Page 1 Facility Location and Strategic Supply Chain Management Prof. Dr. Stefan Nickel Location Theory Chapter 5 – Discrete Location Problems Contents • Introduction • Classification of Discrete Location Problems • Warehouse Location Problem (WLP) • Extensions of the WLP
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Page 1Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Location Theory
Chapter 5 – Discrete Location Problems
Contents
• Introduction
• Classification of Discrete Location Problems
• Warehouse Location Problem (WLP)
• Extensions of the WLP
Page 2Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Chapter 5 – Discrete Location Problems
Introduction
Topic of the chapterCost-oriented location of one or more than one new facilities out of a set of given locations to meet customer demands.
By far most realistic and most flexible class of location problems.
Almost all real-world constraints, demands and guidelines can be considered in these models.
However, these approaches involve an increasing complexity in modeling the problem and especially in solving the resulting model.
Page 3Facility Location and Strategic Supply Chain Management
DemandsE.g. products, servicesThey are independent from the location decision.
Potential locationsGiven a set of potential locations where
- new facilities can be opened or- already existing facilities can be closed.
FacilitiesE.g. central or regional warehouses, transshipment points, factories
Page 4Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
IntroductionCosts
Occur as a result of - the satisfaction of customer demands and - location decisions
Satisfaction of customer demands- transportation, inventory and handling costs- material, raw material or production costs
Are quantity-dependent: variable costs.
Location decisions- estate- , land development- and building costs - closure costs- operating costs (personnel costs, unit costs)
Are both variable and fix (absolute or step-wise).
Page 5Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
IntroductionExample
A company distributes their manufactured goods from its factories through the warehouses to the customers. Due to the increased demand the company has to restructure its distribution network.
Old warehouses which have to be closed and new ones which have to be openedin previously determined locations.
reorganization
Factory Ware-house
Customer Factory Ware-house
store
store
Customer
X
Page 6Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Location Theory
Chapter 5 – Discrete Location Problems
Contents
• Introduction
• Classification of discrete location problems
• Warehouse Location Problem (WLP)
• Extensions of the WLP
Page 7Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Chapter 5 – Discrete Location Problems
Classification of Discrete Location Problems
Time horizonNumber of time periods (e.g. months, years) within the planning horizon.Distinguish
• Static (one-period) modelsLocation decisions must be made at the beginning of the planning horizon.
• Dynamic (multi-period) modelsLocation decisions involve where and when, i.e. in which period, the facilities are located. One has to determine a sequence of changes to be made.
Facility typesDistinguish various specifications of facilities.E.g. regional as well as central warehouses can be located.
Page 8Facility Location and Strategic Supply Chain Management
Is there just one product to plan or several products with different characteristics?
Number of echelonsNumber of echelons in the distribution network.Differentiate
• Single-level modelsThere is just one transportation echelon (e.g. inventory retailer) to distribute the goods Location decisions only at one level (e.g. inventory).
• Multi-level (hierarchical) modelsThere are several transportation echelons (e.g. factory inventory retailer) to distribute the goods Location decisions only for
- one level possible (e.g. inventory)- several levels possible (e.g. inventory and factory)
Page 9Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Classification of Discrete Location ProblemsInteraction among facilities
Material can flow among facilities of the same level, e.g. transports between warehouses are possible. Optimal facility locations depend not only on the spatial distribution of product demand but also on the mutual position of the facilities.
Uncertainty• Deterministic models
all data and parameters are completely known and certain• Probabilistic (stochastic) models
some data or parameters are uncertain
Direction of flowDifferentiate
• Distribution LogisticsGoods flow from factories/suppliers to warehouses/customers
• Reverse LogisticsGoods flow from customers to recycling centers, waste disposals
Page 10Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Classification of Discrete Location ProblemsCapacity restrictions
Limits the quantity, which can be produced, transported, handled, etc.
Causes• technical-economical restrictions (size of the machine)• personal restrictions
Distinguish
• Uncapacitated modelsno or insignificant capacities
• Capacitated models- with single-sourcing (demand is uniquely assigned to the facilities)- without single-sourcing
Page 11Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Classification of Discrete Location ProblemsDelivery mode
• Route deliveryDeliveries from a facility on a higher level to lowerlevel facilities (or vice versa) will be bundled to tours. E.g. several retailers are visited per delivery trip, starting at the depot.
• Direct deliveryDeliveries from a facility on a higher levelto a lower one occur directly and without detours.E.g. a truck serves exactly one customer/retailerat a single tour.
The transportation mode has an influence on the locational decisions.
Page 12Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Location Theory
Chapter 5 – Discrete Location Problems
Contents
• Introduction
• Classification of discrete location problems
• Warehouse Location Problem (WLP)- Introduction- Heuristics- Dualoc Algorithm
• Extensions of the WLP
Page 13Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Chapter 5 – Discrete Location Problems
Warehouse Location Problem (WLP)
Also called Uncapacitated Facility Location Problem (UFLP) or Simple Plant Location Problem (SPLP).
Assumptions• static, single-level, uncapacitated, deterministic• one product, direct delivery, one type of facilities, no interaction
Given• Set of customers I = {1, …, n} with demand bi , i = 1, …,n• Set of potential locations J = {1, …, m} for new facilities• Transportation costs tij per unit from the potential location j ∈ J to a customer i
∈ I. Include costs for material, production, storage, …
Total costs for serving customer i from location j: cij = bi · tij.
