2 Transportation costs
The logistical costs which are most obviously and directly affected by transport decisions are the transportation costs themselves.
If one is hiring the services of professional carriers, the calculation of transportation costs is very simple in theory. Transportation costs consist of the charges that have to be paid to the carrier. There is however the slight problem that the rates are not always transparent and future levels are not always predictable. A comparison of the rates asked by rail companies, joint-cargo companies, shipping companies and airline companies, shows little transparency. If a firm is gearing itself for the future, it will probably find it difficult to conclude longterm contracts that offer complete certainty about transportation prices. But in principle, if the transport services are hired, it is simple: transportation costs are the carriers’ rates.
If a firm provides own-account transportation, then it will be confronted by much the same problems in calculating costs as a transport company. Costs can be fixed or variable; costs can be cut by optimal routing, and by purchasing the right vehicle and replacing it at the right time, etc. In this case the calculation of transportation costs will proceed along the lines of a transport company, as explained in the previous chapters.
In the preceding chapters, attention was also drawn to economies of scale that accrue from the size of the load being transported. It is not the case that a truck with a load capacity of 20 tonnes costs twice as much as a truck with a capacity of 10 tonnes; the transportation of 2 000 tonnes by inland waterway does not cost a hundred times as much as 20 tonnes by road. In terms of transportation cost, pure and simple, the transport user should always go for the largest possible load. It is apparent, however, that they do not do this. And the reason why they do not, is that they also have to take into account other logistical costs, which rise rapidly with the size of the shipment.
Again purely in terms of transportation costs, one would always opt for the slowest mode of transport. Slow transport by ship costs less than super-fast transport by aeroplane, the slow movement by barge is less expensive than the quick movement by road. Here too, there is obviously more at stake than transportation costs. Other logistical factors have to be taken into account.
3 Handling costs
Transportation decisions can have an influence on the handling costs when loading and unloading, or with the transhipment of goods. These costs are borne directly by the shipper if his own staff handles materials, otherwise they will be charged to him by professional operators.
Sometimes the effect of a transportation decision can be considerable. For instance the rates asked by container operators at a seaport for offloading a container onto a barge will be much higher than for offloading the same container onto a truck. Likewise the decision, say, not to transport bulk material directly by road, but instead to cover part of the trip by waterway or by railway, will mean additional handling costs.
In contrast, the effect of a transportation decision in the handling costs is sometimes negligible. Many operators have a flat rate, which within certain limits remains constant per tonne. Then the decision, say, whether to transport goods by barges with a different load capacity can be taken without having to take material handling into account.
4 Inventory costs
Inventory costs are closely tied to transportation decisions. The trade-off between inventory costs and transport costs is a key element in such modern-day trends as just-in-time delivery and zero-based inventory systems. In all of these developments higher transportation costs are accepted in order to cut back inventory costs.
Transport can create inventory, for instance, through a sudden delivery of stock, which cannot be consumed immediately. This creates an inventory cycle. Moreover to protect against the time factor involved in transportation, one needs to build up a safety stock. Of course transportation also produces inventory costs for goods in transit, during their entire transport time.
Before we turn to the different types of inventory, we first want to look at the cost of keeping goods in stock. We need to know this cost to find the right tradeoff between inventory and transport.
The cost of stockholding comprises interest costs, insurance costs or risk costs, depreciation of goods and warehouse costs. In all calculations we shall represent cost by the symbol h : h is the ‘holding cost’, the cost of holding one unit of a particular good in inventory for one year.
1 Interest costs
Interest costs represent the opportunity cost of capital tied up in inventory. This is to be calculated by applying the annual interest rate to the capital, which is tied up per unit of the good. To treat the interest correctly, it is advisable to deduct the expected rise in the price of goods from the interest rate. Since the best guideline often is to assume that price rises will follow the general rate of inflation, it is usually advisable to equate the interest rate with ‘real interest’, i.e. the interest above inflation.
In current markets, real interest rates in Western economies tend to range between 1 % and 3 %, though this varies with macroeconomic conditions and financing structures. That can probably be taken as the percentage basis for calculating the real interest in a firm’s stockholding. Clearly however there will be differences from one firm to the next. A contractor who takes out a bank loan at 8 % interest rate, with inflation at 3 %, pays real interest at 1.08/1.03 = 1.0485, i.e. a real interest rate of 4.85 %.
2 Insurance costs or risk costs
Insurance costs – such as premiums for fire, theft, or natural disaster coverage – should be included in holding cost calculations if they vary with inventory value.
If goods are insured against fire and theft, and if the premium depends on the quantity held in stock, then the additional annual premium per item is also an element in the annual cost h of stockholding. If there is no insurance, then the risk costs from fire and theft will need to be made in a close calculation of h. Usually, insurance costs are insignificant. However, in modern supply chain environments (e.g. increased disruption and climate related risks), this may no longer be the case.
3 Depreciation of goods
One way in which goods depreciate is through physical decline or deterioration. But that is usually not the main factor. Much more important is economic depreciation.
In certain cases economic depreciation can be assessed from the average lifetime of the product. In the computer market, for instance, one might say that each item, irrespective of its age, can become obsolete at any moment due to technical innovation. As a rule of thumb, the average lifetime of an item can be estimated at about three years. If a computer is held in stock for one year, then the economic depreciation costs may be assumed to amount to one third of the value (much more, note, than interest costs, warehouse costs and insurance costs added together).
There are also cases where depreciation of goods is zero. Holding iron ore in inventory, for instance, involves no risk of obsolescence, or indeed of deterioration. All that is required, is to follow the price evolution when calculating the real interest rate.
Likewise, for a car manufacturer planning to build a particular model for years to come there are no depreciation costs in stocking spare parts for that model, since they are bound to be used.
The economic depreciation costs are often the most important factor in the annual inventory costs h. They vary greatly from one product to the next: zero for spare parts in a car factory, very high in designer clothes, where a change of fashion can make an article unmarketable in a matter of weeks.
For this reason it is difficult to make generalizations about the level of depreciation costs. However, with due consideration for the product concerned and market trends, it is possible to make an informed calculation; business managers will usually have a good idea what risk there is of their products becoming obsolescent or otherwise unmarketable.
4 Warehousing costs
Finally, when considering the cost h of holding a unit in inventory for one year, we must also include warehousing costs. This is an easy matter when goods are stored in a public warehouse, as is often the case at seaports, airports, or distribution centres. Then it is simply a question of knowing the annual charge asked by the warehouseman for storing a tonne, or M3, or, more generally, a unit of a product.
When goods are stored in a private warehouse, the annual warehousing costs will include not only leasing or interest costs and depreciation of the building, but also energy, heating, lighting, maintenance costs, etc. Note further, that in calculating the annual cost per unit, it is incorrect to take as a basis the maximum storage capacity of the warehouse. Maximum capacity will not always be maintained; because of fluctuation in stock there are bound to be empty spaces. In order to arrive at the cost of storage per unit, a working rule is to divide the annual cost of the warehouse by the average level of stock, i.e. with these fluctuations taken into account.
