CHOSEN ECONOMICAL ASPECTS OF VESSEL’S
OPERATIONAL SPEED
Kajetan Jackowski
Department of Ship Operation Gdynia Maritime University Poland
ABSTRACT
Determination of ship’s operational speed for various conditions of a voyage is dealt with in this paper. Major economical criteria of speed optimization are considered and analysed, as well as their modification due to occurence of speed limitation in a seaway. Examplary calculations of economically justifiable operational speed are given and graphs of costs and profits, as the functions of speed, are presented. The last paragraph of the paper contains a few practical remarks about dependance of optimal solutions on some external factors.
1. INITIAL REMARKS
Usually, by an operational speed there is meant full sea speed, which is so designed that the working point of ship’s main engine at this speed is technically most advantageous.
However, the scope of the notion “operational speed” may also comprise many other aspects of ship operating, including the main task, that is carriage of cargo for profits. Maximization of profits is an obvious goal. There are periods, however, (for instance ballast trips), when ship operating does not bring any profit. Therefore, in practice, maximization of profits may at times be replaced by minimization of costs, although the goal of increasing profits has always higher priority than keeping the costs down. Also in circumstances, when vessel’s speed should be adjusted according to the changeable requirements (of safety, of punctuality, or of the altered rotation), those adjustments may be treated as temporary modifications of operational speed.
At such approach the operational speed can be defined as the speed which is optimal for the task being performed – optimal in the meaning of assumed criteria and in accordance with alternant priorities attributed to these criteria (depending on
2. OPERATIONAL SPEED AND ECONOMICAL SPEED
In standard conditions of a voyage (if there is no call for extra measures to ensure safety of ship and cargo, or because of an altered rotation of the vessel), the operational speed (VO) can be deemed identical with economical speed(VE), which is most profitable for the ship operator. (In other words VE is the most effective speed from an economical point of view). Also in non-standard conditions, while special measures are taken to ensure safety of ship, or to adapt ship’s speed to new and urgent necessities, they may be deemed the absolute but transient constrains (until perils or emergencies persists) superimposed on the assumed function of economical optimization of the voyage.
3. MAJOR CRITERIA FOR DETERMINING
AN ECONOMICAL SPEED
As it was mentioned before, in order to determine properly ship’s operational speed for the given stage of voyage there must be considered two basic economical rules to be applied, that is m i n i m i z a t i o n of cost of a voyage and m a x i m i z a t i o n of profit–as well as possible interrelations between those rules. (Laytime in ports will not be taken into account).
3.1. Minimum cost criterion
The total cost (Z of a voyage) can be expressed as
Z = n ⋅ Z ,1 (1)
where Z is total daily cost and 1 n d= ⋅(24 )V −1 is the number of days needed to cover the whole distance d (from departure to destination),with a constant speed V. The total daily cost (Z ) is the sum 1
, 1 1
1 z z
Z = +Δ (2)
of daily cost of fuel z1 and other permanent daily costs Δ (i.e. expenses for crew z1 sustenance and wages, for running ship’s electricity plant, for maintenance, etc).
where coefficient c1, for a known cost z1 of daily fuel consumption at full ahead speedVFA,that is z (V1 FA),can be ascertained as
c1 = z (V1 FA) · (VFA)–2 (3a)
With regard to equations (2) & (3) the equation (1) can be transformed as follows: Z (V) = n · (c1 ·V2 +Δ ) = dz1 24⋅ (c1 ⋅ V + Δ ⋅ Vz1 –1), (4)
where d24 = d ⋅ (24)–1.
The total cost, as the function of speedZ (V) for0≤V≤VFA, reaches its minimum ZE, at the point where first derivative d Z /dV= 0, that is for d24⋅ c1– d24⋅ Δ ⋅z1 V–2= 0,
and then finally
V2 = c1⋅ (Δ )z1 –1 = (VE)2 (4a)
Therefore the economical speedVEensuring minimal cost ZEsatisfies equation
VE =
[
c1 ⋅ (Δz1)–1]
½ (5)If all permanent daily costs Δz1 and daily cost of fuel z1 are expressed in some arbitrarily defined covenant financial units[cfu], (for instance: 1cfu = present price of 1 ton of fuel), ship’s speed Vin knots[kn]andthe lengthdof ship’s route in nautical miles [Nm], then dimension of coefficientc1is[cfu]·[kn]2.
