A branchandprice algorithm for an integrated production andinventory routing problem
Jonathan F. Bard
a,
, Narameth Nananukul
b
a
Graduate Program in Operations Research and Industrial Engineering, 1 University Station C2200, The University of Texas, Austin, Tx 787120292, USA
b
Optimize Science, 100 Hepburn Road, Suite 1D Clifton, NJ 070122, USA
a r t i c l e i n f o
Available online 25 March 2010
Keywords:
Production planningLotsizingInventory routingColumn generationBranch and price
a b s t r a c t
With globalization, the need to better integrate production and distribution decisions has become evermore pressing for manufacturers trying to streamline their supply chain. This paper investigates apreviously developed mixedinteger programming (MIP) model aimed at minimizing production,inventory, and delivery costs across the various stages of the system. The problem being modeledincludes a single production facility, a set of customers with time varying demand, a ﬁnite planninghorizon, and a ﬂeet of homogeneous vehicles. Demand can be satisﬁed from either inventory held at acustomer site or from daily product distribution. Whether a customer is visited on a particular day isdetermined by an implicit tradeoff between inventory and distribution costs. Once the decision is made,a vehicle routing problem must be solved for those customers who are scheduled for a delivery.A hybrid methodology that combines exact and heuristic procedures within a branchandpriceframework is developed to solve the underlying MIP. The approach takes advantage of the efﬁciency of heuristics and the precision of branch and price. Implementation required devising a new branchingstrategy to accommodate the unique degeneracy characteristics of the master problem, and a newprocedure for handling symmetry. A novel column generation heuristic and a rounding heuristic werealso implemented to improve algorithmic efﬁciency. Computational testing on standard data setsshowed that the hybrid scheme can solve instances with up to 50 customers and 8 time periods within1h. This level of performance could not be matched by either CPLEX or standard branch and price alone.
&
2010 Elsevier Ltd. All rights reserved.
1. Introduction
Integrating production and distribution decisions is a challenging problem for manufacturers trying to optimize their supplychain. At the planning level, the immediate goal is to coordinateproduction, inventory, and delivery to meet customer demand sothat the corresponding costs are minimized. Achieving this goalprovides the foundations for streamlining the logistics networkand for integrating other operational and ﬁnancial components of the organization. In this paper we analyze a single productionfacility that serves a set of customers with time varying demandover a ﬁnite and discrete planning horizon. The capacity of thefacility is assumed to be limited and each day production isscheduled, a setup cost is incurred. The focus is on the case wherea routing problem must be solved daily to either restockinventory, meet that day’s demand or both. When the associateddecisions are made in a coordinated fashion, we have what isreferred to as the production, inventory, distribution, routingproblem (PIDRP).Lei et al. [1] were the ﬁrst to formulate the PIDRP as a mixedinteger program (MIP) and proposed a twophase solutionapproach that avoided the need to address lotsizing and routingsimultaneously. Boudia et al. [2,3] developed a similar MIP andproposed both a memetic algorithm with population management (MAPA) and a reactive greedy randomized adaptive searchprocedure (GRASP) with pathrelinking [4] as solution methodologies. Following their lead, Bard and Nananukul [5] developed atabu search procedure that provided slightly better results. Theirmodel was based on the MIP in Boudia et al. [3] but is moreefﬁcient in terms of variable deﬁnitions and number of constraints. The same model is used in this paper.Manufacturers who resupply a large number of retailers on aperiodic basis continually struggle with the question of how toformulate a replenishment strategy. One popular approach is abalanced strategy in which an equal proportion of retailers isreplenished each day of the work week. This has the advantage of evening the workload at the warehouse. A second strategyadvocated by many practitioners is synchronized replacement inwhich all retailers are replenished concurrently and goods are
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Computers & Operations Research
03050548/$see front matter
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2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.cor.2010.03.010
Corresponding author. Tel.: +15124713076; fax: +15122321489.