When a firm has overinvested in privately owned warehouses with excess capacity, additional inventory will not mean added warehousing costs. In the short-term this means that levels of stock can be pumped up at no additional warehousing cost. Under these conditions warehousing costs can be ignored, since the variable costs are zero. Nevertheless in the long-term a sustainable warehouse capacity indeed depends on stock levels. As soon as one has the opportunity to dispose of warehouse space, warehouse costs must of course
be included in storage costs. However, the Covid-19 pandemic reshaped the economics of this ‘excess capacity’. Firms now value buffer inventory and resilience over lean just-in-time strategies, meaning that what was once considered idle or ‘free’ capacity may now carry an opportunity cost, such as foregone subletting revenue or reduced asset flexibility.
Moreover, technological change also affects cost structure. In automated warehouses, high fixed costs for equipment and software dominate, while variable labor costs decline sharply. This shifts the balance between fixed and variable cost components of holding costs, making capacity utilization and automation levels critical factors in cost assessment.
Warehousing costs show a characteristic difference compared to other components in the annual holding cost h : viz. during transportation there are no warehousing costs. In contrast, interest costs and depreciation costs apply constantly in transit as well as in storage, while the cost of insuring goods in transit may well be higher than for stored goods in the warehouse.
5 Summary
To summarize then, the annual cost h of holding one unit in inventory comprises interest, insurance or risk, depreciation of goods, and warehousing costs. These costs can be considerable. When correctly evaluated, it stands to reason that firms sometimes take expensive transportation decisions, just to keep down their levels of stock. Such decisions are well grounded, as we shall see in the following chapter. The modern trend towards just-in-time delivery and zero-base inventory systems is perfectly understandable given such high storage costs.
Inventories can be divided into six types2:
1 cycle stock;
2 in-transit inventory;
3 safety stock;
4 speculative stock;
5 seasonal stock;
6 dead stock.
We proceed now to discuss each of these inventory types in more detail. The first three in particular are clearly influenced by transportation, and this effect must be taken into account.
4.1 Cycle stock
When a firm orders goods, this is usually in a quantity that will satisfy its needs over a given period. The goods procured will therefore spend some time in
stock. The evolution of this stock is cyclical. With the arrival of a consignment, the stock will jump by the total amount delivered. Thereafter stocks will decline at the rate at which the goods are consumed. With the arrival of the next consignment, stocks will again jump by the delivered amount, to decline again over a period of time.
If the evolution of stock is presented graphically, we find it presents a saw-tooth profile. The curve makes a vertical jump each time an order arrives. Thereafter stock gradually declines. For the sake of simplicity, the graph assumes that consumption is at a constant rate. Thus stock depletion is shown as a straight line (figure 3.1).
Average cycle stock
Time
It will be seen from the graph above that on average half the order quantity Q is in stock. This conclusion will prove very important in the following chapter when we compare modes of transportation with different load capacities; in the cycle stock, on average half of the order quantity is present.
This result does not depend on the assumption that consumption proceeds at a constant rate. Even when there are random fluctuations in consumption, it continues to hold. However fluctuations must be random, i.e. they must not show any systematic connection with the arrival of consignments. If the rate of consumption reacts to arrivals, the rule is broken. Then it will no longer necessarily be true that half the order quantity is in stock; for instance, in a situation where the ordered quantity is consumed entirely on arrival, then the cycle stock is zero.
Figure 3.1: Cycle stock with order quantity Q
Figure 3.2: Cycle stock with order quantity Q, with demand being treated as stochastic
Post-2020, inventory demand has become increasingly variable, driven by pandemic-induced shifts, surges in e-commerce, and shorter product life cycles. Modern inventory models often treat demand as stochastic, transforming the traditional ‘straight-line’ depletion into a random slope or necessitating a safety stock overlay. This can produce an evolution like in figure 3.2.
Firms now optimize inventory not only for cost, but also under constraints such as carbon footprint or energy usage, influencing order frequency – favoring fewer shipments – and shipment size through consolidation incentives. These factors, in turn, affect the ‘height’ and ‘frequency’ of the classic saw-tooth inventory profile. Yet, in contemporary supply chains, this profile is often irregular due to fluctuating demand, variable lead times, and dynamic replenishment strategies. The rise of stochastic inventory models, real-time tracking, and multi-echelon networks has made more advanced representations essential. Safety stocks, frequent smaller shipments, and collaborative planning are increasingly replacing simple deterministic cycles, though the saw-tooth model remains a foundational tool for illustrating basic replenishment logic. Today, uncertainty is the norm: demand is frequently influenced by exogenous factors such as weather, while lead times can vary independently of carrier intentions, reinforcing the need for flexible and resilient inventory strategies.
As presented so far, cycle stock is held at the place of destination, where the goods are delivered. Clearly, however, cycle stock can also be created at the
place of origin from where goods are dispatched, or even at intermediary depots, where goods are transhipped. Here however it is of much less importance. At most production sources and, for that matter, at most intermediary depots, the annual throughput far surpasses the size of a transportation consignment. It virtually makes no difference to the production cost per tonne that half the quantity of a consignment is available in cycle stock. Exceptions might be the transport of by-products and waste products, and possibly also the collection of goods produced by small firms, or at home. Here, where the production capacity is relatively small, it will make a difference to the average cost per tonne if half a consignment of goods is available in cycle stock.
The holding of stock at the production site in order to accumulate a sufficient quantity for consignment requirements, can be analysed in just the same way as cycle stock at the delivery point – except that now the graphical representation will show a saw-tooth profile in reverse: no longer does stock suddenly jump at delivery, to decline gradually thereafter; instead it is gradually built up through production, to drop suddenly with a delivery consignment. As before, however, half the average consignment quantity remains in cycle stock.
By now it will already be clear that the choice of transport mode will have consequences for inventory costs. If one tries to save on transportation costs by opting for greater load capacity, hence increasing the size of the consignment, one will incur additional costs on cycle stock.
4.2 In-transit inventory
During transportation goods are also in stock, even if they are not available for use. The calculation of in-transit inventory is fairly obvious. Goods are held in inventory for the duration of the in-transit period. All this time they are subject to inventory costs, which, no less than when they are in the storage, comprise interest costs, insurance costs and depreciation. There is this difference, that in-transit inventory generates no warehousing costs. There might, on the other hand, be a slightly higher insurance cost, for there is a greater risk factor during transportation than in the warehouse.
In-transit inventory must not be underestimated. In volume it can surpass cycle stock. For example, if a firm replenishes its stock on a regular monthly basis, an average tonne of goods will remain in cycle stock for only a fortnight. It is quite possible that the total transportation time, say from an overseas origin, exceeds a fortnight. Thus the goods actually spend a longer time in transit than they do in the cycle stock.
