Part I Introduction
7.2 Classification of Transportation Problems
railcars that are considered a single unit with the same origin but perhaps different final destinations. Using blocks in train transportation systems has many economic advantages such as full train loads and the management of longer car strings in yards.
During the long-haul transportation trip, the train may travel on single-track lines, so it is common to meet trains traveling in the opposite direction. In this situ- ation, the train with the higher priority passes first. The train may be stopped in middle train yards where cars and engines are regularly inspected and blocks are separated from one train and put on another. At the final yard, the first blocks are detached from the train and disassembled. Then cars are inspected, put in order, and moved to their final destination to be unloaded. Once a car finishes its delivery trip, it may move to a new pickup point and then be assembled in a new block or it may wait empty for a future assignment.
Managing the main yard’s operations is the most complex activity in a long- haul railway transportation system.
LTL Transportation Network
In an LTL network, small vehicles pick up local traffic at origin points and deliver it to end-of-line terminals. Then local traffic from different parts of the network are grouped and consolidated into larger batches before they begin their long-haul journey. Breakbulks are terminals where arrivals from several origin points are gathered, unloaded, ordered, and consolidated for the rest of the long-haul transport [4]. Breakbulks in LTL networks are the same as main yards in rail-transportation systems. LTL carriers usually have their own terminals, but they use public transportation networks.
Railway and LTL System Differences
The structure of LTL transportation network basically is the same as a rail- transportation network but in simpler scale and with more flexibility in choosing ways to move materials to their destinations. Whereas rail-transportation links are limited, trucks may use any available link of the road and highway network while obeying weight regulations [4]. In LTL systems, terminal operations are generally simple. But in railway systems, more complicated consolidation operations are managed through grouping and consolidation of railcars into blocks and then into trains [7].
Freight transportation plans as decision making problems involve many different variables and constraints. Some of them can be applied for all transportation sys- tems, whereas others are only relevant to specific modes or particular ways of sys- tem operation. For example, the vehicle and drivers scheduling problem is a common decision problem through which there is a least-cost allocation of vehicles and drivers over time. In this way, some constraints such as rules and guidelines on vehicle maintenance and crew rests may be satisfied.
7.2.1 Planning Levels [4]
Transportation systems are among the most complex organizations and involve many components such as human and material resources, complex connections, and balances between decision variables and management policies that directly or indirectly affect different components of the system. To decrease this complexity, researchers have provided a general classification for transportation problems with three planning levels: strategic (long term), tactical (medium term), and operational (short term).
Strategic (Long Term) Planning
Strategic planning involves decisions at the highest level of management and requires long-term investment. Strategic decisions develop general policies and extensively structure the functional strategies of the system. Any physical changes or development in whole network such as locating main facilities (e.g., hubs and terminals) are examples of strategic decision planning. Strategic planning takes place in international, national, and regional transportation systems.
Tactical (Medium Term) Planning
Tactical planning needs medium term investment and is not as critical as strategic planning. This class contains a well-organized allocation and operation of resources to improve system performance. Examples of this category are decision making in the design of service networks, service schedules, repositioning fleets, and traffic routing. Most carriers’ decision making is at this level.
Operational (Short Term) Planning
Operational planning is short term and urgent decision making performed by local management, yard masters, and dispatchers. Decisions at this level do not need large investments. The completion and adjustment of schedules for services, crews, maintenance activities, and routing and dispatching of vehicles and crews are examples of this level.
In the next part, we introduce variants of standard transportation problems and then several important transportation problems, some of which are shipper decision problems and some of which are carrier decision problems.
7.2.2 Variants of the Standard of TPs
The time-minimization transportation problem (TMTP) minimizes the time to transport goods frommorigins ton destination under some constraint of available sources and requested destinations. Such problems especially arise when perishable goods are transported or when it is required to transport essential items such as food and ammunition in the shortest possible time in a war scenario.
The fundamental difference between the cost-minimization transportation prob- lem (CMTP) and TMTP is that the cost of transportation depends on the quantity of commodity being transported but the time involved is independent of this factor.
Many different objective function may be considered for a transportation prob- lem such as minimization of transportation costs, minimization of labor turnover, minimization of risk to a firm or the environment, and minimization of deteriora- tion of perishable goods [8].