Email addresses:
jbard@mail.utexas.edu (J.F. Bard),nananukul@hotmail.com (N. Nananukul).Computers & Operations Research 37 (2010) 2202–2217
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moved into the manufacturer’s warehouse immediately prior todistribution. This creates unbalanced workloads at the warehousebut allows for crossdocking of a signiﬁcant portion of the goods[6]. For justintime suppliers, it is common to partition thecustomers into compact groups and follow the same deliverysequence daily, skipping those locations with absent demand [7].Rather than treating the replenishment strategy in partialisolation, we take a more integrated view and develop a branchandpricebased (B&P) scheme for solving the PIDRP as acomplement to existing metaheuristics. This is our majorcontribution. Secondary contributions include the design of methods for dealing with symmetry during branching and theuse of heuristics to achieve integrality. As part of the solutionprocess, four critical decisions have to be made: how many itemsto manufacture each day, when to visit each customer, how muchto deliver to a customer during a visit, and which delivery routesto use. Because the last decision requires the solution of a vehiclerouting problem (VRP) each day, the PIDRP is evidently NPhard inthe strong sense, so it is not likely that large instances will betractable for more than a few time periods. With this limitation inmind, we propose several compromises to the exact algorithmthat involve the solution of a lotsizing problem to estimatedelivery quantities in each period and the use of a VRP heuristic.In the next section, the literature related to the primarycomponents of the PIDRP is reviewed with an emphasis on recentresearch. In Section 3, a formal deﬁnition of our version of theproblem is given along with a new MIP formulation. The B&Palgorithm is described in Section 4. In Section 5, we discuss ourapproach to dealing with symmetry along with the details of anenhanced branching strategy. Heuristic ideas are outlined inSection 6, and in Section 7, computational results are presentedfor a wide range of problem instances. Section 8 offers severalideas for extending the work.
2. Literature review
There is a vast quantity of literature on each component of thePIDRP so we will only highlight the most relevant work. A vendormanaged inventory replenishment (VMI) system is a goodexample of the type of integration that we wish to address (e.g.,see [8, 9]). In the VMI model, the manufacturer observes andcontrols the inventory levels of its customers, as opposed toconventional approaches where customers monitor their owninventory and decide the time and amount of product to reorder.One of the beneﬁts of VMI is that it permits a more uniformutilization of transportation resources leading to lower distribution costs. Customers beneﬁt from higher service levels andgreater product availability due to the fact that vendors can useexisting inventory data at their customer sites to more accuratelypredict future demand [10, 11].Lei et al. [1] studied a multifacility, heterogeneous ﬂeetversion of the PIDRP that was motivated by a chemicalmanufacturer with international customers. In phase one of theirtwophase methodology, they solve a reduced model in whichtransporter routings were restricted to direct shipments betweenfacilities and customer sites. The results provided a productionschedule and the number of items to be delivered to eachcustomer in each period. In phase two, a routing heuristic wasproposed based on an extended optimal partitioning procedurethat consolidated the lessthantruckload assignments obtainedin phase one into more efﬁcient delivery schedules. The overallapproach contained several novel features but with limitedapplicability to the general problem.As mentioned, Boudia et al. [2,3] proposed both a MAPAand a reactive GRASP with pathrelinking to solve the PIDRP.Their model included a single facility and a set of customerslocated on a grid. Holding costs at the customer sites wereassumed to be negligible compared to the holding costs at thefactory and so were ignored. As in our case, the objective was tominimize the sum of production, holding and transportation costswhile ensuring that all demand was satisﬁed over the planninghorizon. An improved version of the MAPA is presented by Boudiaand Prins [12]. Testing was done on a set of 90 benchmarkproblems.Similarly, Bard