Appendix 1: SOS Voyager mathematical model
In this model, it is assumed that each ship starts its voyage at homeport (open event) and returns to its homeport (close event). In this model let:
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\(S = \left\{ {1,2,3, \ldots ,s_{0} } \right\}\) be the set of ships,
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\(P = \left\{ {1,2,3, \ldots ,p_{0} } \right\}\) be the set of ports of a working trade area,
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\(Q = \left\{ {1,2,3, \ldots ,q_{0} } \right\}\) be the set of cargoes available for transport between ports of this area. It is assumed that cargoes are compatible with the ship carrying them and can be mixed on board the ship with ship stability maintained. Each cargo \(r \in Q\) has a loading event and a discharging event,
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\(L = \left\{ {1,2,3, \ldots ,l_{0} } \right\}\) be a set of loading events, one for each cargo,
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\(D = \left\{ {1,2,3, \ldots ,d_{0} } \right\}\) be a set of discharging events, one for each cargo,
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F = {f} be a one-element set of open event f.
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G = {g} be a one-element set of close event g,
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\(E = L \cup D\) be the set of load and discharge events, combined,
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\(E_{f} = E \cup F\) be the set of open, load, and discharge events, combined,
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\(E_{g} = E \cup G\) be the set of load, discharge, and close events, combined,
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\(E_{fg} = E_{f} \cup G\) be the set of open, load, discharge, and close events, combined.
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pi be port \(p \in P\) identified at event \(i \in E_{fg}\),
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\(Z = \left\{ {1,2,3,4} \right\}\) be an index representing two combined positions: ‘pass or bypass Suez or Panama Canal’ as alternative route position, and ‘laden or ballast’ as ship load position. Z element of ‘1’ represents ship passing canal while in laden position, ‘2’ represents ship bypassing canal while in laden position, ‘3’ represents ship passing canal while in ballast position, and ‘4’ represents ship bypassing canal while in ballast position.
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\(p_{ijz}^{k}\) be the gross profit earned by ship \(k \in S\) on transport link ij while in position \(z \in Z\). Gross profit equals freight plus demurrage (based on reversible or irreversible calculation), minus cooling/heating cost of cargo \(r \in Q\) at \(i \in L\), minus handling cost of cargo \(r \in Q\) at \(i \in E\), minus dispatch (based on reversible or irreversible calculation), minus port dues of port \(p \in P\) at \(i \in E_{f}\), where \(p_{i} \ne p_{j}\), and minus canal/strait dues and fuel consumption of main engine when sailing transport link ij while in position \(z \in Z\), where \(p_{i} \ne p_{j}\),
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\(T_{g}^{k}\) be voyage close day of ship \(k \in S\),
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\(T_{i + }^{k}\) be the delayed days of ship \(k \in S\), when it arrives later than the laycan close date at event \(i \in E\),
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\(C_{i + }^{k}\) be the delayed penalty of ship \(k \in S\), when it arrives later than the laycan close date at event \(i \in E\),
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\(T_{s}^{k}\) be voyage slack days of ship \(k \in S\), when it arrives earlier than the laycan open date, aggregated for all \(i \in E\),
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\(C_{s}^{k}\) be voyage slack cost per day of ship \(k \in S\), when it arrives earlier than the laycan open date,
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\(C_{g}^{k}\) be the cost of fuel consumption of auxiliary engine per day plus daily fixed cost of ship \(k \in S\),