Fig. 1 shows an exemplary graph of function Z (V) for following values: z (V1 FA) = = 22.5cfu, VFA =15.0kn, Δz1=7.5cfu and d=2400Nm.
For these values: c1 = 0.1[cfu]·[kn]2, Z1(VFA) = z (V1 FA) +Δz1 =30.0cfu and total cost function Z (V)=10 · V + 750 · V–1.
Minimum of cost function,ZE=173.2cfu,is achieved at the speed VO ≡VE= 8⅔ kn.
3.2. Maximum profit criterion
From the economical point of view the minimization of total cost of voyage, discussed in subparagraph 3.1, should be considered an ancillary criterion – the principal one is always maximization of profits.
Fig. 1. An example of a voyage cost Z as a function of ship’s speed V 1 Z = Z (V =1 kn) = 760 [cfu] Z9= Z (V= 9 kn) = 173⅓ [cfu] 2 Z = Z (V= 2 kn) = 395 [cfu] Z = Z (V10 = 10 kn) = 175 [cfu] 4 Z = Z (V= 4 kn) = 227½ [cfu] Z12= Z (V= 12 kn) = 182½ [cfu] 6 Z = Z (V= 6 kn) = 185 [cfu] Z = Z (V14 = 14 kn) = 193.6 [cfu] 8 Z = Z (V= 8 kn) = 173¾ [cfu] Z15= Z (VFA = 15 kn) = 200 [cfu] E Z = Z (VE = 8⅔ kn) = 173.2 [cfu] Z20 = Z (V= 20 kn) = 237½ [cfu]
The total profit ΔΣ (i.e. net income) gained in a voyage can be calculated by deducting the voyage costs Z from the total income, that is from the sum Σof all freight revenues:
reduction of operational speed – and that means a loss of time because of prolonged spell of a voyage, time that could be used to earn freight dues in the next voyage. This drawback can be prevented by applying modified criterion of daily profit maximization, that is the profit(Δσ), which is the difference:
Δσ = σ1 − Z1 (7)
between total income per day
σ1 = n−1 ⋅ Σ = (d24 )−1 ⋅ V ⋅ Σ , (7a) and total daily cost Z1.
With the allowance for (2) & (3) it is possible to express the daily profit (7) as the following function of vessel’s speed:
Δσ(V)= 24 · d−1 ⋅ Σ · V − c1 ⋅ V2 − Δz1 (8) Equation(8)describes a second-degree curve, that reaches its maximum at the point where first derivative d(Δσ)/dV = 0, that is for 24 · d–1 · Σ − 2 · c1 · V= 0,
which means that economical speed VE satisfying criterion of maximal daily profit is
VE = 12 · d−1 · Σ · (c1)−1 (8a)
It is obvious that, in order to bring a profitΔσ, the daily income σ1must overcome total daily cost Z at any operational speed V1 O,that is any one between economical speed VE given by (5) and full sea speed VFA. This requirement is tantamount to satisfying inequity:
σ1 > c1· (VFA)2 + Δ (9)z1
For the same data as in case of function Z (V) presented in Fig. 1, inequity (9) takes the form
σ1 > 0.1 · 152+7.5=30[cfu]
and spell of the voyage (at full speed VFA =15kn), in days, is n= 6⅔.