Page 14Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Warehouse Location ProblemCost matrix C = (cij )i=1,…,n, j=1,…,m
• Fixed costs fj for a new facility at location j.Include estate, building and fixed operating costs, but no variable costs.
Schematic illustration
1
2
n
1
2
m
b1
b2
bn
f1
f2
fm
c11
c1n
cnm
c2n
c12
Custo-mers
Pot. Locations
Page 15Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Warehouse Location Problem
Decisions• Number of new facilities• Locations for new facilities• Assignment of customers to the new facilities
Solution of the WLP consists of
p locations X = {x1, …, xp }X: Subset of potential sites J: X ⊆ J.Elements xk ∈ X are indices of potential sites.
Allocation of customers to the locations of new facilities
In uncapacitated problems a customer i ∈ I is always assigned to the most cost-effective new facility j ∈ X
cij = min {cik | k ∈ X}
X induces an allocation.
Page 16Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Warehouse Location ProblemNumber of new facilities
Trade-off between transportation costs and costs for the new facilities
Large number: low transportation costs but high costs for new facilitiesSmall number: high transportation costs but low costs for new facilities
# Facilities
Costs
Transpor-tation Costs
Costs for newfacilities
Total
Costs
Page 17Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Warehouse Location ProblemNumber of new facilities is sensitive to changes in costs.
Example: Daskin (1995)
„Average Distance“: average distance from the customers to the facilities.
Page 18Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Warehouse Location Problem
Other real world examples for the WLPExample 1: supplier selection
Given• bi the demand for product i ∈ I per period• j ∈ J the feasible suppliers• the procurement costs cij of bi quantity units at supplier j• fixed costs fj of business connection
Possible taskChoose the suppliers with minimal costs
Alternative objectiveGiven: delivery time tij for bi quantity units at supplier jChoose at most p suppliers in order to minimize the total delivery time
p – median assignment problem
Page 19Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Warehouse Location ProblemExample 2: Physical Database Design, Index Selection
A database is a set of tables.Tables consists of data fields (columns) and data sets (rows).Data fields j ∈ J can be indexed in order to accelerates requests.
Given• Typical Workload (Set of SQL-Statements like Select, Update, lnsert, Delete)
determined over a Log-File of a period• Answer time gain cij from request i, if field j is indexed
• Maintenance time fj of index j (slowdown of updates at indexing field j)
Single field indexingChoose the fields which should be indexed in order to maximize ATG minus MT
Page 20Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Warehouse Location ProblemExample 3: Location of Bank Accounts
Problem• A company has to pay money to business partners in different cities every
month
• Time between cashing a check and debiting the account, depends on locationj of the debiting account and cashing location i.
Given• Locations i ∈ I of creditors and value bi of monthly payments in i.• Possible bank accounts j ∈ J with monthly account fees fj.• gij equivalent amount of the maturity of a check cut in j and cashed in i.
ObjectiveChoose bank accounts j ∈ J, so that the maturity of checks is maximizedregarding to account fees.
Page 21Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Warehouse Location Problem
Formulation as a mixed integer linear program
Location decisionBinary variable yj, j ∈ J
Allocation decisionVariable xij, i ∈ I, j ∈ J
Page 22Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
WLP as mixed integer linear programMathematical Model
Remark: xij ≥ 0 is sufficient.In uncapacitated problems the customer demand is always completely satisfied by the „cheapest“ new facility.
Each customer i is allocated to a new facility at location j
Minimize transportation costs and costs involved by the facilities
Customer i can only be allocated to location j, if a facility is locatedat j
Page 23Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
WLP as mixed integer linear programInterpretation of the solution (x, y) of the linear program
• costs
• set of potential sites where facilities are located
X := { j | yj = 1, j = 1, …, m}
• customer i is allocated to the new facility at location j, if xij = 1 holds.
Vice versaEach solution X ⊆ J of the WLP with its corresponding allocations induces a solution (x, y) of the linear program
and
Page 24Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
WLP as mixed integer linear programEnumeration of all subsets X from J yields an optimal solution.
HoweverThere are feasible subsets!
ResultThe Warehouse Location Problem is NP – hard.
Possible solution algorithms• Exact algorithms, which determine an optimal solution.
Special algorithms for small problemsDualoc – Algorithm
• HeuristicsGreedy and Interchange Heuristic
Page 25Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Location Theory
Chapter 5 – Discrete Location Problems
Contents
• Introduction
• Classification of Discrete Location Problems
• Warehouse Location Problem (WLP)- Introduction- Heuristics- Dualoc Algorithm
• Extensions of the WLP
Page 26Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Warehouse Location Problem
Heuristics
Algorithms to compute a heuristic (approximate) solution XH.
Characteristics• Highly flexible in order to consider alternative objectives or the incorporation
of complex planning restrictions.• Do not necessarily find an optimal solution.
• In general the quality of a solution is unknown.
Quality of an heuristic solution XH
Relative deviation (in percent), RPD(XH, X*), between the objective function value of XH and the objective function value of an optimal solution X*.
