DISTRIBUTION AND NETWORK MODELS (1/2) Chapter 6 MANGT 521 (B): Quantitative Management.
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Transcript of DISTRIBUTION AND NETWORK MODELS (1/2) Chapter 6 MANGT 521 (B): Quantitative Management.
DISTRIBUTION AND NETWORK MODELS (1/2)
Chapter 6
MANGT 521 (B): Quantitative Management
Chapter 6 (1/2)-2
Chapter 6Distribution and Network Models
1. Transportation Problem• Network Representation• General LP Formulation
2. Assignment Problem• Network Representation• General LP Formulation
3. Transshipment Problem• Network Representation• General LP Formulation
4. Shortest-Route Problem
Chapter 6 (1/2)-3
Transportation, Assignment, and Transshipment Problems
A network model is one which can be represented by a set of nodes, a set of arcs, and functions (e.g. costs, supplies, demands, etc.) associated with the arcs and/or nodes.
Examples of network problems:Transportation, assignment, transshipment, shortest-route, and maximal flow problems of this chapter as well as the minimal spanning tree and PERT/CPM problems (in Project Management courses).
Chapter 6 (1/2)-4
Transportation, Assignment, and Transshipment Problems
Each of the four problems of this chapter can be formulated as linear programs and solved by general purpose LP computer package.
For each of the four problems, if the right-hand side of the linear programming formulations are all integers, the optimal solution will be in terms of integer values for the decision variables.
However, there are many computer packages that contain separate computer codes for these problems which take advantage of their network structure.
Chapter 6 (1/2)-5
1. Transportation Problem
The transportation problem seeks to minimize the total shipping costs of transporting goods from m origins (each with a supply si) to n destinations (each with a demand dj), when the unit shipping cost from an origin, i, to a destination, j, is cij.
The # of constraints in a transportation LP formulation = (# of origins) + (# of destinations) = m + n
The network representation for a transportation problem with two sources and three destinations is given on the next slide.
Chapter 6 (1/2)-6
1. Transportation Problem
Network Representation
22
c1
1c12
c13
c21
c22c23
d1
d2
d3
s1
s2
Sources Destinations
33
22
11
11
Chapter 6 (1/2)-7
1. Transportation Problem
A General LP Model
Using the notation: xij = number of units shipped from
origin i to destination j cij = cost per unit of shipping from
origin i to destination j si = supply or capacity in units at origin i
dj = demand in units at destination jcontinued
Chapter 6 (1/2)-8
1. Transportation Problem
A General LP Model (continued)
To obtain a feasible solution in a transportation problem, “total supply ≥ total demand”
1 1
Min m n
ij iji j
c x
1
1,2, , Supplyn
ij ij
x s i m
1
1,2, , Demandm
ij ji
x d j n
xij > 0 for all i and j
Chapter 6 (1/2)-9
Transportation Problem Example:Foster Generations
Foster Generators operates plants in Cleveland, Ohio; Bedford, Indiana; and York, Pennsylvania. Production capacities over the next three-month planning period for one particular type of generator are as follows:
Chapter 6 (1/2)-10
Transportation Problem Example:Foster Generations
The firm distributes its generators through four regional distribution centers located in Boston, Chicago, St. Louis, and Lexington; the three-month forecast of demand for the distribution centers is as follows:
Chapter 6 (1/2)-11
Transportation Problem Example:Foster Generations
The cost for each unit shipped on each route is also given as follows:
Management would like to determine how much of its production should be shipped from each plant to each distribution center.
Chapter 6 (1/2)-12
Transportation Problem Example:Foster Generations
Network Representation
Chapter 6 (1/2)-13
Transportation Problem Example:Foster Generations
LP Formulation
• The objective of the transportation problem is to minimize the total transportation cost:
• Therefore, the objective function is:
Chapter 6 (1/2)-14
Transportation Problem Example:Foster Generations
• Consider supply constraints• Total # of units shipped from Cleveland:
• Total # of units shipped from Bedford:
• Total # of units shipped from York:
Chapter 6 (1/2)-15
Transportation Problem Example:Foster Generations
• Consider demand constraints• Four demand constraints are needed to
ensure that destination demands will be satisfied:
Chapter 6 (1/2)-16
Transportation Problem Example:Foster Generations
Combining the objective function and constraints into one model provides a 12-variable, 7-constraint LP formulation of the Foster Generators’ transportation problem:
Chapter 6 (1/2)-17
Transportation Problem Example:Foster Generations
Solution Summary
Chapter 6 (1/2)-18
Transportation Problem Example:Foster Generations
Network Representation of Optimal Solution
Chapter 6 (1/2)-19
Transportation Problem Variations
Variations of the basic transportation model may involve one or more of the following situations:1) Total supply not equal to total demand2) Route capacities or route minimums3) Unacceptable routes
Can be easily accommodated with slight modifications
Chapter 6 (1/2)-20
Transportation Problem Variations
1) Total supply not equal to total demand “Total supply > total demand”• No modification in the LP formulation is
necessary.• Excess supply will appear as slack (i.e.
