Suppose you are sending a full truckload to both customers. The truck would go out to C1 and return (roundtrip cost of $6/unit) and then go out to C2 and back (roundtrip cost of $8/unit).
Q:What if you don’t send full truckloads?
A:Then life gets more interesting.
D:Suppose you were sending only a half-truckload to each customer. Then you would get a shipping pattern like Figure 3:
Figure 3: Shipping Half-truckloads
Q:Why have the costs per-unit changed?
A:The truck is no longer making two roundtrips.
D:In Figure 2, the truck was returning to the warehouse, empty, after each delivery. That’s called “deadheading” and delivery companies hate it – it wastes a lot of time and money. In Figure 3, we combine the two shipments into a single run (this is called “piggybacking”)
Q:Why is it called “piggybacking?”
A:The term comes from the picture of a parent carrying a child on his/her back, which is for some reason called a piggyback-ride. Thus, when you put one delivery on top of another, the second load is riding piggyback on the first. This way, there is only one run (the return from C2 to the warehouse) that is empty. So we show the cost of a one-way trip from the warehouse to C1 ($3/unit, half the roundtrip cost of $6/unit), then the cost of continuing on to C2 ($1/unit, half the difference between the C1 roundtrip cost of $6/unit and the C2 roundtrip cost of $8/unit), then the return trip to the warehouse ($4/unit, half the roundtrip cost of $8/unit).
Q:How much did we save?
A:$6/unit.
D:In Figure 2, the roundtrip cost per unit for delivery to both customers is $14/unit ($6/unit + $8/unit). In Figure 3, the roundtrip cost per unit for delivery to both customers is $8/unit ($3/unit + $1/unit +$4/unit).
Q:Does this always happen?
A:No.
D:This only works when you are delivering to both customers and both loads will fit in one truck. That means that the amount you ship (zero, or a small amount, or a large amount) to one customer affects the cost of shipping to another customer. This can also work with a commercial carrier if they will credit you with a volume discount for the first (combined) leg of the trip.
Q:If this happens, then is Additivity wrong, as an assumption?
A:Yes.
Q:Can we still use the Transportation model?
A:Yes.
D:The reasoning is very much like what I laid out for the proportionality assumption. You won’t know if you can piggyback loads until you see the first optimal solution. After you identify potential cost savings, enter the new costs into the transportation table (split them up however seems best to you) and re-solve the model to see if the solution changes.
Q:What if the solution does change?
A:Then you repeat, checking to see if you lost any old opportunities and whether any new opportunities were created.
D:Additivity is simply a warning that the algorithm does not consider the cost savings that are possible from combining loads. If you want those cost savings, you have to do consider that on your own (or buy more sophisticated software to re-run the algorithm for you).
Q:Are the assumptions for the Transportation model any sort of problem?
A:No.
D:The robust nature of the model takes care of Determinism, and the other two – proportionality and additivity – are simply warnings to you that there may be cost savings that the algorithm cannot find.
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