Page 27Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
HeuristicsFormal
However: X* usually unknown⇒ lower bounds for the optimal objective function value
Lower bound LB for ƒ(X*), i.e. LB ≤ ƒ(X*), via• An (optimal) solution of the dual problem• Lagrange – Relaxation• LP – Relaxation (i.e. yj ≥ 0, instead of yj ∈ {0, 1})
Then
Page 28Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Warehouse Location Problem
Location Allocation Heuristics
Distinguish between
Construction heuristicsCompute a first solution of the problem.Classical Algorithms:
Because ω11 < 0, delete location 1 out of J‘ ⇒ J‘ = ∅ stop
Heuristic solution: XH = X2 = {2, 3} and F(XH ) = 81.Value of the LP – Relaxation: 76 ⇒ XH max. 6.5% away from optimal solution.
Greedy Heuristic: Examplex1 = 2
Page 35Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Warehouse Location ProblemInterchange Heuristic
Given a solution X = { x1, …, xp }.Interchange locations step-by-step in order to improve the solution.
If xk ∈ X is a site where a facility is located and j ∈ J‘ = J \ X a site where no facility has been located yet, then obtain a new solution X‘ by
X‘ = X \ { xk } ∪ { j }
If the resulting solution X‘ is better than the last solution X: F(X‘) < F(X), then „accept“ this exchange, i.e. set X = X‘.
Otherwise try to interchange other solution-locations and non-solution-locations in order to improve the solution.
X
J‘
Page 36Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Interchange HeuristicStop the algorithm, if the solution cannot be improved by interchanges of locations.
Sequential choice of exchange pairs.Two possible strategies:
First ImprovementStart with the first location of the solution and interchange with all potential locations, which are not yet in the solution.Repeat this step for the second, third,… location of the solution.Stop, when an exchange provides a better solution.
Best ImprovementDetermine the cost saving for all possible interchanges and realize the interchange with maximal cost saving.
Page 37Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Interchange HeuristicCosts of the new solutionLet
• ui = cij* = min { cij | j ∈ X } the lowest costs to satisfy the demand of customer i from a facility located at a site in X.
• uik = min { cij | j ∈ X, j ≠ xk } the lowest costs to satisfy the demand of
customer i from a facility located at a site inX \ { xk }.
Change in costs by interchanging the locations k ∈ X and j ∈ J‘
Page 38Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Interchange HeuristicFirst Improvement
Do save the interchange of locations xk ∈ X and j ∈ J‘ as soon as holds.
Best ImprovementDo save the interchange for those xk‘ ∈ X and j‘ ∈ J‘, for which
Objective function value from solution X:
Write all information one needs for theinterchange of locations xk ∈ X and j ∈ J‘ in one table
Page 39Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Interchange HeuristicInterchange Algorithm
1. Input X = {x1, …, xp } and J‘ := J \ X.
2. Compute ui = min { cij | j ∈ X } for all i = 1,…,n.3. For k = 1,…,p
Compute uik = min { cij | j ∈ X, j ≠ xk } for all i = 1,…,n.
For all j ∈ J‘If ωj
k > 0, then interchange X = X \ {xk } ∪ { j }, update J‘ = J \ X and go to step 2.
4. X is a heuristic solution of the problem
ImplementationCollect the required data for step 2 and 3 in a single table.
Page 40Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Example: First Improvement StrategyLet the transportationcosts and the costs for new facilities be and
X = {2, 3 } solution of the Greedy-Algorithm, F(X) = 81.
Step 2: Compute ui for all i = 1,…,n.Step 3: Start with k = 1.
One gets the following table
The first interchange-pair provides immediately the first improvement: ωj1 > 0.
Interchange Heuristic
x1 = 2 x2 = 3
Page 41Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
⇒ Interchange X = X \ { x1 = 2 } ∪ { 1 }, i.e. replace location 2 by 1 in the solution.
Improved solution X = {1, 3 } with F(X) = 76.
Begin with step 2 again. One gets the following table
No ωjk > 0 ⇒ the algorithm stops.
Heuristic solution XH = {1,3 } withF(XH) = 76.
Value of the LP – Relaxation: 76
⇒ The obtained heuristic solution XH is optimal.
Interchange Heuristic: Example
x2 = 3x1 = 1
Page 42Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Location Theory
Chapter 5 – Discrete Location Problems
Contents
• Introduction
• Classification of Discrete Location Problems
• Warehouse Location Problem (WLP)- Introduction- Heuristics- Dualoc Algorithm
• Extensions of the WLP
Page 43Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Warehouse Location Problem
Dualoc Algorithm
The Dualoc algorithm is due to Erlenkotter (1978).
It is based on the Branch & Bound algorithm and determines an optimal solution of the mixed integer linear program for the WLP.
It uses extremely efficient algorithms to compute the lower and upper bounds for the subproblems in the root and the nodes of the B&B – tree.
These bounds can be computed as follows:
• heuristics solution of the dual problem of the LP – relaxation lower bound
• construction of an integer solution of the WLPupper bound
Page 44Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Dualoc Algorithm
Efficiency of the Dualoc algorithm is based on the “nice” property of the WLP, that the LP – Relaxation of the mixed integer linear program for theWLP often provides already an integer solution.