In-transit inventory costs have a very different effect from cycle stock costs on transport decisions. Cycle stock costs encourage transportation in small consignments, whereas in-transit inventory costs encourage a faster mode of transport.
This difference is frequently overlooked, because the fastest means of transport also tend to carry the smallest consignments. Since the two elements overlap, the difference easily goes unnoticed. In essence, however, they are quite different.
4.3 Safety stock
Safety stock, or buffer stock as it is sometimes called, is the inventory which is held over and above cycle stock because of uncertainty either about levels of customer demand or about the length of time between the ordering and actual arrival of supplies (order lead time). Since fluctuations cannot be predicted with certainty in either of these, one does not want to plan for supplies to arrive when stocks have run out. The idea is to keep a reserve. Stocks are replenished so as to avert a possible stock-out.
If safety stock and cycle stock are reproduced in the same graph, we arrive at the profile shown in the figure below. Here, replenishments do not arrive at the precise moment when stock reaches zero; instead, orders are placed earlier, and arrival is foreseen at a moment when existing stocks are still at level S. This level constitutes the safety stock. Post-pandemic global disruptions (e.g. Covid-19, the Suez Canal blockage, and war zones) have made lead times highly volatile. As a result, firms now plan for both upside and downside risks, requiring more flexible safety stock policies.
Safety stock is no longer calculated based solely on historical averages and standard deviations. Artificial Intelligence and Machine Learning algorithms are increasingly used to dynamically adjust safety stock based on: demand sensing, supply risk scoring and/or seasonality and promotions.
Figure 3.3: Cycle stock and safety stock
Once again the graph has been drawn assuming a constant sales rhythm so that stock depletion is linear. Needless to say, this is only a representation of the expected pattern of things, the mean as it were. In practice all sorts of random fluctuations can occur so that the rate of stock depletion is speeded up at one moment and slowed down at another. If there are also fluctuations in the order lead time, then there will also be a displacement along the time axis in the vertical supply jumps. These random elements will mean that deliveries do not always coincide with stock level S. Indeed if that were the case there would be no need to keep a safety stock. Because of random fluctuations, existing stock at the arrival of a new consignment will sometimes be greater than S, and sometimes less than S, or even zero. On average, however, considered over a period of order cycles, the level on arrival of supplies will be S. This average is the safety stock.
In order to calculate what safety stock S a firm needs to maintain and when, therefore, it should place its orders, there are four elements that have to be considered: order lead time, demand, acceptability of a stock-out, and the method of stock monitoring.
Order lead time is the period between an inventory manager ordering goods and the arrival of the goods at his warehouse. The longer the order lead time and the more uncertain it is, the greater the required safety stock.
Demand, our second element, means stock depletion. The size of demand and the degree of fluctuation have consequences for levels of safety stock too. The greater the demand and the more uncertain it is, the greater the required safety stock.
The third element is the acceptability of a stock-out. A manager who tolerates a high risk of a stock-out, will need a low level S of safety stock; one who is strongly averse to that risk will need a high level of safety stock.
Finally the fourth factor is the method of stock monitoring. With a system of continuous review the stock is always known and a replenishment order can be placed as soon as the stock level falls below a certain quantity. With a system of periodic review the level of stock is checked at fixed intervals and an order can be placed only at these intervals. With the second method the safety stock will have to be greater than with the first. In the discussion that follows, we shall assume a system of continuous review. As far as inventory costs are concerned, there can be no doubt that it is the best method.
In order to show how safety stock is calculated with continuous review, we shall start with the simple case where there are three possible order lead times, each having the same probability, and two possible demand levels, again, having an equal probability.
Assume that the order lead time can be 1, 2, or 3 days, each with a probability 1/3. Assume also that daily demand is either 95 or 105 units, each with a probability of 1/2. Assume further that order lead time and daily demand are independent of each other: a long order lead time is not systematically coupled to a high demand, or for that matter, a low demand. Further, the successive demand levels are independent of each other: the fact that on one day the level is low gives no indication of the level on the following day. The probability for the following day remains 1/2, both for 95 and 105 units.
We start by taking the levels of total stock consumption and their probabilities during order lead time. This is shown in the following table, where column (2) gives the possible results of stock consumption in ascending order from 95 to 315.
In column (3) of the table we find the probabilities for each of these results. So, for example, the probability of a consumption rate of 95 during order lead time is 1/6: this will occur if the order lead time is 1 day (probability 1/3) and if demand on that day is 95 units (probability 1/2).
The probability of consuming 105 units during order lead time (see the next row in the table) is likewise 1/6. This will occur if the order lead time is 1 day (probability 1/3) and if the demand on that day is 105 units (probability 1/2).
Table 3.1: Probability of stock consumption during order lead time. Order lead time 1, 2 or 3 days (each with probability 1/3). Daily demand 95 or 105 (each with probability 1/2)
Order lead time Level of stock Probability of Probability of Probability of consumption obtaining this obtaining at most obtaining more during order level of stock this level of stock than this level of lead
In the next row we find that the probability of a stock consumption of 190 units during order lead time is 1/12. For this to happen the order lead time must be 2 days (probability 1/3) and on each of these days demand must be 95 units (probability 1/2 × 1/2).
We find the same probability of 1/12 for 210 units (2 days with 105 units) but for 200 units the chances are doubled, since there are two ways in which this can be obtained with the given order lead time: 95 units on day 1 and 105 units on day 2, or vice versa.
The table proceeds row by row until we reach the highest possible consumption during the order lead time, i.e. 315 units. Now there is only a probability of 1/24: the order lead time must be 3 days (probability 1/3) and on each of the days demand must be 105 (probability 1/2 × 1/2 × 1/2).
Column (4) in the table cumulates the probabilities in column (3). Thus the probability of a maximum demand of 105 is 2/6. The probability of a maximum demand of 190 is 5/12, and so on, until one reaches 315 where the probability is 1. One knows for certain that the maximum possible demand during order lead time is 315 units.
Column (5) of the table is arrived at by subtracting column (4) from 1. Thus the probability of 95 units being exceeded is (1 – 1/6) = 5/6. The probability of 105 being exceeded is (1 – 2/6) = 4/6. The probability of 315 units being exceeded is zero.
Calculating the necessary level of safety stock now depends entirely on the risk one is prepared to take of a stock-out. Let us assume that one is prepared to run a 1/24 risk, i.e. that one stock-out may occur for every 24 deliveries. Under these conditions, it will be necessary to reorder as soon as the stock sinks to 305 units, since only at this level the probability of a stock-out during order lead time equals 1/24.
Since we know the reorder level is 305, we can now indicate the safety stock level. The average order lead time is 2 days and the average daily demand is 100 units. Mean consumption during order lead time is therefore 200 units. Since the reorder level is 305 units, on arrival there will be an average of 305 – 200 = 105 units in stock. The safety stock, the expected stock level on arrival, is therefore 105 units.