unused supply or amount not shipped from the origin).
Chapter 6 (1/2)-21
Transportation Problem Variations
1) Total supply not equal to total demand (cont’d) “Total supply < total demand”
• The LP Model of a transportation problem will NOT have a feasible solution.
• Add a dummy origin with supply equal to the shortage amount.
• Assign a zero (0) shipping cost per unit to the dummy origin.
• The amount “shipped” from the dummy origin (in the solution) will not actually be shipped.
• The destination(s) showing shipments being received from the dummy origin will be the destinations experiencing a shortfall, or unsatisfied demand.
Chapter 6 (1/2)-22
Transportation Problem Variations
2) Route capacities or route minimums Also can accommodate capacities or minimum
quantities for one or more of the routes (“capacitated transportation problem”)
Maximum route capacity from i to j: xij < Lij
Fosters Generators example (Example 1):• If the York-Boston route (from origin 3 to
destination 1) had a capacity of 1,000 units because of limited space availability on its normal mode of transportation, the following route capacity constraint should be added to the existing LP model:
x31 < 1,000
Chapter 6 (1/2)-23
Transportation Problem Variations
2) Route capacities or route minimums (cont’d) Minimum shipping guarantee from i to j:
xij > Mij
Fosters Generators example (Example 1):• If the Bedford-Chicago route (from origin 2 to
destination 2) had a previously committed order of at least 2,000 units, the following route minimum constraint should be added to the existing LP model:
x22 > 2,000
Chapter 6 (1/2)-24
Transportation Problem Variations
3) Unacceptable routes Establishing a route from every origin to every
destination may not be possible. Simply drop the corresponding arc from the
network and remove the corresponding variable from the LP formulation.
Fosters Generators example:• If the Cleveland–St. Louis route (from origin 1 to
destination 3) were unacceptable or unusable, the arc from Cleveland to St. Louis (x13) could be removed from the LP formulation.
Chapter 6 (1/2)-25
2. Assignment Problem
Typical assignment problems involve:• Assigning jobs to machines, agents to tasks,
sales personnel to sales, territories, contracts to bidders, etc.
A special case of the transportation problem in which all supply and demand values equal to 1, and the amount shipped over each arc is either 0 or 1; hence assignment problems may be solved as linear programs.
It assumes all workers are assigned and each job is performed.
Chapter 6 (1/2)-26
2. Assignment Problem
An assignment problem seeks to minimize the total cost, minimize time, or maximize profit assignment of m workers to n jobs, given that the cost of worker i performing job j is cij.
The network representation of an assignment problem with three workers and three jobs is shown on the next slide.
Chapter 6 (1/2)-27
2. Assignment Problem
Network Representation
22
33
11
22
33
11c11
c12
c13
c21 c22
c23
c31 c32
c33
Agents Tasks
Chapter 6 (1/2)-28
A General LP Model
Using the notation:
xij = 1 if agent i is assigned to task j
0 otherwise
cij = cost of assigning agent i to task j
2. Assignment Problem
Chapter 6 (1/2)-29
A General LP Model (continued)
2. Assignment Problem
1 1
Min m n
ij iji j
c x
1
1 1,2, , Agentsn
ijj
x i m
1
1 1,2, , Tasksm
iji
x j n
xij > 0 for all i and j
If an agent is permitted to work for multiple (t) tasks at the same time:
1
1,2, , Agentsn
ijj
x t i m
Chapter 6 (1/2)-30
Assignment Problem: Example #1Fowle Marketing Research
Fowle Marketing Research has just received requests for market research studies from three new clients. The company faces the task of assigning a project leader (agent) to each client (task). Currently, three individuals have no other commitments and are available for the project leader assignments. Fowle’s management realizes, however, that the time required to complete each study will depend on the experience and ability of the project leader assigned. The three projects have approximately the same priority, and management wants to assign project leaders to minimize the total number of days required to complete all three projects. If a project leader is to be assigned to one client only, what assignments should be made?