and Nananukul [5] developed a reactive tabusearch algorithm for solving the PIDRP. An essential component of their methodology was the use of an allocation model in the formof a MIP to ﬁnd good feasible solutions that were used as startingpoints for the tabu search. The neighborhood consisted of swapand transfer moves. Pathrelinking was also used in a postprocessing phase to seek out marginal cost reductions. Testing onthe 90 benchmark instances demonstrated the effectiveness of theapproach. In all cases, improvements ranging from 10–20% wererealized in comparison of the results obtained from the abovementioned GRASP, but often at a greater computational cost.When routing is a dominant consideration, the PIDRP is mostsimilar to the inventory routing problem, IRP [13–17] and theperiodic routing problem, PRP [18–21]. Although there has beenmuch research on these two problems, little of it carries over tothe PIDRP. The primary reason relates to the formidable complexity of its structure, as deﬁned by a combination of a capacitatedlotsizing problem (e.g., see [22]) a capacitated, multiperiod VRP.The full PIDRP has so far proven to be beyond the capability of exact methods. By decoupling of the lotsizing and routingdecisions, though, several researchers have had some success inﬁnding good solutions with heuristics. Chandra and Fisher [23],for example, ﬁrst determine a production schedule without regardto the logistics. Next, they develop a distribution schedule foreach planning period based on the results obtained from the ﬁrststage model. This approach worked well when there was enoughinventory in the system to buffer production from the distributionoperations, but consequently led to increased holding costs.Similar to our work, Savelsbergh and Song [24] went beyondthe traditional IRP and included a production component withlimited capacity and the need to route customers over severaldays in their model. While we use outandback trips each day,they allowed for sleeper teams that are on the road for a week ormore to cover wide geographic areas. To ﬁnd solutions, theydeveloped an integer programmingbased algorithm and embedded it in a randomized greedy heuristic with a local searchfeature.In a prior study, Christiansen [25] presented a ship routing andscheduling problem with production and inventory considerations. In the model, a ﬂeet of heterogeneous ships transport asingle product (ammonia) between harbors, some of which areproduction sites and others consumption sites. Since the quantities loaded and discharged are determined by the productionrates of the harbors, possible stock levels, the actual ship visitingthe harbor, and the fact that one company owns all facilities,inventory cost is not an issue. The objective is to minimizetransportation costs while meeting time window constraints andproduction targets. A multicommodity integer programmingformulation is given and solved with a B&P algorithm.The IRP and the PRP are relaxations of the PIDRP, differing inseveral ways. Neither, for example, takes the production decisionand inventory level at the factory into consideration. In addition,the PRP assumes that the delivery patterns deﬁned by deliveryfrequencies or delivery days are given in advance and tries toselect the most suitable pattern for each customer.In their version of the PRP, Mourgaya and Vanderbeck [20]proposed a column generationbased heuristic to ﬁx dates for
J.F. Bard, N. Nananukul / Computers & Operations Research 37 (2010) 2202–2217
2203
ARTICLE IN PRESS
customer visits and to assign customers to vehicles. The dailysequencing decisions were left to an operational model. Zhao et al.[9] studied the integration of inventory control and vehiclerouting for a distribution system in which a set of retailers withconstant rates of demand were resupplied with a single productfrom a central warehouse. The objective was to determineinventory policies and routing strategies such that the longrunaverage costs were minimized and all demand was satisﬁed. Intheir model, and in contrast to ours, no inventory capacityconstraints were imposed on the warehouse or on the retailers.Testing was done on problem instances with 50 and 75 retailers.By analyzing the relationship between the lower bound and theoptimal solution, they were able to establish that the proposedalgorithm in conjunction with a ﬁxed partitioning policy waseffective and robust in almost all cases.