It will be clear even from this very simple example, that transport decisions can have major consequences for safety stock. Suppose one opts for a mode of transportation with greater frequency and punctuality, so that order lead time is guaranteed to be 1 day, not 1 day on average but exactly 1 day. Now the probability of a demand of 95 units during order lead time is 1/2, and the probability of a demand of 105 units is likewise 1/2. With these new probabilities, the risk of a stock-out is reduced to the required 1/24 (or completely eliminated even), by fixing the reorder level at 105 units. The required safety stock is now minimal: reordering at 105, with an average demand of 100 over an order lead time of 1 day, leaves a safety stock of 105 – 100 = a mere 5 units.
That is a spectacular difference to the safety stock in our previous example. A switch to a swift and more reliable mode of transport with a guaranteed order lead time of 1 day implies a drop in the safety stock from 105 to 5 units, i.e. keeping 100 less units in stock. Punctuality and frequency are clearly what one is looking for in transportation, and that explains why these qualities command such high prices.
The mathematical model shown in the present example is of course extremely simple. In particular, the assumption of equal probabilities in lead time and in demand quantities is an oversimplification of what happens in practice. Software for logistical management typically applies more refined probability distributions, most frequently the normal distribution. The required level of safety stock S if demand during order lead time shows a normal distribution, can then be calculated as follows:
S = Kv
where v is the standard deviation of demand during lead time and K is a constant, dependent on the stock-out risk one is prepared to tolerate. So the rule states that the amount of safety stock required is K times the standard deviation. We shall first consider how v is calculated, and then we turn to K.
4.3.1
Calculating v
An obvious way to calculate the standard deviation is to observe the total demand over a number of order lead times. Thus over n lead times the observed demand is L1, L2, ... Ln. The average demand level L is computed from this series:
L = Ri Li/n
Next, the standard deviation is obtained on the basis of the formula:
This method is straightforward and well-known, but it does not show how the standard deviation changes as order lead time increases or becomes more variable, or as demand rises or falls or gets more variable. In fact this method of calculation does not help a lot in taking transportation decisions, which are precisely meant to cause changes in v
It is better therefore to use a formula where the standard deviation is inferred from the data on demand and order lead time. One such formula is the well known equation, derived by Fetter and Dalleck3:
v = Tv Vt 2 + ^ h
where: v = standard deviation of demand during order lead time
T = average order lead time
V = average demand
t = variance of order lead time
v = variance of demand.
There is of course a free choice of units, but they need to be applied consistently. For example, if one selects a day as the time unit and a tonne as the goods unit, then the order lead time T is expressed in days, and the average demand V in tonnes per day; the variance of order lead time t in days and the variance of demand v in tonnes per day. One will thus arrive at a standard deviation v expressed in tonnes.
The average order lead time T and the average demand V are obvious concepts. They need no further explanation. To calculate the variance t in order lead time we can take a series of n order cycles over a certain period, and note the lead time of each T1, T2 ...,Tn. We compute their average T and then obtain the variance of order lead time with the following formula:
Likewise the variance v in daily demand can be measured by noting the demand V1, V2 ..., Vn over an observed number of n days, and then comparing these results with the average V. The variance of demand equals: v = 1 n
Without going into the mathematical proof, we can get an intuitive grasp of Fetter & Dalleck’s equation. For easy understanding let us take the hour as the time unit. Under the radical sign we find the possible fluctuations of lead time demand. These fluctuations are partly due to the average length of the lead time. Every hour in this lead time demand can fluctuate with its hourly variance v. If the lead time takes T hours, total demand can fluctuate with T times that variance. This is represented in the term Tv. The second part in the fluctuations of total demand is not due to the average length T of the lead time, but to its variance t. For every hour more or less in the lead time, the expected total consumption changes by the hourly average V. This effect is represented in the term V2t.
The mathematical proof needs one assumption, which also can be seen intuitively, without really going into the mathematics. The assumption is independence of fluctuations: demand levels in consecutive hours are supposed to fluctuate independently from each other (this assumption is behind the simple addition of variances in Tv) and variations in demand are also supposed to occur inde-
pendently from fluctuations in lead time (this assumption is behind the simple addition of the second term V2t to the first term Tv).
Dependence between fluctuations would clearly invalidate the equation. Positive autocorrelation is an example: a high hourly demand tends to be followed by high demand next hour, while low hourly demand tends to be followed by low demand next hour. Consecutive hours, in other words, tend to ‘resemble’ each other. This increases the variability of lead time stock depletion. The standard deviation of total demand during two hours exceeds the 2v2, counted in Fetter and Dalleck’s equation.
Another example is correlation between lead time and hourly demand. If the occurrence of a long lead time is systematically accompanied by high hourly demand, one obviously faces higher variability of lead time demand. This too is unaccounted for in Fetter & Dalleck’s equation. It assumes independence of fluctuations.
The equation allows one to gauge the importance of speed and punctuality in transport. Less order lead time means a lower T, less variance of order lead time lowers t. The effect on v and on safety stock Kv, will, as one can see, be particularly marked when the supplier’s processing time is low, for then T and t are almost entirely down to transportation, which will accordingly have a relatively greater impact on safety stock. Speed and reliability will pay off especially in a context where the just-in-time principle is followed, where suppliers also keep to a tight delivery schedule.
From the formula for v one can also infer that the order lead time T (function of speed) and variance of order lead time t (function of punctuality) are weighed respectively against the variance v in demand and against the average demand V. If demand shows a relatively big variance (v is big in proportion to V), then speed will matter most. If demand is reasonably stable (v is small in proportion to V), then punctuality in transportation will help most to keep safetystock levels low. As the saying goes: ‘quick service for sloppy customers, reliable service for punctual customers’.
4.3.2 Setting K
The constant K depends on the risk one is prepared to take of a stock-out during lead time. The values for K can be obtained from the accompanying table, or from a table of normal distribution to be found in any handbook of statistics.
If for instance one is prepared to accept a 0.5 risk of a stock-out before the next delivery, then K is zero. No safety stock is required. Orders are placed so that consignments are planned to arrive just as existing stocks reach zero, i.e. with no reserve. In one out of two cases there will be a stock-out before supplies arrive.
If one wants to reduce the risk of a stock-out to 0.45, then one will have to maintain a volume of safety stock such that K = 0.13 times the standard deviation of demand during order lead time. Now 45 % of all arrivals will be too late.
The less one is prepared to take a risk the greater the margin of safety stock. The table descends to the minute risk of 0.0005 (1 in 2 000): for every 2 000 deliveries, there will on average be one single stock-out before supplies arrive. This tiny risk requires a level of safety stock such that K = 3.30 times the standard deviation.
