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The P&T Company Distribution Problem (Section 6.1) 6.2–6.5 Characteristics of Transportation Problems (Section 6.2) 6.6–6.14 Variants of Transportation Problems: Better Products (Section 6.3) 6.15–6.17 Variants of Transportation Problems: Nifty (Section 6.3) 6.18–6.20

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Table of contents chapter 6 transportation and assignment problems l.jpg

The P&T Company Distribution Problem (Section 6.1) 6.2–6.5

Characteristics of Transportation Problems (Section 6.2) 6.6–6.14

Variants of Transportation Problems: Better Products (Section 6.3) 6.15–6.17

Variants of Transportation Problems: Nifty (Section 6.3) 6.18–6.20

Applications of Transportation Problems: Metro Water (Section 6.4) 6.21–6.22

Applications of Transportation Problems: Northern Airplane (Section 6.4) 6.23–6.25

Applications of Transportation Problems: Middletown (Section 6.4) 6.26–6.28

Applications of Transportation Problems: Energetic (Section 6.4) 6.29–6.31

A Case Study: Texago Corp. Site Selection Problem (Section 6.5) 6.32–6.46

Characteristics of Assignment Problems: Sellmore (Section 6.6) 6.47–6.51

Variants of Assignment Problems: Job Shop (Section 6.7) 6.52-6.54

Variants of Assignment Problems: Better Products (Section 6.7) 6.55

Variants of Assignment Problems: Revised Middletown (Section 6.7) 6.56

Transportation & Assignment Problems (UW Lecture) 6.57–6.75

These slides are based upon a lecture to second-year MBA students at the University of Washington that discusses transportation and assignment problems (as taught by one of the authors).

Table of ContentsChapter 6 (Transportation and Assignment Problems)

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P&T Company Distribution Problem

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Shipping Data

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Current Shipping Plan

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Shipping Cost per Truckload

Total shipping cost = 75($464) + 5($352) + 65($416) + 55($690) + 15($388) + 85($685) = $165,595

© The McGraw-Hill Companies, Inc., 2003


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Terminology for a Transportation Problem

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Characteristics of Transportation Problems

  • The Requirements Assumption

    • Each source has a fixed supply of units, where this entire supply must be distributed to the destinations.

    • Each destination has a fixed demand for units, where this entire demand must be received from the sources.

  • The Feasible Solutions Property

    • A transportation problem will have feasible solutions if and only if the sum of its supplies equals the sum of its demands.

  • The Cost Assumption

    • The cost of distributing units from any particular source to any particular destination is directly proportional to the number of units distributed.

    • This cost is just the unit cost of distribution times the number of units distributed.

© The McGraw-Hill Companies, Inc., 2003


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The Transportation Model

Any problem (whether involving transportation or not) fits the model for a transportation problem if

  • It can be described completely in terms of a table like Table 6.5 that identifies all the sources, destinations, supplies, demands, and unit costs, and

  • satisfies both the requirements assumption and the cost assumption.

    The objective is to minimize the total cost of distributing the units.

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The P&T Co. Transportation Problem

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Spreadsheet Formulation

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Network Representation

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The Transportation Problem is an LP

Let xij = the number of truckloads to ship from cannery i to warehouse j (i = 1, 2, 3; j = 1, 2, 3, 4)Minimize Cost = $464x11 + $513x12 + $654x13 + $867x14 + $352x21 + $416x22 + $690x23 + $791x24 + $995x31 + $682x32 + $388x33 + $685x34subject to Cannery 1: x11 + x12 + x13 + x14 = 75 Cannery 2: x21 + x22 + x23 + x24 = 125 Cannery 3: x31 + x32 + x33 + x34 = 100 Warehouse 1: x11 + x21 + x31 = 80 Warehouse 2: x12 + x22 + x32 = 65 Warehouse 3: x13 + x23 + x33 = 70 Warehouse 4: x14 + x24 + x34 = 85andxij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4)

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Integer Solutions Property

As long as all its supplies and demands have integer values, any transportation problem with feasible solutions is guaranteed to have an optimal solution with integer values for all its decision variables. Therefore, it is not necessary to add constraints to the model that restrict these variables to only have integer values.

© The McGraw-Hill Companies, Inc., 2003


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Distribution System at Proctor and Gamble

  • Proctor and Gamble needed to consolidate and re-design their North American distribution system in the early 1990’s.

    • 50 product categories

    • 60 plants

    • 15 distribution centers

    • 1000 customer zones

  • Solved many transportation problems (one for each product category).

  • Goal: find best distribution plan, which plants to keep open, etc.

  • Closed many plants and distribution centers, and optimized their product sourcing and distribution location.

  • Implemented in 1996. Saved $200 million per year.