3. Model formulation
We are given a set of
n
customers geographically dispersed ona grid and a single facility for producing a unique item. Distancesare calculated with the Euclidean metric and accordingly satisfythe triangle inequality. Over a planning horizon of
t
periods, eachcustomer
i
has a ﬁxed nonnegative integral demand
d
it
in period
t
that must be fulﬁlled; i.e., shortages are not permitted (for adiscussion of backordering, see, e.g. [26]). If production takesplace at the facility in period
t
, then a setup cost
f
t
is incurred,
t
¼
0,1,
y
,
t
1. A limited number of items (up to
C
) can beproduced in each time period and a limited number
I
P
max
can bestored at a unit cost of
h
P
. The deﬁnition of a period depends onthe context of the problem, and might be a shift, a day, or a week.We use a day in the description of the model.In constructing delivery schedules, each customer can bevisited at most once per day and each of
y
homogeneous vehiclescan make at most one trip per day. A limited amount of inventory,
I
C
max
,
i
, can be stored at customer
i
’s site at a unit cost of
h
C i
, buttransshipments between customers are not permitted (cf. [27]).An additional assumption is that all deliveries take place at thebeginning of the day and arrive in time to satisfy demand for atleast that day if necessary. All production on day
t
is available fordelivery only on the following morning (this is common in foodproduction and distribution; e.g., see [28]) and all inventory ismeasured at the end of the day. Demand on day
t
can be met outof deliveries from the factory on day
t
and ending inventory onday
t
1 at the customer site. Initial customer inventory on day 0simply reduces demand on subsequent days, while initialinventory at the factory must be eventually routed. At the endof each day it is assumed that all vehicles return to the factoryempty, and at the end of the planning horizon all inventory isrequired to be zero. While the latter requirement was imposed toreduce the number of parameters in the experimental design,practical considerations often argue for positive, ﬁnal inventorylevels, especially for short planning horizons. Nevertheless,restricting the ﬁnal level to zero has no effect on algorithmiccomplexity or performance with respect to runtimes.Our goal is to construct a production plan and deliveryschedule that minimize the sum of all costs while ensuring thateach customer’s demand is met over the planning horizon.Implicit in the solution will be a daily distribution of surplusitems to be placed in inventory. Under the common assumptionthat unit production costs are constant, the decision to overproduce and hold items in inventory will be a function of thesetup cost at the factory, system holding costs, production andstorage capacities, vehicle routes, and daily demand (foran analysis of synchronized ordering under ﬁxed order intervals,see [6]).In the development of the model, we make use of the followingnotation.
Indices and setsi
,
j
indices for customers, where 0 corresponds to thefactory
t
index for periods or days
N
set of customers;
N
0
¼
N
[
{0}; 
N

¼
nT
set of days in the planning horizon;
T
0
¼
T
[
{0} and 
T

¼
t
Parametersd
it
(integral) demand of customer
i
on day
t D
max
it
upper bound on the maximum amount to be delivered tocustomer
i
on day
t D
max
t
upper bound on the maximum amount that can beloaded on a vehicle on day
t
y
number of available vehicles
Q
(integral) capacity of each vehicle
I
P
max
maximum inventory that can be held at the factory
I
C
max
,
i
maximum inventory that can be held by customer
iC
(integral) daily production capacity of the factory
c
ij
cost of going from customer
i
to customer
j f
t
setup cost at factory on day
t h
P
unit holding cost at the factory
h
C i
unit holding cost at customer
i
site
Decision variables x
ijt
1 if customer
i
immediately precedes customer
j
on adelivery route on day
t
; 0 otherwise
y
it
load on a vehicle immediately before making a deliveryto customer
i