Column k of the table4 shows how the volume of safety stock will have to be increased in order to reduce the risk of a stock-out by 0.0001 (= 1 in ten thousand). For example, starting from a risk level of 0.4, one would need to increase the volume of safety stock by k = 0.00026 times the standard deviation of demand during order lead time. Starting from a risk level of 0.1 this rises to 0.00057 times the standard deviation. And at the extremely small risk of 0.005 it is only possible to reduce the risk by a further 0.0001 if the safety stock is increased by 0.00692 times the standard deviation. The more one wants to shrink the risk, the greater the additional cost involved in reducing it by a further 0.0001.
Table 3.2: Stock-out risk and safety stock (normal distribution of demand during order lead time)
Tolerated risk of stock-out Required value of K
(increase
For the inventory manager faced with decisions on levels of safety stock, it is useful to know the values of k, for they show the additional safety stock required
for removing a particular risk. One can then compute how far one should go in reducing risk.
In practice the precise calculations are hardly ever made. Of course inventory managers weigh the costs of providing additional stock against the costs of a potential stock-out, but they usually do this by intuitive methods, where the acceptable stock-out risk is set by a rule of thumb. The exact calculation proceeds on the basis of the parameter k. This allows one to weigh the costs of additional stock against the economic damage of a stock-out. It is beneficial to reduce stock-out risk so long as additional stock costs less than damage of a stock-out. The optimal safety stock can be found on the basis of the equation
hkv = 0,0001 Nz
where: h = cost of holding one unit of the good in stock for one year k = increase in K in order to reduce risk at replenishment by 0.0001 (see table)
v = standard deviation of demand during order lead time
hkv = additional yearly costs of increasing safety stock by kv
N = number of replenishments per year
z = cost of stock-out
0,0001 Nz = yearly benefit of avoiding stock-out costs, when safety stock is increased by kv.
The left side of our equation gives the marginal costs of holding additional stock, the right side gives the marginal benefits of avoiding stock-outs. Both sides are expressed on an annual basis. As long as the marginal costs are less than the marginal benefits, it pays to go on increasing safety stock. The optimum is reached when marginal costs and marginal benefits are equal.
The optimum therefore has the characteristic k = 0.0001 Nz/hv, and the optimal safety stock can be arrived at by consulting the table for this value of k. The difficulty in applying this rule is that one needs to know the economic cost z of a stock-out. In estimating z there will always be an element of intuition and subjectivity.
Even if the costs z associated with a stock shortage are hard to evaluate, the rule allows one to draw useful conclusions. It appears that, regardless of the cost z of the shortage, the optimal value of k is inversely proportional to the annual stock holding cost h and to the standard deviation of demand during the delivery time v. It is directly proportional to the number of replenishments per year N and the stock out cost z. This says a great deal about the adjustment of the safety stock to changed circumstances. For example, if the annual costs of stockholding h should rise because of a general price increase, but one has at the same time good reason to think that the cost of a stock-out z will go up proportionally, then one should leave the volume of safety stock as it is.
An important element in the calculation is the standard deviation v of demand during order lead time. Clearly we need to consider the total lead time, of which transport is just one component. This is very well illustrated in table 3.3, giving the average time and the variance for 19 consecutive stages in a trans-ocean supply5
Table 3.3: Major elements in typical total trans-ocean supply chains between inland and transocean inland points (time in hours)
Adapted from E. Frenkel, ‘The Economics of Total Trans-ocean Supply Chain Management’, International Journal of
The expected time of the total supply chain can obviously be obtained as the sum of the third column (588). Likewise, the variance of the total supply time can be obtained as the sum of the variances in the last column (1 146). This however is less obvious. Simply adding the variances will be a correct procedure only if the ‘covariances’ between the 19 elements are zero. We will not dwell on this technical issue. If the various elements fluctuate independently, their covariances are zero.
Example: Reducing safety stock by improving transportation
One way to reduce safety stock is by improving transport reliability – for instance, by shifting focus to land transport (option 17). The question is: should we aim for speed (shorter lead times) or reliability (lower variability in lead time)?
To illustrate, let us start from the safety stock formula, which considers uncertainty in both demand and lead time, where K equals 2.51(based on a 99.4 % service level, or a 6 in 1 000 stockout risk), average demand (V) of 20 units per week and variance in demand (v) of 10.
Case 1: Speed – halving the lead time
If we halve the average lead time from 60 days to 30 days, then the safety stock becomes:
Case 2: Reliability – halving the lead time variance
Now, suppose instead we keep the average lead time the same, but cut the variance in half, from 64 to 32. This improves delivery reliability. Using:
While both strategies reduce safety stock, improving reliability (lower lead time variance) often has a greater and more sustainable impact than speed alone. This is especially true when demand is stable – a more predictable delivery schedule reduces the need to buffer with excess inventory.
4.4 Speculative stock
It is sometimes the case that operators want to amass stock because they think the price of goods will rise. This stock is not cycle stock, it is not in-transit inventory that simply exists during transportation, nor is it safety stock, which is necessary to deal with unforeseen demand.
Now there might already be speculative elements present in the cost calculation of cycle stock, in-transit inventory, and safety stock. The annual inventory cost h will include real interest, which we get by subtracting the expected rise in the price of goods from the nominal interest. If a big price rise is expected, this will lead to lower annual inventory costs h and, in turn, to a decision to increase stocks. With negligible or perhaps even negative real interest there will be less objection to large arrivals which push up the cycle stock; there will be less aversion to slow modes of transport which keep large volumes of goods in-transit, and there will also be less objection to non-punctuality which necessitates high levels of safety stock. In a way, then, the speculative element can be taken care of, as already explained, in the calculation of cycle stock, in-transit inventory, and safety stock, by counting the right interest rate in the annual inventory cost h.
Exceptionally, the anticipated price rise can be so great, that not only the real interest, but also the entire cost h is negative: this negative cost arises with an expected price rise beyond all warehouse costs, insurance costs, interest costs, etc. This changes the calculation completely. Inventory ceases to be a burden and becomes a financial bonus; now maximum use will be made of warehouse space to accumulate speculative stock, while safety stock disappears totally from the calculation. In-transit inventory is profitable, because of the expected price rise and negative holding costs. In spite of these negative costs, however, cycle stock remains a drawback: the cyclical variation prevents one from utilizing the entire storage space for speculative stock. If supplies are procured in large consignments, then a considerable amount of warehouse space will need to be cleared in time for the arrival of the next delivery. The adjustment in the calculations is fairly straightforward, and need not detain us further.
4.5 Seasonal stock
It is sometimes necessary to build up stock because production of goods is subject to seasonal variations that are different from the demand fluctuation. Clearly seasonal stock cannot be equated with order cycle stock. On the contrary, seasonal stock might well render order cycle stock irrelevant.
For instance a manager who orders from a production point with seasonal stock will, if he bears the inventory cost at both ends, have little reason to take order cycle stock into account. There is in any case seasonal cycle stock available over the entire year. The only difference the order quantity makes is to the place where the stock is being held. Given the availability of this seasonal stock, it would make sense to count the order cycle stock, only if he had to bear the costs of stockholding at the place of destination while those at the point of production are not charged to him.