    For more details, see 1997 Jan-Feb Interfaces article, “Blending OR/MS, Judgement, and GIS: Restructuring P&G’s Supply Chain”, downloadable at www.mhhe.com/hillier2e/articles

© The McGraw-Hill Companies, Inc., 2003


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Better Products (Assigning Plants to Products)

The Better Products Company has decided to initiate the product of four new products, using three plants that currently have excess capacity.

Question: Which plants should produce which products?

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Transportation Problem Formulation

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Spreadsheet Formulation

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Nifty Co. (Choosing Customers)

  • The Nifty Company specializes in the production of a single product, which it produces in three plants.

  • Four customers would like to make major purchases. There will be enough to meet their minimum purchase requirements, but not all of their requested purchases.

  • Due largely to variations in shipping cost, the net profit per unit sold varies depending on which plant supplies which customer.

    Question: How many units should Nifty sell to each customer and how many units should they ship from each plant to each customer?

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Data for the Nifty Company

Question: How many units should Nifty sell to each customer and how many units should they ship from each plant to each customer?

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Spreadsheet Formulation

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Metro Water (Distributing Natural Resources)

Metro Water District is an agency that administers water distribution in a large goegraphic region. The region is arid, so water must be brought in from outside the region.

  • Sources of imported water: Colombo, Sacron, and Calorie rivers.

  • Main customers: Cities of Berdoo, Los Devils, San Go, and Hollyglass.

Question: How much water should Metro take from each river, and how much should they send from each river to each city?

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Spreadsheet Formulation

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Northern Airplane (Production Scheduling)

Northern Airplane Company produces commercial airplanes. The last stage in production is to produce the jet engines and install them.

  • The company must meet the delivery deadline indicated in column 2.

  • Production and storage costs vary from month to month.

Question: How many engines should be produced in each of the four months so that the total of the production and storage costs will be minimized?

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Spreadsheet Formulation

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Optimal Production at Northern Airplane

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Middletown School District

  • Middletown School District is opening a third high school and thus needs to redraw the boundaries for the area of the city that will be assigned to the respective schools.

  • The city has been divided into 9 tracts with approximately equal populations.

  • Each school has a minimum and maximum number of students that should be assigned.

  • The school district management has decided that the appropriate objective is to minimize the average distance that students must travel to school.

    Question: How many students from each tract should be assigned to each school?

© The McGraw-Hill Companies, Inc., 2003


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Data for the Middletown School District

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Spreadsheet Formulation

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Energetic (Meeting Energy Needs)

  • The Energetic Company needs to make plans for the energy systems for a new building.

  • The energy needs fall into three categories:

    • electricity (20 units)

    • heating water (10 units)

    • heating space (30 units)

  • The three possible sources of energy are

    • electricity

    • natural gas

    • solar heating unit (limited to 30 units because of roof size)

      Question: How should Energetic meet the energy needs for the new building?

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Cost Data for Energetic

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Spreadsheet Formulation

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Location of Texago’s Facilities

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Potential Sites for Texago’s New Refinery

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Production Data for Texago

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Cost Data for Shipping to Refineries

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Cost Data for Shipping to Distribution Centers

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Estimated Operating Costs for Refineries

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Basic Spreadsheet for Shipping to Refineries

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Shipping to Refineries, Including Los Angeles

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Shipping to Refineries, Including Galveston

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Shipping to Refineries, Including St. Louis

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Basic Spreadsheet for Shipping to D.C.’s

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Shipping to D.C.’s When Choose Los Angeles

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Shipping to D.C.’s When Choose Galveston

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Shipping to D.C.’s When Choose St. Louis

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Annual Variable Costs

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Sellmore Company Assignment Problem

  • The marketing manager of Sellmore Company will be holding the company’s annual sales conference soon.

  • He is hiring four temporary employees:

    • Ann

    • Ian

    • Joan

    • Sean

  • Each will handle one of the following four tasks:

    • Word processing of written presentations

    • Computer graphics for both oral and written presentations

    • Preparation of conference packets, including copying and organizing materials

    • Handling of advance and on-site registration for the conference

      Question: Which person should be assigned to which task?

© The McGraw-Hill Companies, Inc., 2003


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Data for the Sellmore Problem

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Spreadsheet Formulation

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The Model for Assignment Problems

Given a set of tasks to be performed and a set of assignees who are available to perform these tasks, the problem is to determine which assignee should be assigned to each task.

To fit the model for an assignment problem, the following assumptions need to be satisfied:

  • The number of assignees and the number of tasks are the same.

  • Each assignee is to be assigned to exactly one task.

  • Each task is to be performed by exactly one assignee.

  • There is a cost associated with each combination of an assignee performing a task.

  • The objective is to determine how all the assignments should be made to minimize the total cost.

© The McGraw-Hill Companies, Inc., 2003


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The Network Representation

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Job Shop (Assigning Machines to Locations)

  • The Job Shop Company has purchased three new machines of different types.

  • There are five available locations where the machine could be installed.