on day
t p
t
production quantity on day
t z
t
1 if there is production on day
t
; 0 otherwise
I
P t
inventory at the production facility at the end of day
t I
C
it
inventory at customer
i
at the end of day
t w
it
amount delivered to customer i on day
t Model
f
IP
¼
Minimize
X
t
A
T
X
i
A
N
0
X
j
A
N
0
c
ij
x
ijt
þ
X
t
A
T
f
t
z
t
þ
X
t
A
T
0
\
f
t
g
h
P
I
P t
þ
X
t
A
T
\
f
t
g
X
i
A
N
h
C i
I
C
it
ð
1a
Þ
subject to
I
P t
¼
I
P t
1
þ
p
t
X
i
A
N
w
it
,
8
t
A
T
ð
1b
Þ
I
C
it
¼
I
C i
,
t
1
þ
w
it
d
it
,
8
i
A
N
,
t
A
T
ð
1c
Þ
X
i
A
N
w
it
r
I
P t
1
,
8
t
A
T
ð
1d
Þ
p
t
r
Cz
t
,
8
t
A
T
0
\
f
t
g ð
1e
Þ
p
0
Z
X
i
A
N
ð
d
i
1
I
C i
0
Þ
I
p
0
ð
1f
Þ
X
j
A
N
0
j
a
i
x
ijt
r
1
,
8
i
A
N
,
t
A
T
ð
1g
Þ
X
i
A
N
0
i
a
j
x
ijt
¼
X
i
A
N
0
i
a
j
x
jit
,
8
j
A
N
,
t
A
T
ð
1h
Þ
X
j
A
N
x
0
jt
r
y
,
8
t
A
T
ð
1i
Þ
y
jt
r
y
it
w
it
þ
D
max
t
ð
1
x
ijt
Þ
,
8
i
A
N
,
j
A
N
0
,
t
A
T
ð
1j
Þ
J.F. Bard, N. Nananukul / Computers & Operations Research 37 (2010) 2202–2217
2204
ARTICLE IN PRESS
w
it
r
D
max
it
X
j
A
N
0
x
ijt
,
8
i
A
N
,
t
A
T
ð
1k
Þ
0
r
I
P t
r
I
P
max
,
0
r
I
C
it
r
I
C
max
,
i
,
8
i
A
N
,
t
A
T
\
f
t
g
;
I
P
t
¼
I
C i
t
¼
0
,
8
i
A
N
ð
1l
Þ
x
ijt
A
f
0
,
1
g
,
0
r
y
it
r
Q
,
w
it
A
Z
n
t
þ
,
8
i
a
j
A
N
0
,
t
A
T
;
z
t
A
f
0
,
1
g
,
p
t
A
Z
t
þ
,
t
A
T
0
\
f
t
g
,
p
t
¼
0
ð
1m
Þ
where
D
max
it
¼
min
Q
,
X
t
l
¼
t
d
il
( )
and
D
max
t
¼
min
Q
,
X
i
A
N
X
t
l
¼
t
d
il
( )
The objective function in (1a) minimizes the sum of transportation costs, production setup costs, holding costs at the factory, andholding costs at the customer sites. Because all demand must bemet, the variable component of the production costs, which areassumed to be linear and constant, can be omitted, as can anyinitial inventory costs. Constraints (1b) and (1c) are inventoryﬂow balance equations in which it is assumed that the initialinventories
I
P
0
and
I
C i
0
are given for all customers
i
A
N
. Constraints(1d) limit the total amount available for delivery on day
t
to theamount in inventory at the factory on day
t
1. The speciﬁcamount delivered to customer
i
is limited by the parameter
D
max
it
in (1k), which is the smaller of the vehicle capacity
Q
or thecumulative demand from day
t
to the end of planning horizon
t
.When customer
i
is not visited on day
t
, the righthand side of (1k)will be zero so no delivery is possible.Constraint (1e) bounds production on day
t
to the capacity of the factory. A simple way to tighten this constraint is to replace
C
with
C
t
¼
min
f
C
,
I
P
max
,
D
max
t
þ
1
g
, where the third term is the demand of all customers for the remainder of the planning horizon. Theassumption that items produced on day
t
are only available fordelivery on day
t
+1 implies that
p
t
r
I
P
max
and
p
t
¼
0. It is possibleto strengthen the latter inequality by subtracting from
I
P
max
thereduction in inventory due to deliveries on day
t
to get
p
t
r
I
P
max
ð
I
P t
1
P
i
A
N
w
it
Þ
, but this constraint is dominated by(1b). To ensure that demand on day 1 can be met, it is necessary toinclude (1f) which allows production on day 0. If
I
P
0
¼
I
C i
0
¼
0, then
p
0
Z
P
i
A
N
d
i
1
or the problem is infeasible.Constraints (1g)–(1j) represent the routing aspect of theproblem. Constraints (1g) and (1h) ensure that if customer
i
isserviced on day
t
, then it must have a successor on its route,which may be the factory. Route continuity is enforced by (1h);i.e., if a vehicle arrives at customer
j
on day
t
, then a vehicle mustdepart customer
j
on day
t
. Logic, and the fact that the ﬂeet ishomogeneous, requires that it be the same vehicle.The number of vehicles that leave the factory on any day
t
islimited to the number available,
y
, as indicated by (1i). Finally, (1j)keeps track of the load on the vehicles and guarantees that on day
t
, if customer
i
is the immediate predecessor of customer
j
on aroute, then the load on the vehicle before visiting customer
j
mustbe less than or equal to the load just before visiting customer
i
minus the amount delivered, which is represented by the variable