4.6 Dead stock
Dead stock is stock that has become unsaleable. Transport decisions can play a part in this: for instance, delivery in large quantities or by a slow form of transportation can mean an increased likelihood of stock becoming obsolescent. In fact, however, this risk should already have been included when calculating the annual cost h of holding goods in stock. In a correct calculation this will include risk of depreciation. It is not necessary to include this factor twice, as it were, by separately adding the risk of obtaining dead stock.
If cycle stock, in-transit inventory, and safety stock have been calculated at the proper level of the annual cost h, then dead stock has been fully taken into account.
5 Stock-out costs
Stock-out costs must not be confused with stockholding costs. By stock-out costs we mean the losses that are incurred from a shortage of stock: losses resulting from a fall in customer service, from machines standing empty, from disruption of production, etc. In a number of logistical decisions stock-out costs and stockholding costs have a contrary effect. For example, the decision to lower the volume of safety stock will mean lower stockholding costs but higher stock-out costs. And for this reason in our analysis of safety stock, we explicitly took into account the costs of a stock-out, which we called z.
In practice it is extremely difficult to make a direct estimate of z – the costs resulting from a stock-out, but often z can be inferred indirectly from the behaviour of the inventory manager. All we have to do is observe what stockholding
costs are deemed to provide sufficient protection against the risk of a stock-out. If the manager maintains a high level of safety stock, then he obviously believes that a stock-out will involve high costs. If he is satisfied with a modest level of safety stock, he is clearly convinced that the results of a stock-out will be less than catastrophic. Thus by opting for a particular level of safety stock, he indirectly signals his opinion on stock-out costs.
What we do, therefore, is take the manager’s policy at face value, and deduce the implied cost of a stock-out. Our calculation assumes normal distribution of demand over order lead time. The rule for setting the optimal safety stock then is to find k in the table of the normal distribution, and to fix the safety stock at a level where
k = 0.0001 Nz/hv
As explained before, k is the fraction of the standard deviation v which must be added to the safety in order to reduce the risk of a stock-out by 0.0001, N is the number of arrivals per year, z the cost of a stock-out, and h the cost of holding one unit in stock for a year.
Originally the rule enabled us, starting from a known stock-out cost z, to deduce the value of k. But it can be used just as well to make the reverse calculation, i.e. from the known value k (set by the manager who chooses a safety stock) we can now deduce the cost z of a stock-out. Mathematically this is simple. All we need do is replace k on the left by z, as follows:
z = khv/0.0001 N
This is easily translated into economic terms. The numerator khv is the annual cost of additional safety stock: stock is increased by kv units, each costing h per year. The denominator is the number of stock-outs avoided, i.e. a risk reduction of 0.0001 for each of the N deliveries in a year. Thus the quotient gives us the additional stockholding costs one is prepared to lay out in order to prevent one stock-out.
We need now to observe what level of safety stock the manager actually sets. Let us assume that he holds 1.55 times the standard deviation in safety stock. Given this, we look up k in the table of the normal distribution and find k = 0.00084. The cost of a stock-out is then equivalent to:
z = 0.00084 hv/0.0001 N
In this way, the cost z of a stock-out is expressed in just three variables; the stock holding cost of one unit of a good for one year h, the standard deviation of demand during order lead time v, and the number of deliveries per year N. Each of these variables is measurable, so the cost of a stock-out can indeed be
calculated. The calculation, note, hinges on the observed decision of the manager to select a level of safety stock such that k = 0.00084. This decision in effect gives away the cost of a stock-out. If he had gone for more safety stock, i.e. with k greater than 0.00084, then this would imply a higher stock-out cost; if, on the other hand, he had opted for less safety stock, i.e. with a lower value for k, then he is clearly convinced that a stock-out will cost him less.
This is a very handy way to estimate the cost of a stock-out. One only has to extrapolate from the manager’s decision to go for a particular value of k, and the rest is plain sailing. Of course the method falls short once the manager himself begins to have doubts about the volumes of his safety stock and specifically asks for advice on this matter. Still, it proves to be an excellent method when a transportation analyst has to come up with advice on transport without getting involved in stockholding policy. Then it pays simply to take existing stockholding policy as given and to draw conclusions from it.
6 Packaging costs
Transport decisions also have an influence on packaging costs. It is easy to see that bulk and tanker transportation require less expensive packaging than smaller lots by parcel carriers. Opting for container transport can also save on costs, for the packaging will be less durable and less expensive than that required for individual lots in the traditional parcel delivery service.
The choice between road, rail, sea, and air transport can also make a difference to packaging – certainly where the transportation of dangerous goods is concerned, where the legal requirements vary according to the mode of transport.
Packaging costs are therefore important, but within the field of transport decisions their calculation is a fairly straightforward matter, and there are no special methodological aspects that need going into here.
7 Costs of order processing and administration
If deliveries are in small consignment then costs of order processing and administration can even outweigh the costs of transportation and stockholding. Because of this, firms seek to group orders and deliver in larger consignments. Clearly, these costs are incidental when bulk orders are concerned or when it involves valuable goods where the risk costs in stocking alone may exceed the processing and administration costs several times over.
The general evolution shows that processing and administration are gradually becoming less important elements in a firm’s transport decisions. This is partly
due to the massive switch to computers in logistical management, which means these areas are much less labour-intensive than before.
8 Start-up costs
These are the costs incurred when a concern has to switch its activities to a different operational basis. Transportation and inventory decisions that impose frequent stock replenishment and short production cycles will push up start-up costs, whereas those that involve long cycles will lower them.
It was pointed out when discussing stockholding costs, that delivery in large consignments increases the cost of cycle stock. On average, half the volume of the consignment will be present in the cycle stock. The effect on start-up costs is quite the opposite though. If an arrival of goods means readjusting the operational procedure with additional costs, then it is more economical to deliver in large consignments and to handle the goods in one fell swoop, possibly then placing them in stock. Because of the large quantity involved, there is a saving on start-up costs – perhaps even as a trade-off for the cycle costs created by that.
Start-up costs, order-processing costs, and administrative costs are often bunched together in logistical analysis, into a single fixed cost per consignment. These costs are then proportional to the number of consignments, hence they form an important element when determining the optimal size of a consignment.
9 Costs of customer service
Transport decisions can influence the level of customer service, not only through delivery times and stock-outs, but more generally through the treatment of goods and the way customers are looked after.
Attention to customer service may be given high priority in cases where transportation is geared to the servicing requirements of individual customers. In such cases, just about all firms prefer to provide own-account transport, or failing that they will pass on consignments to a select group of professional hauliers with experience in those goods. The alternative, using random hauliers selected on the basis of price alone, would involve too high a risk of customer loss. Even for goods of no special value, requiring no special care or expertise, service standards are perceived as important. Within the European Union, for instance, where the transportation market is gradually being opened up to competition between hauliers from different member countries, the language of the driver remains an important factor for the customer and a reason to select from hauliers within the same country.