Page 45Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Dualoc Algorithm
Overview of the Dualoc AlgorithmFormulate the WLP as a mixed integer LP
Formulate the dual problem to the LP – relaxation
Compute a heuristic solution of the dual LPlower bound for the WLP
Derive a primal solution from the dual oneupper bound for the WLP
lower bound = upper
bound ?Adjust dual solution
Stopprimal
solution optimal
yes no Modified dual solution ?
no
Branch in the B & B – tree
yes
Page 46Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Dualoc Algorithm
The WLP as a mixed integer linear program
LP – RelaxationRelaxation of all integer constraints
Page 47Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Dualoc Algorithm
The dual LP
LP – relaxation (LP) ofthe mixed integer linearprogram for the WLP …
… and the correspondingdual linear program (DP)with the dual variablesvi and wij.
Page 48Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
The Dual LPElimination of variables wij
For a feasible choice of the vi , feasibleand as small as possible values wij canbe found easily.
The wij are bounded from below by constraints(1) and (2):
⇒ Replace wij by max {0, vi – cij } in the dual LP.
Simplified (Reduced) dual LP
Page 49Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
The Dual LPIt holds
vi = 0, i = 1, …, n (i.e. v = 0) is a feasible solution of the (RDP).(Assuming that fj, cij ≥ 0, ∀ i, j)
Let (x‘, y‘) and (x, y) be optimal solutions of (LP) and (WLP), respectively. Then
Weak duality theorem of linear optimizationLet (x‘, y‘) be a feasible solution of (LP).Then for each feasible solution v of (RDP) holds
⇒ RD(v) is a lower bound for the optimal objective function value of (LP) and (WLP).
Page 50Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Dualoc Algorithm
Overview of the Dualoc AlgorithmFormulate the WLP as a mixed integer LP
Formulate the dual problem to the LP – relaxation
Compute a heuristic solution of the dual LPlower bound for the WLP
Derive a primal solution from the Dual oneupper bound for the WLP
lower bound = upper
bound ?Adjust dual solution
Stopprimal
solution optimal
yes no Modified dual solution ?
no
Branch at B & B – tree
yes
Page 51Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Dualoc Algorithm
Dual Ascent Algorithm
(RDP) can be solved optimally using the simplex algorithm.Alternative:
Dual Ascent – AlgorithmSimple, fast and efficient heuristic, to determine a feasible, but not necessarilyoptimal solution of (RDP).
ObservationIn many cases, the solution of the Dual Ascent Algorithm is already optimalfor the dual problem.
ApproachStarting with an initial solution v, sequentially increase each vi by a small amount until no further increase is possible anymore.
Page 52Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Dual Ascent AlgorithmDefine
as the slack of the j-th constraint, i.e. the j-th facility of (RDP).
For each solution v of (RDP) the constraints sj ≥ 0 must hold.
If v = 0 holds, it follows sj = fj for the slack sj of each constraint.
Now sort the cost elements cij of each customer i∈I in non-decreasing order and denote them ci
k , k = 1,…,m.
Define for each i∈ I the index k(i) := min {k | cik ≥ vi } as the index of the
first cost element that is larger than vi.
Page 53Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Dual Ascent AlgorithmNow try to increase sequentially each vi to the next larger value ci
k(i).Only allowed, if no slack sj becomes negative.
Let ∆ := cik(i) - vi be the value of the intended increase of vi.
Effect of the increase on the slack sj , j∈ J:None, if vi < cij, because
and hence
∆, if vi ≥ cij, because
i.e. the slack sj decreases by ∆:
Page 54Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Dual Ascent AlgorithmDefine:
Set of all constraints j whose slack sj would decrease by an increase of vi.
For an intended increase of vi by ∆ it must hold:
and therewith:
⇒ increase vi by
Let I‘ ⊆ I be the set of customer indices i for which vi still can be increased.
Page 55Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Dual Ascent AlgorithmDual Ascent Algorithm
1. InitializationLet ci
1 ≤ ci2 ≤ … ≤ ci
m ≤ cim+1 := ∞ be an ordering of the cost elements, i∈ I
Set I‘ := I and vi := ci1 for all i∈ I.
Compute Ji, i∈ I and sj, j∈ J.Determine k(i) := min {k | ci
k ≥ vi }, i∈ I . If vi = cik(i) then k(i) := k(i) + 1.
2. IterationFor all i∈ I‘
Determine ∆i . If ∆i = 0 then set I‘ = I‘ \ { i } and choose the next i ∈ I‘.Set ∆ = min {∆i , ci
k(i) - vi }.If ∆ = ∆i then set I‘ = I‘ \ { i }, else set k(i) := k(i) + 1.Set vi := vi + ∆ and for all j ∈ Ji set sj := sj – ∆.Update Ji.
3. StopIf I‘ =∅, i.e. no vi can be increased anymore, Stop. Else Step 2
Page 56Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
ExampleLet the transportationcosts and the costs for new facilities be and
I.e. 6 customers and 5 potential locations.
1. InitializationSort the cost elements: e.g. for i = 1
Page 60Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
I‘ ≠ ∅ ⇒ start a new iteration. Table after i = 1,…,6
Remarks• v5 could not be increased to the third largest cost value 8 in row 5, because
otherwise s4 would become negative.