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\(C_{0}^{k}\) be voyage fixed cost of ship \(k \in S\), not considered elsewhere,
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\(x_{ijz}^{k}\) be the problem decision variable. It equals 1 if ship \(k \in S\) sails transport link ij while it is in position \(z \in Z\), and it equals zero otherwise. If \(x_{ijz}^{k} = 1\) and \(i \in E\), cargo \(r \in Q\) is loaded on board ship k, where i is its loading port, or discharged from the ship if i is its discharging port. Likewise, if \(x_{ijz}^{k} = 1\) and \(j \in E\), cargo \(r \in Q\) is loaded on board ship k, where j is its loading port, or discharged from the ship if j is its discharging port,
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yi be another problem decision variable, alternative to \(x_{ijz}^{k}\). It equals 1 if \(x_{ijz}^{k} = 0\) for all ships sailing all transport links to pick up cargo \(r \in Q\) at \(i \in L\), and it equals zero otherwise. Cargo \(r \in Q\) at \(i \in L\) is transported by a chartered-in ship in this case. Variable yi is introduced to represent the possibility of carrying a cargo at event i by a chartered-in ship rather than the owned ships, where yi = 1 in this case. If this happens, all the variables representing the possibility of carrying the cargo by the owned ships should equal to zero. For the chartered-in ship, let Pi be its voyage gross profit, ti be its voyage time, Ci0 be its voyage fixed cost, and ti0 be its voyage fixed time. Each one of these parameters is to have a value ≠ 0 if the chartered-in ship is taken as an alternative and a value = 0 otherwise,
It is required to maximize the sum of voyage gross profit per day for all ships, given by:
$$\begin{aligned} G & = \mathop \sum \limits_{k \in S} \left( {\mathop \sum \limits_{i \in Ef} \mathop \sum \limits_{j \in Eg} \mathop \sum \limits_{z \in Z} p_{ijz}^{k} x_{ijz}^{k} - \mathop \sum \limits_{i \in E} C_{i + }^{k} T_{i + }^{k} - C_{s}^{k} T_{s}^{k} - C_{g}^{k} T_{g}^{k} - C_{0}^{k} } \right) / T_{g}^{k} \\ & \quad + \mathop \sum \limits_{i \in L} \left( { P_{i }^{ } y_{i}^{ } - C_{i0}^{ } } \right)/(t_{i }^{ } y_{i}^{ } + t_{i0}^{ } ). \\ \end{aligned}$$
(1)
Subject to:
Flow constraints
Using the above-mentioned denotations, the flow constraints can be formulated as follows:
-The flow constraints, which restrict the flow of transport links for each ship, originating from open event, to only one link at most, given by:
$$\mathop \sum \limits_{j \in Eg } \mathop \sum \limits_{z \in Z} x_{fjz}^{k} \le 1,\quad k \in S,$$
(2)
-Flow constraints, which restrict the flow of transport links for each ship towards event \(e \in E\) to be equal to the flow of transport links outward from this event, given by:
$$\mathop \sum \limits_{i \in Ef} \mathop \sum \limits_{z \in Z} x_{iez}^{k} = \mathop \sum \limits_{j \in Eg} \mathop \sum \limits_{z \in Z} x_{ejz}^{k} , \quad e \in E,\quad {\text{and}}\quad k \in S,$$
(3)
-Flow constraints, which restrict the flow of transport links for each ship towards load event \(l \in L\) of cargo \(r \in Q\) to be equal to the flow of transport links towards discharging event \(d \in D\) of same cargo, given by:
$$\mathop \sum \limits_{i \in Ef} \mathop \sum \limits_{z \in Z} x_{ilz}^{k} = \mathop \sum \limits_{i \in E} \mathop \sum \limits_{z \in Z} x_{idz}^{k} ,\quad l \in L,\quad d \in D,\quad l\quad {\text{and}}\,d\,{\text{are}}\,{\text{of}}\,{\text{same}}\,{\text{cargo}}\, r \in Q\quad {\text{and,}}\quad k \in S,$$
(4)
-Flow constraints, which prohibit the flow of transport link of each ship in two opposite directions, given by:
$$\mathop \sum \limits_{z \in Z} x_{ijz}^{k} + \mathop \sum \limits_{z \in Z} x_{jiz}^{k} \le 1,i, j \in E, \quad {\text{and}}\quad k \in S,$$
(5)
-Flow constraints, which restricts the flow of transport link of each ship passing by the en-route bunkering port to only one link (optional):