10 Location costs
A firm’s costs can be influenced greatly by its location. This is not only a matter of land prices, but also of availability of a qualified staff, wage levels, delivery costs and the cost of public utilities. Even taxes can depend on location.
Location theory has a long tradition, going back to von Thünen, of models for calculating optimal location. Some of these models consider exclusively the costs of transport, thereby selecting the location, which generates the lowest transport expenditure. Yet it is clear that the overall effect of the location factor on logistic costs needs to be taken into account.
We shall not be pursuing location costs further, however. It is a matter that really only arises once in a while, when a firm has to chose a location, which will then be fixed for the long-term future. We will focus on the short-term, within a fixed location.
11 Just-in-time supply and zero stocks
Some recent developments in logistical management lie outside the field of transport and hence, outside the scope of a handbook such as this one. Two major trends, however, do have important implications for transportation: justin-time supply and the drive to reduce stocks to zero level.
Behind these trends there lie three factors:
1 greater awareness of stockholding costs;
2 lowering of start-up costs and order costs per consignment;
3 better planning of demand with less variance6.
These factors lead, as one can see by applying the standard rules of stockholding, to lower safety stocks and to smaller orders.
There are no generally agreed definitions of either ‘just-in-time’ or ‘zero stocks’. Many vague definitions are in circulation and frequently terms are bandied about with no one quite knowing whether they are meant to be synonyms or not. Indeed there are those who make no distinction between ‘just-in-time’ and ‘zero stocks’. One often gets the impression that the terms are being used to motivate staff or promote new attitudes, rather than as a tool for technical analysis.
In order to arrive at a clear analysis, it is first necessary to strip the concepts of their psychological connotations and to consider them in their strict sense. We shall then see that there is indeed a difference between just-in-time delivery and zero stocks.
11.1 Just-in-time supply
In a strict sense just-in-time supply (or JIT) can be defined as supplying without holding safety stock, and without incurring stock-outs. Thus, arrivals cannot be early, for then there would be stock left, nor can they be late, for then there would be a stock-out. As the name says, arrivals must be ‘just-in-time’7
Defined in this way, JIT is an aim which can rarely be achieved, for to do so without error over a length of time, the standard deviation of lead time demand would have to be zero. In practice, it is extremely difficult to attain absolute certainty about demand. Generally speaking, JIT will have served its purpose if it achieves significant reductions in safety stock, even if it does not eliminate it completely.
To illustrate this effort to eliminate safety stock, we consider the standard deviation v in demand during order lead time. One way to calculate the standard deviation, upon which the level of safety stock depends, was, we said, via the formula
v = Tv Vt 2 + ^ h
The standard deviation can be lowered by changing the four variables beneath the root sign: the average lead time T, the variance of demand v, the mean average demand V, and the variance of lead time t.
Obviously proponents of JIT are not going to propose a reduction in the mean demand V. Nor is it their first priority to shorten the average lead time T. Their chief aim is to reduce the two variances: variance of demand v, through strict planning of consumption, and variance of lead time t, through increased punctuality in supply and transport. A glance at the formula will show that making the two variances zero, is enough to make the whole expression zero. Thus v and hence the required safety stock will also be zero.
In the effort to achieve these reductions, transportation plays a vital role with regard to t. Through a strict compliance with quoted delivery times, the variance of t will be very low. True this is only one component in the total order lead time; if t is to be reduced to zero, the supplier too will have to deliver punctually.
It can easily be seen from the formula that punctuality in transport becomes increasingly important as the other elements are optimized. The standard deviation v and thus the corresponding safety stock are the outcome of the square root of a sum. Saving a variance of one day in transit time will only have a slight effect on v if the variance of demand v and the average lead time T remain high, or if the unreliability of the supplier keeps variance t at a high level. But saving a variance of one day when it was the last source of uncertainty can have a very marked effect on v.
Thus JIT delivery is a policy to be recommended in a context where JIT production also prevails, i.e. where it is part of an entire JIT environment with strict planning from suppliers to customers. JIT production, however, is a much broader area than JIT transport, and it would take us well beyond the scope of this book. It entails a complete turn-about in production, which is no longer based on stockholding but on direct orders from the customer; that is to say, with demand certain beforehand, and with variance v equal to zero. Its implementation also involves a strict timing of production. If, within this perfectly planned context, transportation also achieves absolute punctuality, then safety stock disappears entirely.
The analysis of the formula poses no great problems. It is easy to compute the impact of the various elements T, t, and v, at different values, and weigh them against each other. Much more difficult is the practical challenge of achieving shorter delivery times, and reduced or eliminated variances. This is easier said than done. Putting JIT supply into practice is complex and stressful. The rewards, it must be said, are substantial. From the case studies presented in the following chapter, one can see what costs are tied up in safety stock and stock-outs. Through JIT supply these costs can be avoided.
11.2 Zero stock
Through JIT supply one eliminates safety stock, but other stocks remain: cycle stock, in-transit inventory, possibly also speculative stock and seasonal stock. If the goal is zero stock, all these others will have to vanish.
Here, there are a number of approaches within logistical management, including, for instance, techniques for levelling out seasonal fluctuation and replacing speculative stock by forward purchases. Not all these methods fall within the field of transport.
Transport does play a role with regard to in-transit inventory and cycle stock. Intransit inventory is proportional to the duration of the journey; it can therefore be reduced by using faster modes of transport. Cycle stock is proportional to the size of loads; it can therefore be reduced by more frequent deliveries of smaller consignments.
Obviously neither in-transit inventory nor cycle stock can actually be reduced to zero. Even in the most advanced logistical organization, with JIT production and a high-speed delivery service, the fact is that transportation still takes a certain time, and in-transit costs remain. Furthermore, deliveries almost always involve more than one unit at a time, so newly arrived goods spend at least a short time in cycle stock before they are consumed.
While JIT supply, without any safety stock, is feasible in principle, it must be said that the overall goal of zero stock can never be fully achieved; nor, indeed, would one want to achieve it. Zero stock remains a limit one strives towards, without actually getting there.
It is essential to keep a clear distinction between the various demands that are made of the transport firm: transit time, size of consignments and punctuality, are three essentially different properties. They need not all go together, and they have a fundamentally different impact on stockholding costs. To keep down the cost of in-transit inventory, what matters is the average transit time, not the size of the load or punctuality. In contrast, to reduce cycle stock costs, the only one that matters is the size of consignments; to reduce safety stock, what counts is punctuality and transit time, not size.
Distinguishing between these three properties is important when it comes to making a logistical analysis of transportation decisions, as we shall see in the next chapter.
Footnotes
1 Standard works on logistics are for example BALLOU (1999), BOWERSOX (1978), KINNON (1989), CHRISTOPHER (2022) and RUSHTON, CROUCHER & BAKER (2022). An excellent manual is CHOPRA and MEINDL (2001).