w
it
. The value of the parameter
D
max
t
is speciﬁed to be as small aspossible while ensuring that (1j) is always feasible. Because theload on each vehicle is monotonically decreasing as customers arevisited, (1j) provides the added beneﬁt of eliminating subtoursthat do not include the factory. If a vehicle returned to apreviously visited customer
j
on day
t
, the result would be that
y
jt
o
y
jt
, a logical inconsistency. After all deliveries are made, theﬂeet returns to the factory empty so
y
0
t
can be set to 0 for all
t
A
T
.To conclude the formulation, variables are deﬁned in (1l) and(1m). It is straightforward to generalize model (1), to includemultiple facilities and multiple products.The size of model (1) is determined largely by constraint (1j)and the number of binary variables,
x
ijt
, both of which grow at arate proportional to
O
(
n
2
t
). The majority of the other constraintsonly grow at a rate proportional to
O
(
n
t
). A small problem with 3vehicles, 30 customers, and a 5day planning horizon containsroughly 5200 constraints, 4600 binary variables, and 500continuous variables. Initial attempts to solve instances of thissize with CPLEX 8.1 were not encouraging. When a time limit of 90min was imposed, optimality gaps between 7% and 10% werethe norm.
4. Solution methodology
This experience led ﬁrst to development of a tabu searchalgorithm [5] and then to an exact method based on branch andprice (B&P), which we discuss in this section. In simple terms, B&Pcombines (i) Dantzig–Wolfe (D–W) decomposition extended toaccommodate integer variables, and (ii) standard branch andbound (B&B) [29,30]. Below we outline how initial feasiblesolutions are obtained and then describe the principal components of our B&P algorithm.
4.1. Initial solutions
All exact optimization methods depend on good bounds tospeed convergence. As a forerunner to this work, we developed athreephase methodology centered on tabu search to solve thePIDRP. In phase 1, an initial solution is found by solving arelaxation of (1a)–(1m) called the
allocation model
. In thisformulation the routing variables (
x
ijt
) and constraints (1g)–(1j)are removed and an aggregated vehicle capacity constraint isadded for each time period. Because the actual cost of making adelivery to customer
i
on day
t
cannot be determined withoutincluding the routing constraints, the cost term
P
ijt
c
ij
x
ijt
in (1a) isreplaced with a lower bounding approximation.The solution to the allocation model provides customerdelivery quantities
w
it
for all
i
¼
1,
y
,
n
and
t
¼
1,
y
,
t
. In phase 2,these values become the demand for
t
independent routingproblems. An efﬁcient CVRP subroutine also based on tabu search[31] is called to ﬁnd solutions. In phase 3, a full tabu searchalgorithm is used to improve the allocations and routings found inphase 2.
4.2. Column generation
D–W decomposition like Lagrangian relaxation is mosteffective when the feasible region can be partitioned into a setof complicating or aggregated constraints and a set of easy ordisaggregated constraints. For the current problem, it is natural toseparate the production and inventory constraints (1b)–(1f),which cut across two time periods each, from the routing anddelivery constraints, which can be written separately for eachtime period. We use the former along with the objective functionin (1a) to create the D–W master problem
ð
MP
Þ
.After removing the production and inventory constraints, weare left with (1g)–(1k) and the variable deﬁnitions, whichconveniently decompose by time period giving
t
subproblems.Points in the feasible region of subproblem
t
ð
SP
t
Þ
correspond toall feasible schedules on day
t
, where a schedule is a set of
y
orfewer tours specifying the delivery sequence and the amount tobe delivered if a customer is serviced on that day. In thereformulated model, each column in
MP
corresponds to afeasible schedule for the
n
customers. In a solution, at most onecolumn can be selected for each day.