Time Minimizing Solid Transportation Problem
The cost minimizing solid transportation problem (CMSTP) is [8]:
Minimize
z5 Xn
i51
Xm
j51
Xp
k51
cijkxijk ð7:1Þ
subject to Xn
j51
xijk5ajk; Xm
j51
xijk5bki
Xp
k51
xijk5eij ð7:2Þ
Xm
j51
ajk5Xp
i51
bki; Xp
k51
bki; 5Xm
j51
eij; Xn
i51
eij5 Xp
k51
ajk ð7:3Þ
Xm
j51
Xp
k51
ajk5 Xp
k51
Xn
i51
bki5Xn
i51
Xm
j51
eij; xij$0 ð7:4Þ
where
iis the number of origin points providing typekof goods, jis the number of destinations,
xijkis the number of typeksent from theith origin to thejth destination,
cijkis the cost of transporting the unit item of thekth from theith supply point to thejth destination,
ajkis the requirement at thejth destination of typekof goods, bkiis the availability of typekof goods at theith supply point,
eijis the total quantity of goods to be sent from theith supply point to thejth destination.
The TMTP form of this problem is as given below.
Minimize
½Maxtijk:xijk.0 ð7:5Þ
subject to Equations (7.27.4), where tijk is the time of transportation type k of goods from theith source to thejth destination.
It is assumed that all carriers start simultaneously. The convexity of objective function has been demonstrated in CMSTPs but not in TMTPs.
Pricing of Bottlenecks at Optimal Time in a Transportation Problem
The conventional transportation problem deals with minimizing the cost of trans- porting a homogeneous product from various supply points to a number of desti- nations without caring for the time of transportation. By increasing the bottleneck at time T, we can have a less-cost transportation schedule. However, a commis- sioning of the project is influenced by the bottleneck. Thus, the larger bottleneck has to be valued by comparing its compact on the functioning of the project with the saving in transportation cost. Assuming that the impact of the bottleneck flow on the functioning of the project is known, a convergent iterative procedure was proposed by Malhotra and Puri [9] which finds all various efficient pairs at timeT.
Minimize X
iAI
X
jAJ
cijxij ð7:6Þ
subject to the following constraints:
X
jAJ
xij5ai;ai.0;iAI; X
iAI
xij5bj;bj.0;jAJ xij$0;iAI;jAJ
9>
>>
=
>>
>;
ð7:7Þ
where
idenotes the index set of supply points, jthe index set of destination,
xijthe number of the product transported from theith supply point to thejth destination, cijthe unit cost of transportation on the (i,j)th route.
Minimize X
iAI
X
jAJ
c0ijxij ð7:8Þ
subject to Equation (7.7), wherec0ijis different for different values oftij.
c0ij5
1 iftij5T 0 iftij,T Niftij.T 8<
: ð7:9Þ
Bi-Criteria Transportation Problem
A bi-criteria transportation problem is a kind of problem with two linear objectives that are the minimization of the total transportation cost and minimization of the total deterioration of goods during transportation [10]. The mathematical model of problem is formulated as below.
Minimizez5(z1,z2) z15 X
iAI
X
jAJ
cijxij ð7:10Þ
z25 X
iAI
X
jAJ
dijxij ð7:11Þ
subject to X
jAJ
xij5ai; ai.0; iAI X
iAI
xij5bj; bj.0; jAJ xij$0; ði;jÞAI3J
9>
>>
>=
>>
>>
;
ð7:12Þ
where
idenotes the index set of supply points, jis the index set of destination,
xij is the number of the product transported from the ith supply point to the jth destination,
cijis the unit cost of transportation on (i,j)th route,
dijis the cost of deterioration of a unit while transporting fromitoj, aijis the availability of the product at theith supply point,
bjis the demand at the destinationj.
Observing that the nondominated set in the decision space has the larger number of extreme points compared with the extreme points of the nondominated set in the criteria space, Aneja and Nair [10] also have developed an algorithm to determine the efficient extreme points in the criteria space. They have solved the same trans- portation problem again and again but with unlike objectives. In continual iterations, the objective function is the positive weighted average of the two linear objectives under consideration.
7.2.3 Carrier Decision-Making Problems
A carrier decision-making problem is a problem whose objective function is defined in the same direction of maximization of a carrier’s profits. Some of the carrier decision-making problems are crew-assignment problems, vehicle allocation and scheduling problems, terminal design problems, allocation and operation pro- blems, freight-traffic assignment problems, service network design problems, and fleet-composition problems [1].
As mentioned before, most of a carrier’s decision-making problems belong to the tactical level.
Dynamic Driver Assignment Problem
The first carrier decision-making problem introduced here is the dynamic driver assignment problem (DDAP) that arises in TL trucking. Crews are assigned to vehicles in order to support the planned operations [11].