2 Cf. LAMBERT & STOCK (1993), p. 403 ff.
3 The expression under the square root sign represents the variance in demand during order lead time. The formula was first applied to inventory management by FETTER, R. & DALLECK, W.C., (1961), pp. 105-108, and later followed by many authors. See, for instance, Lambert and STOCK, (1993) p. 415 or BALLOU (1985) p. 388. The formula applies only if there is independence both between lead time and daily demand and between successive daily demand levels. For a broader treatment see, for instance, ZINN, MARMORSTEIN & CHARNES (1992).
4 The given value of k is a linear approximation of a non-linear function. In order to calculate k, the normal density function is evaluated at the initial point of the interval and then kept constant. A manager can compile a similar table based on his own experiences. In Excel, K is calculated as =NORM.S.INV(1 - A2). The increase in K for a risk reduction of 0.0001 is calculated as =NORM.S.INV(1 - (A2 - 0.0001)) - NORM.S.INV(1 - A2).
5 FRENKEL, E., 1999. We have adapted the last column of the table. In the original publication, it contains the standard deviation. For our purpose, here, it is more convenient to mention the variance, which is the square of the standard deviation. A manager can compile a similar table based on his own experiences.
6 Cf. WINSTON (1995), p. 951.
7 Just-in-case inventory arises primarily from fluctuations in lead times, for example due to traffic congestion or other supply chain disruptions. See CHOI et al. (2023).
Bibliography
BALLOU, R.H., Business Logistics Management, Prentice Hall, Upper Saddle River, 1999, 681 p.
BOWERSOX, D.J., Logistical Management, Macmillan, New York/London, 1978, 528 p.
CHOPRA, S. & MEINDL, P., Supply Chain Management: Strategy, Planning and Operation, Prentice Hall, Upper Saddle River, 2001, 457 p.
CHOI, T.Y., NETLAND, T.H., SANDERS, N., SODHI, M.S., WAGNER, S., ‘Just-in-time for supply chains in turbulent times’, Productions and Operations Management, 2023, Vol. 32, No. 7, pp. 23312340.
CHRISTOPHER, M., Logistics & Supply Chain Management (5th Ed.), Pearson Education Limited, Harlow, 2016, 328 p.
ELLRAM, L. & TATE, W., ‘Cost avoidance: not everything that counts is counted’, Journal of Business Logistics, 2021, Vol. 42, No. 4, pp. 406-427.
ENZ, M. & LAMBERT, D., ‘A supply chain management framework for services’, Journal of Business Logistics, 2023, Vol. 44, No. 1, pp. 11-36.
FETTER , R. & DALLECK, W.C., Decision Models for Inventory Management, Irwin, Homewood, 1961.
FRENKEL, E., ‘The Economics of Total Trans-ocean Supply Chain Management’, International Journal of Maritime Economics, July-Sept 99, pp 61-69.
GEVAERS, R., VAN DE VOORDE, E. & VANELSLANDER, T., ‘Cost modelling and simulation of last-mile characteristics in an innovative B2C supply chain environment with implications on urban areas and cities’, Procedia – Social and Behavioral Sciences, 2014, Vol. 125, pp. 398-411.
HESHMATI, S., VERSTICHEL, J., ESPRIT, E. & VANDENBERGHE, G., ‘Alternative e-commerce delivery policies. A case study concerning the effects on carbon emissions’, EURO Journal on Transportation and Logistics, 2019, Vol. 8, pp. 217-248.
JACOBI, D., Guide to Supply Chain Management, Profile Books, London, 2009, 262 p.
KAPADIA, S., ‘To avoid product shortages, big retailers are scrapping reactive methods for AI, June 3, 2025, Business Insider. https://www.businessinsider.com/walmart-target-use-ai-to-preventinventory-shortages-2025-6?utm_source=chatgpt.com
KINNON, A.C., Physical Distribution Systems, Routledge, London/New York, 1989, 316 p.
LAMBERT, D.M. & STOCK, J.R., Strategic Logistics Management, Irwin, Homewood-Boston, 1993, 862 p.
LIU, Y., YIN, M. & HANSEN, M., ‘Economic costs of air cargo flight delays related to late package deliveries’, Transportation Research Part E: Logistics and Transportation Review, 2019, Vol. 125, pp. 388-401.
MASORGO, N., DOBRZYKOWSKI, D. & FUGATE, B., ‘Last-mile delivery: A process view, framework, and research agenda’, Journal of Business Logistics, 2024, Vol. 45, No. 4, e12397.
PAHWA, A. & JALLER, M., ‘A cost-based comparative analysis of different last mile strategies for e-commerce delivery’, Transportation Research Part E, 2024, Vol. 164, Article 102783.
RUSHTON, A., CROUCHER, P. & BAKER, P., The handbook of logistics and distribution management. Understanding the supply chain (7th ed.), Kogan Page, London, 2022, 824 p.
SALANI, M. & BATTARRA, M., ‘The opportunity cost of time window violations’, EURO Journal on Transportation and Logistics, 2018, Vol. 8, pp. 343-361.
SIVANKUTTY, S., & ELAHI, B. ‘Proposing a New Dynamic Safety Stock Adjustment System: Balancing Service Level Targets and Financial Constraints’, in: Proceedings 9th North American Conference on Industrial Engineering and Operations Management, IEOM Society International, Washington D.C., 2024.
SNOEK, A. & WINKENBACH, M., ‘The value of physical distribution flexibility in serving dense and uncertain urban markets’, Transportation Research Part A, 2020, Vol. 136, pp. 151-177.
WINSTON, W.L., Operations Research: Applications and Algorithms, Wadsworth, Belmont, 1994, 1318 p.
YALCIN, H. & DAIM, T., ‘Logistics, supply chain management and technology research: An analysis on the axis of technology mining’, Transportation Research Part E, 2022, Vol. 168, Article 102943.
YANG, L., LIU, K., ZHANG, J. & ZELBST, P., ‘Inventory management with actual palletized transportation costs and lost sales’, Transportation Research Part E, 2024, Vol. 184, Article 103462.
YU, W., Wong, C. Y., JACOBS, M. A., & CHAVEZ, R. ‘What are the right configurations of just-intime and just-in-case when supply chain shocks increase?’, International Journal of Production Economics, 2024, Vol. 276, 2024, Article 109352.
ZHAO, S., YANG, H., ZHENG, J., & LI, D., ‘A two-step approach for deploying heterogeneous vessels and designing reliable schedule in liner shipping services’, Transportation Research Part E: Logistics and Transportation Review, 2024, Vol. 182, Article 103416.
ZINN, W., MARMORSTEIN, H. & CHARNES, J., ‘The effect of autocorrelated demand on customer service’, Journal of Business Logistics, 1992, No. 1, pp. 173-192.