  • Some of these locations are more desirable for particular machines because of their proximity to work centers that will have a heavy work flow to these machines.

    Question: How should the machines be assigned to locations?

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Materials-Handling Cost Data

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Spreadsheet Formulation

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Better Products (No Product Splitting)

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Middletown School District (No Tract Splitting)

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The Transportation Problem

  • A common problem in logistics is how to transport goods from a set of sources (e.g., plants, warehouses, etc.) to a set of destinations (e.g., warehouses, customers, etc.) at the minimum possible cost.

  • Given

    • a set of sources, each with a given supply,

    • a set of destinations, each with a given demand,

    • a cost table (cost/unit to ship from each source to each destination)

  • Goal

    • Choose shipping quantities from each source to each destination so as to minimize total shipping cost.

© The McGraw-Hill Companies, Inc., 2003


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The Network Representation

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Transportation Problem Example

A company has two plants (in Seattle and Atlanta) producing a certain product that is to be shipped to three distribution centers (in Sacramento, St. Louis, and Pittsburgh).

  • The unit production costs are the same at the two plants, and the shipping costs per unit are shown in the table below.

  • Shipments are made once per week.

  • During each week, each plant produces at most 60 units and each distribution center needs at least 40 units.

Question: How many units should be shipped from each plant to each distribution center?

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Spreadsheet Solution

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Shipping from D.C.’s to Customers

The same company ships one of its products from its three distribution centers to four different customers

  • The shipping costs per unit are shown in the table below.

  • Shipments are made once per week.

  • During each week, each distribution center has received 40 units.

  • Customer demand is also shown in the table below.

Question: How many units should be shipped from each distribution center to each customer?

© The McGraw-Hill Companies, Inc., 2003


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Spreadsheet Solution

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Managing the Whole Supply Chain(Plant to D.C. to Customer)

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Site Selection

  • The lease is up on their distribution center in St. Louis. They now must decide whether to sign a new lease in St. Louis, or move the distribution center to a new location.

  • One possible new location is Omaha, Nebraska, which is offering a better deal on the lease.

    Question: Should they move their distribution center to Omaha?

© The McGraw-Hill Companies, Inc., 2003


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Spreadsheet Solution to Site Selection

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Distribution System at Proctor and Gamble

  • Proctor and Gamble needed to consolidate and re-design their North American distribution system in the early 1990’s.

    • 50 product categories

    • 60 plants

    • 15 distribution centers

    • 1000 customer zones

  • Solved many transportation problems (one for each product category).

  • Goal: find best distribution plan, which plants to keep open, etc.

  • Closed many plants and distribution centers, and optimized their product sourcing and distribution location.

  • Implemented in 1996. Saved $200 million per year.

    For more details, see 1997 Jan-Feb Interfaces article, “Blending OR/MS, Judgement, and GIS: Restructuring P&G’s Supply Chain”, downloadable at www.mhhe.com/hillier2e/articles

© The McGraw-Hill Companies, Inc., 2003


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The Assignment Problem

  • The job of assigning people (or machines or whatever) to a set of tasks is called an assignmentproblem.

  • Given

    • a set of assignees

    • a set of tasks

    • a cost table (cost associated with each assignee performing each task)

  • Goal

    • Match assignees to tasks so as to perform all of the tasks at the minimum possible cost.

© The McGraw-Hill Companies, Inc., 2003


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Network Representation

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Assignment Problem Example

The coach of a swim team needs to assign swimmers to a 200-yard medley relay team (four swimmers, each swims 50 yards of one of the four strokes). Since most of the best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five fastest swimmers and their best times (in seconds) they have achieved in each of the strokes (for 50 yards) are shown below.

Question: How should the swimmers be assigned to make the fastest relay team?

© The McGraw-Hill Companies, Inc., 2003


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Algebraic Formulation

Let xij = 1 if swimmer i swims stroke j; 0 otherwisetij = best time of swimmer i in stroke jMinimize Time = ∑ i ∑ jtijxijsubject to each stroke swum: ∑ ixij = 1 for each stroke j each swimmer swims 1: ∑jxij ≤ 1 for each swimmer iandxij ≥ 0 for all i and j.

© The McGraw-Hill Companies, Inc., 2003


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Spreadsheet Formulation

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Bidding for Classes

  • In the MBA program at a prestigious university in the Pacific Northwest, students bid for electives in the second year of their program.

  • Each of the 10 students has 100 points to bid (total) and must take two electives.

  • There are four electives available:

    • Quantitative Methods

    • Finance

    • Operations Management

    • Accounting

  • Each class is limited to 5 students.

    Question: How should students be assigned to the classes?

© The McGraw-Hill Companies, Inc., 2003


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Points Bid for Electives

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Spreadsheet Solution(Maximizing Total Points)

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Spreadsheet Solution(Maximizing the Minimum Student Point Total)

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