It means that shipment of cargo shall be profitable for freight rates ensuring total income
Σ > n · Z (V1 FA) = 200[cfu], (9a)
where daily profit is defined by relationship:
Δσ(V)= 0.01 · Σ · V − 0.1 · V2 − 7.5 (9b)
Graphs of function (9b), for five various cases of total income Σ [cfu]: (1) for Σ= 100cfu, (2) for Σ= 200cfu, (3) for Σ=300cfu, (4) for Σ=400cfuand(5) for Σ=500cfu, (where income is expressed in covenant financial units [cfu]), are shown
Fig. 2. An examplary graph of daily profit Δσ as a function of operational speed VO Maximal levels of daily profit:
function (1) for Σ = 100 [cfu] → max(Δσ) = – 5 [cfu] (minimal loss), at VE = 5kn function (2) for Σ = 200 [cfu] → max(Δσ) = +2.5 [cfu] (max. profit), at VE = 10kn function (3) for Σ = 300 [cfu] → max(Δσ) = +15.0 [cfu] (max. profit), at VE = 15kn function (4) for Σ = 400 [cfu] → max(Δσ) = +32.5 [cfu] (max. profit), at VE = 20kn function (5) for Σ = 500 [cfu] → max(Δσ) = +55.0 [cfu] (max. profit), at VE = 25kn
3.3. Hazard factors and speed modifications
Fig. 3. An examplary graph of daily profit Δσ as a function of operational speed VO with the constrain: Vmax = 10kn = ⅔VFA
function (1) for Σ = 100 [cfu] → max(Δσ) = – 6.4 [cfu] (minimal loss), at VE = 2.2(2)kn function (2) for Σ = 200 [cfu] → max(Δσ) = – 3.1 [cfu] (minimal loss), at VE = 4.4(4)kn function (3) for Σ = 300 [cfu] → max(Δσ) = +2.5 [cfu] (max. profit), at VE = 6.6(6)kn function (4) for Σ = 400 [cfu] → max(Δσ) = +10.3 [cfu] (max. profit), at VE = 8.8(8)kn function (5) for Σ = 500 [cfu] → max(Δσ) = +20.3 [cfu] (max. profit), at VE = 11.1(1)kn Fig. 3presents the graphs of daily profitΔσ(V) for the same data as in the case of Fig. 1 and Fig. 2, but in a situation,where due to prevailing external conditions the maximal speed drops, despite setting full ahead, to Vmax = ⅔VFA = 10kn (or to proportionally slower speeds for lower telegraph settings). Making allowance for it means multiplying coefficientc1 by (VFA ⁄Vmax)2 and then the general equation (8) takes the form:
Δσ(V)= 24 · d−1 · Σ · V − c1 · (VFA ⁄ Vmax)· V2 −Δ , z1 (10) The application of multiplier (VFA ⁄ Vmax)2 to formerly adopted data gives
and profit function: Δσ(V)= 0.01 · Σ · V − 0.225 · V2 − 7.5 (10b) It must be noted, however, that if the reduction of speed is not forced by external factors, but is a result of navigator’s decision due to imminent perils (for example fog, or risk of collision), then the assumed daily profit function is given by (8), but within limited range of admissible operational speeds.
4. CONCLUDING REMARKS
Comparison of function (9b) with (10b) indicates, that externally forced limitations of speed are essential for the choice of optimal operational speed of a vessel.
(To set an example: for total freight incomeΣ=300cfu and no forced speed limitations, the economical speedVE =VFA =15.0kn =Vmax;in the case of restriction Vmax = ⅔VFA = 10kn the economically optimal speed will change toVE = 6.6(6)kn≠VFA =10.0kn = Vmax). Of key importance is also ratio of total income (Σ) to the sum of all voyage costs (Z ). The higher the income level in comparison to the level of cost, the more profitable is time saving instead of cost reduction.In this question an important indication ensuing from (8a) is the value of economical speed, VE, which maximizesdaily profit Δσ. If VE is greater thanVFA, then the maximum of daily profit, max(Δσ), is attained at the highest possible speed Vmax and thus the operational speed should be VO= Vmax. However, in the case of total income only several per cent greater than the sum of incurred costs(i.e. slightly above the level of profitability) the economical speed VE is lower than attainable maximal speed (Vmax) − and this means that maximization of profit, to get max(Δσ) per each day of the voyage, is simultaneous with some reduction of daily cost of fuel consumption.
REFERENCES
[1] Caplan B., The Economics and Philosophy of the Cruise Ship, Library of Economics and Liberty, July 21, 2005.
[2] Heggstad K. M., The Economic Speed of Warships and Patrol Vessels, Naval Engineers Journal, March 18, 2009.
[3] Jurdziński M., Nawigacyjne planowanie podróży, Wydawnictwo Morskie, Gdańsk 1989. [4] Klanac A., Nikolić P., Kovac M., McGregor J., Economics and Environmental Impact
of Ship Speed Reduction, Proceedings of XIX SORTA Conference, 2010.