In this problem, a fully loaded vehicle is assigned to a driver in a scheduled operation. It may take several days for a vehicle to be fully loaded because the cus- tomer’s demands are not known in advance and are received randomly. Here each driver is supposed to be assigned to just one demand at one time [1].
The DDAP is formulated as a minimum-cost problem during which, the cost of driver’s assignments (for empty moving from waiting location to pick-up point) is minimized. It is assumed that n drivers are waiting for assignment to fully loaded vehicles. Let the set of drivers be shown by D5{1, . . ., n} and the set of ready fully loaded vehicles by V5{1, . . .,m}. If the maximum number ofD andVare not equal (n6¼m), the extra number is compensated by is compensated by a dummy value 0. Oncen,m, a dummy driver 0 is added toD, whereas ifm,n, a dummy load 0 is added to V. In this case (m,n), the DDAP is formulated as follows.
Minimize X
iAD
X
jAL
cijxij ð7:13Þ
subject to X
iAD
xij51; jAL\f g0 X
jAL
xij51; iAD xijAf0;1g; iAD; jAL
ð7:14Þ
where
cijis the predefined cost for driveriwho drives vehiclej,
xijis a binary variable equal to 1 if driveriis assigned to vehiclejand 0 otherwise.
There are also many other known problems in the crew-scheduling management category, such as optimizing the scheduling of terminal employees or the quantity of the reserve crew [12].
Fleet-Composition Problem
Carriers always try to decrease the investment in their own crew. In other words, paying attention to the variety of demands over a year, they avoid having their own maximum number of vehicles needed in peak periods during a year. They usually have a base number of their own vehicles to answer their usual demands, and they hire additional required vehicles in peak periods. In this way, carriers can save money by creating a balance point at which the total cost of their own and hired vehicles are minimized. For this problem, similarity of vehicles is an assumption [1].
Let a year includentime periods; the number of own vehicles is shown byv. If
i5{1, . . .,n} is the set of time periods, then vi is the number of needed vehicles
during the time period t. The main variable mis the number of time periods per year in which carriers need to hire vehicles (vi.v). There are two types of costs for carriers’ own vehicles during a time period: fixed cost (cf) and variable cost (cv), respectively, whilechis the cost of a hired vehicle in the same time period.
This problem is formulated in a simple minimization model without any constraint as follows.
Minimize
ciðvÞ5cfðvÞ1cvminðvi;vÞ1chðvi2vÞ ð7:15Þ To find the annual cost, we just need to add up the costs of different periods of a year. The formulation is changed in Equation (7.16).
Minimize Xn
i51
ciðvÞ5Xn
i51
cfðvÞ1Xn
i51
cvminðvi;vÞ1 X
i5vi.v
chðvi2vÞ ð7:16Þ
Let Cfbe the annual fixed cost, while Cvand Chare the annual variable costs for own and hired vehicles, respectively. Now we can formulate this problem in a simpler form in Equation (7.17).
Minimize
CðvÞ5nCfv1Cv
Xn
i51
minðvi;vÞ1Ch
X
i:vt.v
ðvi2vÞ ð7:17Þ
Because this transportation cost problem has a simple linear formulation without any constraint, the optimal annual cost is found by setting the derivation of c(v)
(with regard tov) equal to 0 (Equation 7.18). Derivative ofc(v) is found in the fol- lowing equation, so the best value ofmis obtained in Equation (7.19).
nCf1Cvm2Chm50 ð7:18Þ
Figure 7.2 demonstrates the variety of demands over the year. Minimum annual cost is obtained when the area below and above the linevt5vbe equal. This condi- tion is achieved if the Equation (7.18) is satisfied.
m5n Cf
Ch2Cv ð7:19Þ
Vehicle Allocation and Scheduling Problem
Carriers can best use their vehicles by applying an optimum allocated schedule to respond the maximum number of demands. A vehicle-allocation problem is a kind of carrier’s decision problem formulated as a minimum-cost flow problem in which carriers decide which demand should be responded to and which one should be rejected, which vehicle should be moved to a new pickup point, and which one should wait for a future assignment [1].
In this problem, it is supposed that all demands have been known in advance and all vehicles are the same in type, size, and capacity. Lett5{1,. . .,T} be the set of time periods assumed to divide the planning horizon and N be the set of demand pickup or delivery points. For this problem, there are several parameters:
dijt,iAN, jAN,t51, . . .,T, is an available demand delivered from pickup point i
to destinationjduring time periodt;τij,iAN,jANis the travel time from pointito pointj;cij,iAN,jAN, is the cost of moving an empty vehicle from pickup pointi to pointj;pij,iAN,jAN, is the profit obtained by delivering a shipment from origin i to destination j; and mit,iAN, t51, . . ., T, is the available number of vehicles ready to be moved from pointiin time periodt.
vt
t Hired vehicles
Owned vehicles
Figure 7.2 Fleet compositions when demand varies over the year [1].