• For v6 even ∆6 = 0.⇒ Change: I‘ = {1,…,5}
Dual Ascent Algorithm: Example
Page 61Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
I‘ ≠ ∅ ⇒ start a new iteration. Table after i = 1,…,5
Merely v2 could be increased to the fourth largest cost value 4.Change: I‘ = {2 }
I‘ ≠ ∅ ⇒ start a new iteration. However, v2 can not be increased anymore (s2 = 0) ⇒ stop
Solution: v = (4,4,9,6,5,4) with RD(v) = 32.
Dual Ascent Algorithm: Example
Page 62Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Warehouse Location Problem
Overview of the Dualoc AlgorithmFormulate the WLP as a mixed integer LP
Formulate the dual problem to the LP – relaxation
Compute a heuristic solution of the dual LPlower bound for the WLP
Derive a primal solution from the Dual oneupper bound for the WLP
lower bound = upper
bound ?Adjust dual solution
Stopprimal
solution optimal
yes no Modified dual solution ?
no
Branch at B & B – tree
yes
Page 63Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Dualoc Algorithm
Construction Algorithm
Based on a feasible solution v of (RDP), construct an integer solution (x, y)of (WLP).For this purpose consider the complementary slackness conditions.
TheoremLet (x, y) and (v, w) be feasible solutions of the LP – relaxation (P) and the dual LP (DP), respectively. (x, y) and (v, w) are optimal if:
Page 64Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Construction AlgorithmRemark: The fourth constraint is always satisfied for each solution.
Set and
into the complementary slackness conditions in order to get the following conditions for optimality.
Optimality conditionsLet v be a feasible solution of (RDP). If an integer feasible solution (x,y) of (WLP) can be found, so that the following conditions for optimalityhold,
then both solutions are optimal.
Page 65Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Construction AlgorithmProof
(x, y) is also a feasible solution for the LP – Relaxation (P)⇒ (x, y) optimal for (P) (complementary slackness conditions hold)⇒ F(x, y) = F‘(x, y) (objective function value identical for (P) and
(WLP))⇒ (x, y) optimal for (WLP)
Remark: the optimality conditions are just sufficient conditions.
Determination of the best integer solution for which (3) – (5) holds, is a combinatorial problem.Use a simple heuristic.
IdeaSelect a feasible solution (x, y) of (WLP),for which the conditions (3) and (4) hold.If (5) also holds, then (x, y) is optimal for (WLP).
Page 66Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Construction AlgorithmFor a feasible solution of (WLP) one has to determine
1. locations y of new facilities and2. allocations x of customers to these facility locations.
From (3) follows: yj > 0 just for sj = 0.
TherewithDefine J+ = { j ∈ J | sj = 0 } and set yj := 0 for all j ∉ J+.
J+ is the set of candidates of potential locations for the inclusion into the solution.
Actual selection of locations out of the set J+ is done by the analysis of the permitted allocations.
Page 67Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Construction AlgorithmCustomer i can only be assigned to alocation j ∈ J+, if cij ≤ vi holds.
If for customer i exists• only one location j ∈ J+ with cij ≤ vi, then xij = 1 must hold and therewith yj = 1
(because every customer must be allocated to one location).
• several locations j ∈ J+ with cij ≤ vi, then only one of these has to be a part of the solution. If none of them is a part of the solution yet, then choose the one with lowest costs j*: xij* = 1 and yj* = 1.
If for the solution (x, y) constraint (5) holds, then it is optimal.Otherwise one can not decide whether this solution is optimal (i.e. the solution may or may not be optimal!).
Denote X = { j ∈ J | yj = 1 } as the set of all potential locations, selected by the heuristic.
Page 68Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Construction Algorithm
Construction AlgorithmGiven: feasible solution v of (RDP).1. Initialization
Set X = ∅. Compute J+ = { j ∈ J | sj = 0 }.
2. Selection of locationsFor all i∈ I
If there is exactly one location j ∈ J+ for customer i with cij ≤ vi and j ∉ X,then set X = X ∪ { j } and yj = 1.
For all i∈ IIf there are several locations j ∈ J+ for customer i with cij ≤ vi, but none of these j ∈ X, then set X = X ∪ { j* } and yj* = 1 for the j* with cij* = min { cij | cij ≤ vi}.
3. AllocationAllocate every customer i to the location j ∈ X with lowest costsxij* = 1 for the j* with cij* = min { cij | cij ≤ vi, j ∈ X }.
Page 69Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
ExampleFeasible dual solution v = (4, 4, 9, 6, 5, 4)of the Dual Ascent – Algorithm, withs = (1, 0, 3, 0, 6) and RD(v) = 32.
2. Selection of locationsExactly one location j ∈ J+ with cij ≤ vi for customer i ?
i = 1: solely c12 ≤ v1 ⇒ X := {2},i = 2: solely c24 ≤ v2 ⇒ X := {2, 4} = J+.
One can already stop, since all permitted locations are already selected.⇒ y2 = y4 = 1 and y1 = y3 = y5 = 0.
3. Assignment x12 = x24 = x32 = x42 = x54 = x64 = 1. All other xij = 0.
Construction Algorithm
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Value of the solution: F(x, y) = 35 > 32 = RD(v).
Optimality constraint (5) is not satisfied, sincev4 > c42 and v4 > c44
and no more than one x4∗ is allowed to be positive. Here: x42 = 1 and x44 = 0.
⇒ no information about optimality of the solution (x, y).