$$\mathop \sum \limits_{i \in Ef} \mathop \sum \limits_{j \in Eg} \mathop \sum \limits_{z \in Z} x_{ijz }^{k} = 1,\quad k \in S,\quad {\text{where}}\,{\text{link}}\,{\text{i,j}}\,{\text{passes}}\,{\text{by}}\,{\text{the}}\,{\text{en - route}}\,{\text{bunkering}}\,{\text{port,}}$$
(6)
-Flow constraints, which restrict the flow of transport links for ship \(k \in S\) towards load event \(l \in L\) of cargo \(r \in Q\) to be equal to one if the cargo is booked and be carried by this ship, given by:
$$\mathop \sum \limits_{i \in Ef} \mathop \sum \limits_{z \in Z} x_{ilz}^{k} = 1,\quad l \in L,\quad l\,{\text{is}}\,{\text{of}}\,{\text{same}}\,{\text{cargo}}\,r \in Q,\quad {\text{and}}\quad k \in S,$$
(7)
-Flow constraints, which restrict the flow of transport links for ship \(k \in S\) towards discharging event \(d \in D\) of cargo \(r \in Q\) to one if the cargo is already carried by this ship, given by:
$$\mathop \sum \limits_{i \in E} \mathop \sum \limits_{z \in Z} x_{idz}^{k} = 1,\quad {\text{d}} \in D,\quad d\,{\text{is}}\,{\text{of}}\,{\text{same}}\,{\text{cargo}}\, r \in Q,\quad {\text{and}}\quad k \in S,$$
(8)
-Flow constraints, which restrict the flow of transport links of all ships towards loading event \(l \in L\) of cargo \(r \in Q\) plus their alternative decision of acquiring a charter-in ship, to only one at most, given by:
$$\mathop \sum \limits_{k \in S} \mathop \sum \limits_{i \in Ef} \mathop \sum \limits_{z \in Z} x_{ilz }^{k} + h_{l}^{ } y_{l}^{ } \le 1, \quad l \in L,\quad h_{l} = 1\quad {\text{if}}\,y_{l} \,{\text{is}}\,{\text{taken}}\,{\text{as}}\,{\text{an}}\,{\text{alternative}}\,{\text{decision}}\,{\text{and}}\,h_{l} \, = \,0\,{\text{otherwise}}.$$
(9)
Capacity constraints
Let:
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wi be weight of cargo \(r \in Q\) at event \(i \in E\), in mt,
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vi be volume of cargo \(r \in Q\) at event \(i \in E\), in cum (if non-container),
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ni be number of TEU of cargo \(r \in Q\) at event \(i \in E\) (if container),
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\(W_{i}^{k}\) be the remaining dwt capacity of ship \(k \in S\) after load or discharge of cargo \(r \in Q\) at event \(i \in E\), in mt,
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\(W_{0}^{k}\) be the min weight remaining on board ship \(k \in S\) which keeps the ship in laden position,
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\(V_{i}^{k}\) be the remaining volume capacity of ship \(k \in S\) after load or discharge of cargo \(r \in Q\) at event \(i \in E\), in cum (if non-container),
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\(N_{i}^{k}\) be the remaining TEU capacity of ship \(k \in S\). after load or discharge of cargo \(r \in Q\). at event \(i \in E\) (if container),
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Wk be the dead weight capacity of ship \(k \in S\),
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Vk be the volume capacity of ship \(k \in S\) (if non-container),
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Nk be the TEU capacity of ship \(k \in S\) container),
Using the above-mentioned denotations, the capacity constraints can be formulated as follows:
-Load remaining weight constraints which restrict remaining weight on board each ship at end event \(j \in E\) to be at least equal to remaining weight at start event \(i \in L\) of any transport link minus weight of cargo \(r \in Q\) at \(i \in L,\) given by:
$$W_{j}^{k} \ge W_{i}^{k} - w_{i} \mathop \sum \limits_{z \in Z} x_{ijz}^{k} ,\quad i \in L,\quad j \in E,\quad {\text{and}}\quad k \in S,\quad {\text{where}}\quad \mathop \sum \limits_{z \in Z} x_{ijz}^{k} = 1,$$
(10)
Constraints (10) can be re-written as follows:
$$M\left( {1 - \mathop \sum \limits_{z \in Z} x_{ijz}^{k} } \right) + W_{j}^{k} \ge W_{i}^{k} - w_{i} \mathop \sum \limits_{z \in Z} x_{ijz}^{k} , \quad i \in L,\quad j \in E\quad {\text{and}}\quad k \in S,$$
where M is a large number. So \(W_{j}^{k} \ge W_{i}^{k} - w_{i} \sum\limits_{z \in Z} {x_{ijz}^{k} }\) will hold true only when \(\sum\limits_{z \in Z} {x_{ijz}^{k} } = 1\).