There are two types of decision variables in this problem: xijt, iAN, jAN,
t51, . . ., T, denotes the number of vehicles starting their delivery services from
origini and ending at destination jin time periodt; andyijt,iAN,jAN,t51, . . ., T, indicating the number of vehicles moving empty from pointito pointj at time period t. This minimum-cost flow problem can be formulated as a maximum- profit problem.
Maximize XT
t51
X
iAN
X
jAN;j6¼i
ðpijxijt2cijyijtÞ ð7:20Þ
subject to X
jAN
ðxijt1yijtÞ2 X
kAN;k6¼i:t.τki
xkiðt2τkiÞ1ykiðt2τkiÞ
2yiit21
5mit;iAV;tAf1;. . .;Tg ð7:21Þ
xijt#dijt; iAN; jAN; tAf1;. . .;Tg ð7:22Þ
xijt$0; iAN; jAN; tAf1;. . .;Tg ð7:23Þ
yijt$0; iAN; jAN; tAf1;. . .;Tg ð7:24Þ
where the objective function (Equation 7.20) is the total profit over the whole plan- ning horizon derived by revenues minus costs.
Constraint (7.21) confirms that the number of entered vehicles at pointi during time periodt must equal the exact valuemit.Constraint (7.22) states that the num- ber of vehicles moved from pointi to pointj in time periodtshould not be more than the number of demands between these points during this period. Regarding this,dijt2 xijtrepresent loads should be rejected.
7.2.4 Shipper Decision-Making Problems
A shipper decision-making problem is a problem with an objective function defined in the same direction as the maximization of a shipper’s profits. Some of the ship- per decision-making problems are transportation mode selection, a shipment con- solidation and dispatching problem, a commodities load planning and packing problem, and a carrier-type decision problem [1].
Shipment Consolidation and Dispatching
Shipment consolidation and dispatching problem is a kind of shipper decision- making problem often faced by producers (if they do their delivery activities them- selves) and contracted shippers. The manufacturer or shipper has to choose the best
way for timely delivery of orders to customers during a time horizon divided by T periods. They should find the most suitable transportation mode for each shipment.
They also should choose the best design for consolidation of shipments and esti- mate the start time of dispatching. In this way, any related scheduling critical factor should be considered [1].
Shipment consolidation and dispatching problem is formulated as a minimiza- tion model. Herein there are several parameters for each vehicle i: a destination diAN, a weight wi$0, a ready time ri(the period in which vehicle i is ready for delivery), and a deadlineki(the period in which vehicleishould be reached to des- tinationdi).
A shipper may use LTL carriers or hire one-way truck trips. For rented trucks, consider a set of route Rwith the following characteristics: Sris the set of cessa- tions during router;fris the fixed cost for using router;qris the available capacity of router. Letτir,iAI,rAR, be the number of time periods taken to deliver ship- ment ion router. By using LTL carriers,gi is the cost of delivery for shipmenti, andτ0iis the number of time periods this delivery takes.
This problem has three kinds of binary variables: (1)xirt,iAI,rAR,t51,. . .,T, is equal to 1 if shipmentistarts its delivery trip on time periodtduring routerand 0 otherwise; (2)yrt,rAR,t51,. . .,T, is equal to 1 if routeris used in time period tand 0 otherwise; and (3)wiis equal to 1 if shipmentiis delivered by LTL carriers and 0 otherwise (this variable is definable only whenri1τ0i,ki).
Minimize X
rAR
XT
t51
fryrt1 X
iAI
giwi ð7:25Þ
subject to X
i:ri#t#di2τir;siASr
wixirt#qryrtrAR; t51;. . .;T ð7:26Þ
X
r:siASr
X
t:ri#t#ki2τir
xirt1wi51iAI ð7:27Þ
xirtAf0;1g; iAI; rAR; t51;. . .;T ð7:28Þ
yrtAf0;1g; rAR; t51;. . .;T ð7:29Þ
wiAf0;1g; iAI ð7:30Þ
In objective function (Equation 7.25), the total cost for delivering shipments is minimized. Constraint (7.26) states that for each routerand time periodt51,. . ., T, if LTL carriers are used, the total weight carried on routerduring time periodt should not be more than capacity qr. Constraints (7.27) confirm that delivery of
each shipment should be assigned to only one of transportation operators (LTL car- riers or rented trucks).