Construction Algorithm : Example
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Warehouse Location Problem
Overview of the Dualoc AlgorithmFormulate the WLP as a mixed integer LP
Formulate the dual problem to the LP – relaxation
Compute a heuristic solution of the dual LPlower bound for the WLP
Construct primal solution out of the Dualupper bound for the WLP
lowerbound = upper
bound ?Adjust dual solution
Stopprimal
solution optimal
yes noModified dual
solution ?
no
Branch at B & B – tree
yes
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Dual Adjustment Algorithm
The dual ascent algorithm does not always finds the optimal dual solution.⇒ It is possible to improve the solution.
If the dual ascent algorithm and the construction heuristic cannot find a verifiable optimal solution for (WLP), the following heuristic can be used to adjust the dual solution in order to eliminate all violations of the complementary slackness conditions.
If 2 or more of these constraints vi > cij are violated for one customer i, then decrease the value of vi, to reduce the number of violations.
Through this decrease it should be possible to increase at least one other dual variable vi by the same amount.
Search for an alternative solution, which is, if possible, better than the previous one and involves fewer violations of condition (5).
Dualoc Algorithm
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Let
be the set of locations where customer i has to be allocated, to satisfy condition (5), i.e. xij = 1 ! (since yj = 1 for all j ∈ X.)
Condition (5) is violated, if | Jix | > 1.
Reduce | Jix | by decreasing vi to the next smaller cij.
Thereby at least one violation is eliminated.But the objective function value RD(v) of the dual problem is also reduced.
Dual Adjustment Algorithm
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Example (cont.)
It holds: v4 > c42 and v4 > c44
⇒ J4x = { 2, 4 }.
Decrease v4 = 6 to the next smaller value,that is c44 = 3: v4 = 3
⇒ J4x = { 2 }.
The slack of s2 and s4 increases each by 3.Now it is possible to increase e.g. v5 by 3.
In generalDecreasing vi increases the slack of all constraints j ∈ Ji
x.
Use this slack to increase other dual variables vk. But only, if they are not blocked by other constraints.
Dual Adjustment Algorithm
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Because | Jix | > 1, there exist j*, j** ∈ Ji
x , with
cij* = min { cij | j ∈ Jix } and cij** = min { cij | j ∈ Ji
x, j ≠ j* }.
Decreasing of vi increases the slack of the constraints j* and j**.
LetIjx = { i ∈ I | j ∈ X is the only location with vi ≥ cij }, for all j ∈ X
be the set of customers, which solely can be allocated to location j.
If Ijx ≠ ∅ holds, the dual variables vi, i ∈ Ijx, cannot be blocked by other constraints.
Dual variables vk, k ∈ Ij*x, Ij**x, are „good“ candidates, but no guaranty for a better dual solution.
Dual Adjustment Algorithm
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Now determine an index i with | Jix | > 1, decrease vi to the next smaller
value cij and adjust the slacks of the constraints j ∈ Jix.
Thereafter use the dual ascent algorithm 3 times to increase the value of other dual variables.
But with the following changes:• use in the first step as set I‘ ⊆ I of customer indices i for which vi can
still be increasedI‘ := Ij*x ∪ Ij**x
• at the second time extend I‘ by i: I‘ := I‘ ∪ { i }• and at the last time to total I: I‘ := I.
If one gets another (better) dual solution v, use the construction heuristic to construct a new integer solution (x, y) for (WLP).
Dual Adjustment Algorithm
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Dual Adjustment Algorithm
Dual Adjustment AlgorithmGiven: feasible solution v of (RDP) and (x, y) of (WLP).
IterationForall i∈ I
If | Jix | ≤ 1 holds or Ij*x ∪ Ij**x = ∅, then continue with the next i.
Determine the next smaller cij: ∆ := max { cij | cij < vi, j ∈ J }.Set sj := sj + (vi – ∆) for all j ∈ Ji
x .Set vi := ∆.
Perform the dual ascent – algorithm successively with- I‘ := Ij*x ∪ Ij**x,- I‘ := I‘ ∪ { i } and- I‘ := I
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Example (cont.)Feasible dual solution v = (4, 4, 9, 6, 5, 4)with s = (1, 0, 3, 0, 6) and RD(v) = 32.
2. Selection of locationsExactly one location j ∈ J+ with cij ≤ vi for customer i ?
i = 1: solely c11 ≤ v1 ⇒ X := {1},i = 3: solely c31 ≤ v3, 1 already included in X. i = 4: solely c44 ≤ v4 ⇒ X := {1, 4}.
If there are several locations j ∈ J+ with cij ≤ vi for customer i, but none of thesej ∈ X, then set X = X ∪ { j* } and yj* = 1 for the j* for which holdscij* = min { cij | cij ≤ vi }.
The solution already consists of a location for each customers i.⇒ y1 = y4 = 1 and y2 = y3 = y5 = 0.
3. Assignmentx11 = x21 = x31 = x44 = x54 = x64 = 1. All other xij = 0.
Dual Adjustment Algorithm: Example
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Condition (5) holds⇒ (x, y) is optimal.
It also holds, thatthe objective function value F(x, y) of the solution (x, y) for the (WLP) and of the dual LP are equal.
F(x, y) = 33 = RD(v).
⇒ (x, y) is optimal.