J.F. Bard, N. Nananukul / Computers & Operations Research 37 (2010) 2202–2217
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ARTICLE IN PRESS
Informally speaking, the idea behind B&P is to have thesubproblems act as schedule generators guided by the values of the dual variables associated with the LP solution to the restricted
MP
; i.e., one that contains only a subset of feasible schedules. Ateach major iteration, an optimality check is made by implicitlypricing out
MP
. This is done by solving the subproblems to seewhether one or more schedules can be identiﬁed that hasnegative reduced cost. When added to the restricted
MP
, suchschedules in the form of columns will improve the current LPsolution, which serves as a lower bound on the solution to thesrcinal MIP; i.e., model (1). If all reduced costs are nonnegativebut some variable values are fractional in the
MP
solution, B&B isperformed.
4.2.1. Master problem
We ﬁrst derive
MP
and then show how its solution isused to construct the objective function of the pricing subproblems. Let
k
be the column index and let all previous notation bethe same. The following new symbols will be used in theformulation of
MP
.
Parameters and setsc
kt
cost of the schedule for day
t
associated with column
kW
k
it
amount delivered to customer
i
on day t in the scheduleassociated with column
k X
k
ijt
mapping parameter; 1 if customer
i
is the immediatepredecessor of customer
j
on day
t
in the scheduleassociated with column
k
, 0 otherwise
K
(
t
) set of columns for day
t Decision variables
l
kt
(binary) 1 if the schedule associated with column
k
on day
t
is selected, 0 otherwise
Master problem
ð
MP
Þ
f
IP
¼
Minimize
X
t
A
T
X
k
A
K
ð
t
Þ
c
kt
l
kt
þ
X
t
A
T
0
\
f
t
g
f
t
z
t
þ
X
t
A
T
0
\
f
t
g
h
P
I
P t
þ
X
t
A
T
\
f
t
g
X
i
A
N
h
C i
I
C
it
ð
2a
Þ
subject to
X
k
A
K
ð
t
Þ
X
i
A
N
W
k
it
l
kt
þ
I
P t
I
P t
1
p
t
¼
0
,
8
t
A
T
ð
2b
Þ
X
k
A
K
ð
t
Þ
W
k
it
l
kt
I
C
it
þ
I
C i
,
t
1
¼
d
it
,
8
i
A
N
,
t
A
T
ð
2c
Þ
X
k
A
K
ð
t
Þ
X
i
A
N
W
k
it
l
kt
I
P t
1
r
0
,
8
t
A
T
ð
2d
Þ
X
k
A
K
ð
t
Þ
l
kt
r
1
,
8
t
A
T
ð
2e
Þ
p
t
Cz
t
r
0
,
t
A
T
0
\
f
t
g ð
2f
Þ
p
0
Z
X
i
A
N
ð
d
i
1
I
C i
0
Þ
,
I
p
0
ð
2g
Þ
l
kt
A
f
0
,
1
g
,
8
t
A
T
,
k
A
K
ð
t
Þ
;
z
t
A
f
0
,
1
g
,
p
t
Z
0
,
8
t
A
T
0
\
f
t
g ð
2h
Þ
0
r
I
P t
r
I
P
max
,
0
r
I
C
it
r
I
C
max
,
i
,
8
i
A
N
,
t
A
T
\
f
t
g
;
I
P
t
¼
I
C i
t
¼
0
,
8
i
A
N
ð
2i
Þ
The objective function in (2a) minimizes the total production,inventory, and distribution cost over the planning horizon. Theonly change from (1a) is that the routing cost has been replacedwith the cost of a schedule but these are really the same.Constraints (2b)–(2d), (2f)–(2g) are equivalent of (1b)–(1f),respectively, but written in terms of the schedule variables
l
kt
instead of the srcinal variables (
w
it
,
x
ijt
). The parameters
ð
W
k
it
,
X
k
ijt
Þ
are derived from the solution of the pricing subproblems, whichare presented next. The convexity constraint (2e) ensures that atmost one delivery schedule is selected for each day.When the integrality constraint on the
l
kt
variables in (2h) isrelaxed, the solution to (2) provides a lower bound on the optimalsolution to (1). With the D–W procedure, this bound is usuallytighter than the bound obtained by solving the LP relaxation of (1)directly.