-Load remaining volume constraints, which restrict remaining volume on board each non-container ship at end event \(j \in E\) to be at least equal to remaining volume at start event \(i \in L\) of any transport link minus volume of cargo \(r \in Q\) at event \(i \in L\) given by:
$$V_{j}^{k} \ge V_{i}^{k} - v_{i} \mathop \sum \limits_{z \in Z } x_{ijz}^{k} ,\quad i \in L,\quad j \in E,\quad {\text{and}}\quad k \in S,\quad {\text{where}}\quad \mathop \sum \limits_{z \in Z} x_{ijz}^{k} = 1,$$
(11)
-Load remaining TEU constraints, which restrict remaining TEU on board each container ship at end event \(j \in E\). to be at least equal to remaining TEU at start event \(i \in L\) of any transport link minus TEU of cargo \(r \in Q\) at event \(i \in L\) given by:
$$N_{j}^{k} \ge N_{i}^{k} - n_{i} \mathop \sum \limits_{z \in Z} x_{ijz}^{k} ,\quad i \in L,\quad j \in E,\quad {\text{and}}\quad k \in S,\quad {\text{where}}\quad \mathop \sum \limits_{z \in Z} x_{ijz}^{k} = 1,$$
(12)
-Discharge remaining weight constraints, which restrict remaining weight on board each ship at end event \(j \in E\) to be at least equal to remaining weight at start event \(i \in D\) of any transport link plus weight of cargo \(r \in Q\) at event \(i \in D,\) given by:
$$W_{j}^{k} \ge W_{i}^{k} + w_{i} \mathop \sum \limits_{z \in Z } x_{ijz}^{k} ,\quad i \in D,\quad j \in E,\quad {\text{and}}\quad k \in S,\quad {\text{where}}\quad \mathop \sum \limits_{z \in Z} x_{ijz}^{k} = 1,$$
(13)
-Discharge remaining volume constraints, which restrict remaining volume on board each non-container ship at end event \(j \in E\) to be at least equal to remaining volume at start event \(i \in D\) of any transport link plus volume of cargo \(r \in Q\) at event \(i \in D,\) given by:
$$V_{j}^{k} \ge V_{i}^{k} + v_{i} \mathop \sum \limits_{z \in Z} x_{ijz}^{k} , \quad i \in D,\quad j \in E,\quad {\text{and}}\quad k \in S,\quad {\text{where}}\quad \mathop \sum \limits_{z \in Z} x_{ijz}^{k} = 1,$$
(14)
-Discharge remaining TEU constraints, which restrict remaining TEU on board each container ship at end event \(j \in E\) to be at least equal to remaining TEU at start event \(i \in D\) of any transport link plus TEU of cargo \(r \in Q\) at event \(i \in D,\) given by:
$$N_{j}^{k} \ge N_{i}^{k} + n_{i} \mathop \sum \limits_{z \in Z} x_{ijz}^{k} , \quad i \in D,\quad j \in E,\quad {\text{and}}\quad k \in S,\quad {\text{where}}\quad \mathop \sum \limits_{z \in Z} x_{ijz}^{k} = 1,$$
(15)
-Weight capacity constraints, which restrict remaining weight on board each ship after discharge of all cargoes at, end event \(g \in G\) so that it does not exceed ship dwt capacity, given by:
$$W_{i}^{k} \ge W^{k} ,\quad i \in D,\quad {\text{and}}\quad k \in S,\quad {\text{where}}\quad \sum\limits_{z = 3,4} {x_{igz}^{k} = 1} ,\quad g \in G,$$
(16)
-Volume capacity constraints, which restrict remaining volume on board each non-container, ship after discharge of all cargoes at end event \(g \in G\) so that it does not exceed ship volume capacity, given by:
$$V_{i}^{k} \ge V^{k} ,\quad i \in D,\quad {\text{and}}\quad k \in S,\quad {\text{where}}\quad \mathop \sum \limits_{z = 3,4} x_{igz}^{k} = 1,\quad g \in G,$$
(17)
-TEU capacity constraints, which restrict remaining TEU on board each container ship after discharge of all cargoes at, end event \(g \in G\) so that it does not exceed ship TEU capacity, given by:
$$N_{i}^{k} \ge N^{k} ,\quad i \in D,\quad {\text{and}}\quad k \in S,\quad {\text{where}}\quad \mathop \sum \limits_{z = 3,4} x_{igz}^{k} = 1,\quad g \in G,$$
(18)
-Laden-or-ballast load position constraints which restricts ship load position to either laden or ballast. Ship is assumed to be in laden position on transport link ij if \(i \in L\), and is considered so if \(i \in D\) and remaining weight on board the ship at this event is greater or equal to the min remaining weight \(W_{0}^{k}\), which is given by:
$$W_{i}^{k} \ge W_{0}^{k} ,\quad i \in D,\quad {\text{and}}\quad k \in S,\quad {\text{where}}\quad \mathop \sum \limits_{z = 1,2} x_{ijz}^{k} = 1,\quad j \in E,$$