Dual Adjustment Algorithm: Example
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Warehouse Location Problem
Overview of the Dualoc AlgorithmFormulate the WLP as a mixed integer LP
Formulate the dual problem to the LP – relaxation
Compute a heuristic solution of the dual LPlower bound for the WLP
Construct primal solution out of the Dualupper bound for the WLP
lower bound = upper
bound ?Adjust dual solution
Stopprimal
solution optimal
yes no Modified dual solution ?
no
Branch at B & B – tree
yes
Page 83Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Branch & Bound Algorithm
If it is not possible to find a verifiable optimal solution of the (WLP), then the branch & bound algorithm can be used to• try to prove the optimality of the best found integer solution or• determine an other optimal solution.
Branching is done by fixing a binary variable yj to 0 or 1 respectively. It can be implicitly achieved by changing the fixed costs of the facility that has to be located:
• yj = 1: Set fixed costs fj := 0the location will be included in the solution.
• yj = 0: Set fixed costs fj := M (M a very large number)the location will not be in the solution.
Solve the subproblems in the nodes again by using a combination of the dual ascent, construction algorithm as well as dual adjustment.
Dualoc Algorithm
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Location Theory
Chapter 5 – Discrete Location Problems
Contents
• Introduction
• Classification of Discrete Location Problems
• Warehouse Location Problem (WLP)
• Extensions of the WLP
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Chapter 5 – Discrete Location Problems
Extensions of the WLPClassical extensions of the Warehouse Location Problem:
• Capacitated WLP
• Closing of Facilities
• Budget restrictions
• Different facility types
• Multi-Product and Multi-Activity Models
• Multi-Echelon Models
• Multi-Period (dynamic) Models
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Extensions of the WLP
The Capacitated WLP
Also known as Capacitated Facility Location Problem (CFLP).
Capacity restrictions for facilities limit the quantities, which can be• produced• transported• handled, etc..
The „single assignment“ property no longer holds for capacitated WLP:
In a solution of the problem, the demand of a customer is not necessarily completely satisfied by a single facility.
That is a customer can simultaneously be assigned pro-rata to several new facilities.
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Extensions of the WLPRedefine the assignment variable xij as:
xij denotes the is a fraction of demand of customer i, which is satisfied by a new
facility at location j (i.e. transported from j to i).
Therewithbi xij denotes the amount of the product, which is delivered from a new facility at
location j to customer i.
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Extensions of the WLPMaximal Capacity
Maximal capacity Kjmax of a new facility at the potential location j ∈ J.
Extension
The amount of goods, which is delivered from a new facility at location j to all customers, has to be smaller than the capacity of the facility.
Implications- Amount of goods equal to zero, if no facility was built.
- If goods are delivered, then the facility has to be located.
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The Capacitated WLPMinimal Capacity Utilization
Often there is a minimal capacity utilization Kjmin of facilities required, e.g.,
because of profitability.
Extension
Implication- If a facility is located, then the outgoing amount of goods has to exceed a
minimal capacity utilization
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The Capacitated WLPSingle-Sourcing
Often a unique allocation (called Single-Sourcing) of customers to facilities is required.
Causes for Single-Sourcing- closer customer loyalty, easier logistics- better traceability in case of deficiency
Model extension
Demand of a customer i is either completely satisfied by a new facility at location j, or not at all.
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The Capacitated WLPMathematical model
This problem is much more difficult to solve than the WLP.In particular with Single-Sourcing.
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Extensions of the WLP
Closing of facilities
Consideration of facilities, which already exist at the beginning of the planning period and may be closed.
DefineJo = Set of potential locations where a new facility can be built.
Jc = Set of locations where a facility is already in use, but which may be closed.
foj and fc
j denote costs for opening and closing, respectively, of facilities at location j ∈ Jo or Jc, respectively.
Costs for closing fcj may be positive or negative
Positive: costs for demolition, disposal, severance payments for staffNegative: sales profit
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Closing of facilitiesLocation decisions
Opening
Closing
Extensions of the modelCosts for opening and closing
Supply is just possible from open facilities
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Closing of facilities
Mathematical model
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Extensions of the WLP
Budget restrictions
Only variable costs are to be minimized.
Fixed costs fj of facilities (e.g. for construction, infrastructure) are financed by limited budget B.
However, consider variable costs gj for operating the facilities.
Model extensionOperating costs for the facilities and the transportation of goods
Budget constraint
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Budget restrictionsMathematical model
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Extensions of the WLP
Different facility types
For the construction of a new facility at a location one can choose from several different building types.
Example• regional or central warehouses• production plants for temporary fabrication or finishing
In case of capacity facilities also differentiate between various sizes of facilities of the same type.Set K of possible facility types.
Location decision: binary variable yjk, j ∈ J, k ∈ K
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Different facility types
Fixed costsFixed costs fjk for the opening of a facility from type k at location j.Model extension
Maximum capacity Max. capacity Kjk of a facility of type k at location j.Model extension
Amount of goods bounded by the capacity of type k of the new facility at locationj.
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Different facility typesBuilding restriction
Model extension
At most one type of facility can be placed at location j.
Forbid the opening of a facility of type k ∈ K at location j ∈ J by fixing the corresponding decision variable in the model to zero: yjk = 0.