4.2.2. Pricing subproblems
To formulate the objective function for
SP
t
, let
a
1
t
be the dualvariable associated with the factory inventory constraints (2b),
a
2
it
the dual variable associated with customer
i
’s inventory constraints (2c),
a
3
t
the dual variable associated with the deliveryconstraints (2d), and
a
4
t
the dual variable associated with theconvexity constraints (2e). Only
a
3
t
and
a
4
t
are required to be nonnegative. For each day
t
, the reduced cost of column
k
in
MP
is
c
kt
¼
c
kt
a
1
t
X
i
A
N
W
k
it
X
i
A
N
a
2
it
W
k
it
þ
a
3
t
X
i
A
N
W
k
it
þ
a
4
t
ð
3
Þ
where
c
kt
¼
X
i
A
N
0
X
j
A
N
0
c
ij
X
k
ijt
To put (3) into a form that is usable in
SP
t
, the parameters
ð
W
k
it
,
X
k
ijt
Þ
must be expressed in terms of the original problemvariables. Making the appropriate substitutions, collecting terms,and removing the column index
k
, gives
c
t
¼
X
i
A
N
0
X
j
A
N
0
c
ij
x
ijt
X
i
A
N
ð
a
1
t
þ
a
2
it
a
3
t
Þ
w
it
þ
a
4
t
ð
4
Þ
The ﬁrst term on the righthand side of (4) represents therouting cost on day
t
while the remaining terms are theadjustments imposed by the
MP
dual values. In formulatingthe subproblems, we retain all the routing and delivery constraints (1g)–(1k) of the srcinal model.Subproblem t
ð
SP
t
Þ
f
SP
t
¼
Minimize
X
i
A
N
0
X
j
A
N
0
c
ij
x
ijt
X
i
A
N
ð
a
1
t
þ
a
2
it
a
3
t
Þ
w
it
þ
a
4
t
ð
5a
Þ
subject to
X
j
A
N
0
j
a
i
x
ijt
r
1
,
8
i
A
N
ð
5b
Þ
X
i
A
N
0
i
a
j
x
ijt
¼
X
i
A
N
0
i
a
j
x
jit
,
8
j
A
N
ð
5c
Þ
X
j
A
N
x
0
jt
r
y
ð
5d
Þ
y
jt
r
y
it
w
it
þ
D
max
t
ð
1
x
ijt
Þ
,
8
i
A
N
,
j
A
N
0
ð
5e
Þ
w
it
r
D
max
it
X
j
A
N
0
x
ijt
,
8
i
A
N
ð
5f
Þ
x
ijt
A
f
0
,
1
g
,
0
r
y
it
r
Q
,
w
it
Z
0 and integer
,
8
i
a
j
A
N
0
ð
5g
Þ
Although it is common to try to solve (5) to optimality, thismay be too time consuming for large instances. All that is reallyneeded, though, is a feasible solution whose objective functionvalue in (5a) is negative. If one is found, then the correspondingcolumn is added to the set
K
(
t
) in
MP
. Alternatively, if thecomputations are terminated before feasibility is achieved or if the smallest reduced cost of a feasible solution is nonnegative,then no column is generated by the subproblem.
J.F. Bard, N. Nananukul / Computers & Operations Research 37 (2010) 2202–2217
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