(19)
Time constraints
Let:
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ai be laycan open day of cargo \(r \in Q\) at event \(i \in E\),
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bi be laycan close day of cargo \(r \in Q\) at event \(i \in E\),
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\(t_{i}^{k}\) be the number of days taken to handle cargo \(r \in Q\) at event \(i \in E\) by ship \(k \in S\) plus waiting days at port \(p \in P\) at event \(i \in E\),
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\(t_{ijz}^{k}\) be the number of days taken to sail the transport link from event \(i \in E_{f}\) to event \(j \in E_{g}\) by ship \(k \in S\)
while it is in position \(z \in Z\), plus waiting days at sea, where \(p_{i} \ne p_{j}\),
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\(T_{i}^{k}\) be the arrival day of ship \(k \in S\) at event \(i \in E_{fg}\), assuming \(T_{f}^{k} = 0\),
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\(T_{0}^{k}\) be the voyage fixed days of ship \(k \in S\), not considered elsewhere,
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Tk be the total allowable days of ship \(k \in S\),
Using the above-mentioned denotations, the time constraints can be formulated as follows:
-Event arrival time constraints which restrict arrival day at end event \(j \in E_{g}\) to be at least equal to arrival day at start event \(i \in E_{f}\) of any transport link plus handling days of cargo \(r \in Q\) at \(i \in E_{f}\), waiting days in port \(p \in P\) at \(i \in E_{f}\), sailing days on link ij, and waiting days at sea, given by:
$$T_{j}^{k} \ge T_{i}^{k} + t_{i} + \mathop \sum \limits_{z \in Z} t_{ijz }^{k} x_{ijz}^{k} ,\quad i \in E_{f} ,\quad j \in E_{g} , \quad {\text{and}}\quad k \in S,\quad {\text{where}}\quad t_{ifz}^{k} = 0, \quad {\text{and}}\quad \mathop \sum \limits_{z \in Z} x_{ijz}^{k} = 1,$$
(20)
-Event time precedence constraints, which control arrival times, so that arrival day at discharge event \(d \in D\) succeeds arrival day at load event \(l \in L\) of cargo \(r \in Q\), given by:
$$T_{d}^{k} \ge T_{l}^{k} ,\quad l \in L,\quad d \in D,\quad l\,{\text{and}}\,d\,{\text{are}}\,{\text{of}}\,{\text{same}}\,{\text{cargo}}\, r \in Q,\quad {\text{and}}\quad k \in S,\quad {\text{where}}\quad \mathop \sum \limits_{i \in E} \mathop \sum \limits_{z \in Z} x_{idz}^{k} = 1$$
(21)
-Time window constraints, which restrict ship arrival day at event \(j \in E\) so that it does not violate cargo laycan open and close days at this event, given by:
$$T_{j}^{k} \ge a_{j} ,\quad j \in E,\quad {\text{and}}\quad k \in S,\quad {\text{where}}\quad \mathop \sum \limits_{i \in Ef} \mathop \sum \limits_{z \in Z} x_{ijz}^{k} = 1,$$
(22)
$$T_{j}^{k} \le b_{j } + T_{j + }^{k} ,\quad j \in E,\quad {\text{and}}\quad k \in S,\quad {\text{where}}\quad \mathop \sum \limits_{i \in Ef} \mathop \sum \limits_{z \in Z} x_{ijz}^{k} = 1,$$
(23)
-Closing time constraints which restrict final closing day for each ship so that it equals total cargo handling days and waiting days in port, sailing days and waiting days at sea, waiting days before cargo open day, and voyage fixed days, given by:
$$\mathop \sum \limits_{i \in Ef} \mathop \sum \limits_{j \in Eg} \mathop \sum \limits_{z \in Z} (t_{i}^{k} + t_{ijz}^{k} ) x_{ijz}^{k} + T_{s}^{k} + T_{0}^{k} = T_{g}^{k} ,\quad k \in S,$$
(24)
-Allowable closing time constraints, which restrict closing day for each ship to a maximum allowable day, given by:
$$T_{g}^{k} \le T^{k} , \quad g \in G,\quad k \in S,\quad {\text{where}}\quad \mathop \sum \limits_{z = 3,4} x_{igz}^{k} = 1\quad {\text{and}}\quad i \in D,$$
(25)
Non-negativity and integrality constraints
-Non-negativity constraints of continuous variables, given by:
$$W_{i}^{k} ,V_{i}^{k} ,N_{i}^{k} ,\quad T_{i}^{k} \ge 0,\quad i \in E_{g} ,\quad k \in S,\quad T_{s}^{k} \ge 0,\quad k \in S,$$
(26)
-integrality constraints of integer variables, given by:
$$x_{ijz}^{k} = 0,1,\quad i \in E_{f} ,\quad j \in E_{g} ,\quad k \in S,$$
(27)
$$\begin{aligned} & \mathop \sum \limits_{z \in Z} x_{ijz}^{k} \le 1,\quad i \in E_{f} ,\quad j \in E_{g} ,\quad k \in S, \\ & y_{i}^{ } = 0,1,\quad i \in L. \\ \end{aligned}$$
(28)
Appendix 2: The new stochastic cargo soft time windows of SOS Voyager mathematical model
The new stochastic version of the cargo soft time windows mentioned in “Appendix 1”—formula (22) and (23), can be described using the following simple denotations, assuming one ship and one cargo:
For either the cargo loading or discharging ports, the arrival time of this ship is unconfirmed and assumed to be a random variable having a known probability distribution. The probability distribution is the marginal distribution of the arrival time. Let:
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A be the random laycan open date, either for the cargo loading or discharging port, expressed in days,
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a be the deterministic-equivalent laycan open date, either for the cargo loading or discharging port, expressed in days,
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B be the random laycan close date, either for the cargo loading or discharging port, expressed in days,
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b be the deterministic-equivalent laycan close date, either for the cargo loading or discharging port, expressed in days,
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Tb+ be the late days when the ship arrives after B,
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Pa be the least probability the ship owner stipulates the ship arrives after A,
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Pb be the least probability the ship owner stipulates the ship arrives before B + Tb+,
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y be the ship arrival date, expressed in days.
In the stochastic model, the probability of the ship arriving at date A or after; \({\text{Prob}}.\left\{ { y \ge A} \right\}\), must be greater or equal to Pa, as indicated by:
$${\text{Prob}}{.} \left\{ { y \ge A} \right\} \ge P_{a} ,$$
(29)
Whereas, the probability of the ship arriving at date B + Tb+ or before; \({\text{Prob}}.\left\{ { y \le B + T_{b + } } \right\}\), must be greater or equal to Pb, as indicated by:
$${\text{Prob}}., \left\{ { y \le B + T_{b + } } \right\} \ge P_{b}$$
(30)
The stochastic constraint (29) and (30) are considered when the ship arrival time A or B is a random variable and Pa or Pb is a probability value.
To elaborate more, take constraint (30) as an example. It says: the probability of arriving at or before the laycan close date must be at least equal to Pb. If at B = b the descending cumulative probability of ship arrival time has a value just greater or equal to Pb, then (30) can be expressed as:
$$y \le \underline {b} + T_{b + }$$
(31)
Constraint (31) is the deterministic-equivalent constraint to the one given by the deterministic model, where b is the deterministic laycan close date. The difference between them is that b is the confirmed value of the laycan close date, while b in (31) is the deterministic-equivalent value of the random laycan close date, as described earlier. To illustrate, assume for discrete arrival time B, Prob. {B < 5 days} = 0.0, Prob. {B = 5 days} = 0.2, Prob. {B = 10 days} = 0.5, Prob. {B = 15 days} = 0.3, and Prob. {B > 15 days} = 0.0. According to the additive rule of the probability theory, the arrival time descending cumulative probability distribution reads: Prob. {B ≥ 5 days} = 0.2 + 0.5 + 0.3 + 0.0 = 1.0, 0.8 ≤ Prob. {B ≥ 10 days} < 1.0, and 0.3 ≤ Prob. {B ≥ 15 days} < 0.8. Now suppose Pb = 0.9. This value falls in second class, which implies a deterministic-equivalent arrival time value of 10 days (neither 5 nor 15 days), i.e. at b = 10.
Use the same illustration mentioned above to convert cargo quantities in Table 4 and arrival times in Table 5 to deterministic-equivalent quantities and arrival times as shown in Table 6.