E.g. because infrastructure insufficient, needed personnel not on the spot,…
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Different facility types
Mathematical model
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Extensions of the WLPMulti-Product and Multi-Activity Models
Customers have a demand for different products, services or activities with various characteristics.Necessary, if product aggregation not reasonable.
E.g. due to • The case that transportation or inventory requirements are too different
(pallets refrigerating store house)• Completely different service levels (first aid emergency doctor)
Availability• All products / activities are available at each location without restrictions • Not all locations offer the complete range
Costs• Production / storage of a product or offering a service possibly generate
additional product-dependent fixed costs.
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Multi-Product and Multi-Activity ModelsSet P of products.
Definexijp = Rate of the demand for product p of customer i, which is satisfied by a
new facility at location j.
CostsTransportation costs cijp and customer demands bip are product-dependent, p ∈ P.
Model extensionCosts for satisfying the customer demands
Customer demands for all products satisfied
Page 103Facility Location and Strategic Supply Chain Management
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Multi-Product and Multi-Activity ModelsProduct availability
Binary variable zjp, j ∈ J, p ∈ P
Fixed costs gjp for offering product p in a new facility at location j.
Model extensionCosts for making services and goods available
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Multi-Product and Multi-Activity ModelsCustomers can only be served by facilities which offer the product
Locations can only offer products, if a facility has been located
Forbid that a product p ∈ P is not available at location j ∈ J by fixing the corresponding decision variable in the model to zero:
zjp = 0.
Reasons- infrastructure insufficient- economically not reasonable- no qualified staff on the spot
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Multi-Product and Multi-Activity ModelsSingle-Sourcing
Total demand of a customer is satisfied from the same location.E.g. for better in case of deficiencies, closer customer loyalty, …
Auxiliary variables for Single-Sourcing
Model extensionIf customer i is (pro-rata) assigned to facility j, then he obtains products from
there
Customers can obtain products only from exactly one location
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Multi-Product and Multi-Activity Models
Mathematical model
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Extensions of the WLP
Multi-Level Models
(Production) distribution network often includes several stages of facilities each of the same or similar type.
Example: distribution of goods over three transport stages
Levels classically are arranged hierarchically, i.e. goods always flow just from one stage to the next one.
But also transportation links within levels or beyond levels possible.
factories centralstore customersregional
store
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Multi-Stage ModelsLocation decisions on
• One stage possible (e.g. warehouses)• Several stages possible (e.g. warehouses and factories)
Define for each stage a corresponding set of location variables.
Two-stage problem with location decisions on one stage
first stage: customers i ∈ Isecond stage: potential locations j ∈ J for warehousesthird stage: factories k ∈ K
Transport variables and transportation costs factories warehousezjk = Amount of the product, which will be transported from factory k to warehouse jtjk = Transportation costs per quantity unit for factory k warehouse j
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Multi-Stage ModelsExtensions
Transportation costs from factory to warehouse
Quantity of goods from factory to warehouse = transportation volume from factory to customer
Flow conservation:“What goes in, must come out”
zjk bi xij
j
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Multi-Stage ModelsMathematical model
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Extensions of the WLP
Multi-Period (dynamic) Models
Planning decisions about the opening (or closing) of facilities and their consequences often range over several years.Therefore it is important to know “where" but in particular also “when”facilities are to be opened.
Divide the planning period into several periods (e.g. months, years).
year t year t + 1factories customerswarehouse factories customerswarehouse
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Multi-Period (Dynamic) ModelsApplications
Long-term constant increase in demand (of distribution).
When is the best moment for the opening of the new factory or inventory?
Stepwise reorganization of the (production) distribution network.
Do not „move“ all warehouses at once, but during a certain time period.
Consideration of high demand variabilitiySeasonal rental of facilities, if demand peaks exceed the capacity.
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Multi-Period (Dynamic) ModelsVarious problem issues
Alternative 1If a facility is opened it remains in use for the rest of the planning period.Typically for facilities with high capital expenditure.
Alternative 2A facility remains open only for the duration of the period.E.g. when storage areas or production capacities were only rented.
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Multi-Period (Dynamic) ModelsSet T of time periods.Transportation costs cijt and customer demands bit are time-dependent,t ∈ T.
Definexijt = Part of the demand from customer i, which is satisfied in period t from a
new facility at location j.
Model extensionCosts for satisfying the customer demands for the whole planning period
Customer demand satisfied in each period
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Multi-Period (Dynamic) ModelsAlternative 1
Location decision: Binary variable yjt, j ∈ J, t ∈ T
Fixed costs fjt and operating costs gjt for the opening and the operation, respectively, of a facility at location j in period t.
Set
Costs for construction and operation of a facility at location j in period t for the rest of the planning period.
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Multi-Period (Dynamic) ModelsExtensions
Costs for building a facility
Satisfaction of customer demands only from an already opened facility
Only open a facility at most once
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Multi-Period (Dynamic) ModelsAlternative 2
Location decision: Binary variable yjt, j ∈ J, t ∈ T
Operating costs gjt of a facility at location j during period t (e.g. lease, personnel costs) .No fixed costs.
ExtensionsOperating costs
Just „active“ facilities can satisfy customer demands
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Multi-Period (Dynamic) Models
Mathematical models
Alternative 1
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