# Problem formulation of optimal dispatching for laden and empty containers

The following notation is used in this section:

*N:* the container fleet size in the system, which is a constant (container

leasing is not considered).

a random variable representing the customer demands arriving at port *i* during one week.

the number of demands arrived at port *i* during the (и—l)th week, which is a sample of

*Sj **„:* the inventory level of empty containers at port *i* at the beginning of the nth week.

y, the number of laden containers dispatched from port *i* to the other in

the nth week.

*Xj„:* the number of empty containers repositioned from port *i* to the other in the «th week.

r_{(}: the vessel carrying capacity from port *i* to the other port in each week.

Cf: the empty container inventory holding cost per unit per week at port *i. Cf:* the empty container repositioning cost per unit from port *i* to the other. *Cf:* the lost-sale penalty cost per unit at port *i.*

It is assumed that and ^{are} independent because they represent the weekly demands at two different ports. The variables у,- „ and *Xj„* are decision variables. The variables *(s„,* $2,н) represent the inventory state in the dynamic system. It should be pointed out that + 5_{2},1 ^{=} N, which reflects the initial distribution of the container fleet over two ports. The sequence of events in each week is described as follows:

i.) At the beginning of the week, the vessel operator makes decisions on how many empty containers to meet customer demands that have been received in the previous week.

ii.) At the beginning of the week, the operator makes decisions on how many empty containers to be repositioned to the other port.

iii. ) At the beginning of the week, the above container dispatching decisions

are executed by taking into account the current inventory level of empty containers and the vessel carrying capacity; the vessels carry laden and empty containers and sail to the other port.

iv. ) At the beginning of the week, unmet customer demands will be lost and

incur lost-sale penalty.

v. ) At the end of the week, the laden and empty containers arrive at the

destination ports.

vi. ) At the end of the week, the laden containers are unpacked and become

the inventories of empty containers; the arrived empty containers also become inventories.

vii. ) At the end of the week, the operator receives customer demands for the

current week.

To simplify the narrative, let и; ,, := Xj ,, + yj ,,, which represents the sum of empty and laden containers dispatched from port *i* to the other port in week *n.* The evolution of the system state (sj *S**2** _{n})* can be then described by

where Si „+i is the inventory level of empty containers at port 1 at the beginning of the (n+l)th period. The inventory level of empty containers at port 2 at the beginning of the (n+l)th period is given by S2,n+i ^{= s}*2,n ~ ^{u}2,n* +

*+*

^{u},n*Щ,п*

^{or} = f^{4}

Due to the constraints of the shipping capacity and the empty container availability, we have 0 < *щ „* < min(r„ s,Note that meeting customer demands is always more important than repositioning empty container. This implies that customer demands should be satisfied as much as possible before repositioning empties. This is intuitively true as laden container movements will generate revenue, whereas empty container repositioning will only incur costs. Since at the beginning of the nth period, the customer demands to meet are given by *d n* and (/2,10 which represent the total demands received during the (n—l)th period at terminals 1 and 2 respectively, we can derive x, „ and у,- „ as follows:

It follows that the unmet demands (or lost sales) are given by - у,- „ = max(0, *dj „* — «, ,,). Thus, instead of considering decisions for laden and empty containers separately, we can simplify the formulation by only considering the decisions и,-

The one-step cost function (i.e. the costs incurred during the »th week) can be given by

The first term on the right-hand side (RHS) represents the empty container inventory holding costs; the second term on the RHS represents the lost-sale penalty costs; and the third term on the RHS represents the empty container repositioning costs.

The problem is to find the optimal dispatching policy *u:=* {«] ,„ «2,,, | *n* = 1, 2, ...} by minimising the long-run average cost,