Chapter 6 (1/2)-31
Management must first consider all possible project leader–client assignments and then estimate the corresponding project completion times.
Estimated completion times (in days)
Assignment Problem: Example #1Fowle Marketing Research
Chapter 6 (1/2)-32
Network Representation
Assignment Problem: Example #1Fowle Marketing Research
Chapter 6 (1/2)-33
Assignment Problem: Example #1Fowle Marketing Research
LP Formulation Using the notation:
xij = 1 if project leader i is assigned to client j 0 otherwise
where i = 1, 2, 3, and j = 1, 2, 3 The completion times for three project leaders:
Thus, the objective function is:
Chapter 6 (1/2)-34
Assignment Problem: Example #1Fowle Marketing Research
LP Formulation (cont’d) Constraints reflect the conditions that each project
leader can be assigned to at most one client and that each client must have one assigned project leader. Thus:
• “# of project leaders = # of clients”: All the constraints could be written as “=“
• When “# of project leaders ≥ # of clients”: All the project leader constraints must be written as “≤“
Chapter 6 (1/2)-35
Assignment Problem: Example #1Fowle Marketing Research
Combining the objective function and constraints into one model provides a 9-variable, 6-constraint LP formulation of the Fowle Marketing Research’s assignment problem:
Chapter 6 (1/2)-36
Computer Solution Output
Assignment Problem: Example #1Fowle Marketing Research
Value of the optimal solution
Optimal solutionThe change in the optimal value of the solution per unit increase in the RHS of the constraint.
Terry is assigned to client 2 (x12 = 1), Carle is assigned to client 3 (x23 = 1), and McClymonds is assigned to client 1 (x31 = 1). The total completion time required is 26 days.
Chapter 6 (1/2)-37
Assignment Problem: Example #1Fowle Marketing Research
Solution Summary
Chapter 6 (1/2)-38
Assignment Problem Variations
Similar to transportation problem1) Total # of agents not equal to the total
number of tasks 2) Unacceptable assignments
Can be easily accommodated with slight modifications
Chapter 6 (1/2)-39
Assignment Problem Variations
1) Total # of agents not equal to the total number of tasks “# of agents > # of tasks”• No modification in the LP formulation is
necessary.• Extra agents will appear as slack (i.e.
unassigned agents).
Chapter 6 (1/2)-40
An electrical contractor pays his subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Given below are the distances between the subcontractors and the projects.
ProjectsSubcontractor A B C Westside 50 36 16
Federated 28 30 18 Goliath 35 32 20
Universal 25 25 14
How should the contractors be assigned so that totalmileage is minimized?
Assignment Problem: Example #2
Chapter 6 (1/2)-41
Network Representation50
36
16
2830
18
35 32
2025 25
14
West.West.
CC
BB
AA
Univ.Univ.
Gol.Gol.
Fed. Fed.
ProjectsSubcontractors
Assignment Problem: Example #2
Chapter 6 (1/2)-42
Linear Programming Formulation
Min 50x11+36x12+16x13+28x21+30x22+18x23
+35x31+32x32+20x33+25x41+25x42+14x43
s.t. x11+x12+x13 < 1
x21+x22+x23 < 1 x31+x32+x33 < 1 x41+x42+x43 < 1 x11+x21+x31+x41 = 1 x12+x22+x32+x42 = 1 x13+x23+x33+x43 = 1 xij = 0 or 1 for all i and j
Agents
Tasks
Assignment Problem: Example #2
Chapter 6 (1/2)-43
The optimal assignment is:
Subcontractor Project Distance Westside C 16
Federated A 28Goliath (unassigned) Universal B 25
Total Distance = 69 miles
Assignment Problem: Example #2
Chapter 6 (1/2)-44
Assignment Problem Variations
1) Total # of agents not equal to the total number of tasks (cont’d) “# of agents < # of tasks”
• The LP Model of a assignment problem will NOT have a feasible solution.
• Add enough dummy agents to equalize the number of tasks.
• The objective function coefficients for the dummy agents would be zero (0).
• No assignments will actually be made to the clients receiving dummy project leaders.
Chapter 6 (1/2)-45
Assignment Problem Variations
2) Unacceptable assignments When an agent does not have the experience
necessary for one or more of the tasks Simply remove the corresponding decision
variable from the LP formulation.