Transportation and Assignment Models

Transportation and Assignment Models
© The McGraw-Hill Companies, Inc., 2003

Outline Network models in general
Transportation, Assignment and Transshipment models belong to a special class of linear programming problems called network flow problems Characteristics of Transportation models Characteristics of Assignment models Variations on a theme © The McGraw-Hill Companies, Inc., 2003

Network Optimization Problems
Many optimization problems can be represented by a graphical network representation. Arcs Nodes Some examples: Distribution problems Routing problems Maximum flow problems Designing computer / phone / road networks Equipment replacement Slides are based upon a lecture from a 2nd-year MBA elective course “Modeling with Spreadsheets” at the University of Washington (as taught by one of the authors). The lecture covers network optimization problems. © The McGraw-Hill Companies, Inc., 2003

Transportation problem
Frequently arises in planning for distribution of goods and services from several supply locations to several demand locations. Examples? Characteristics (typical) Quantity of goods available at each supply location is limited. Quantity of goods needed at each of several demand locations is known. Usual objective of a transportation problem is to minimize the cost of shipping goods from the origins to the destinations. Variations on the Transportation Problem theme Total supply not equal to total demand. Maximization objective function. Route capacities or route minimums. Unacceptable routes. © The McGraw-Hill Companies, Inc., 2003

Example: Supply Chains – the generic model
Raw materials supplier Manufacturing plant Distribution center Customers/ Retailers upstream downstream © The McGraw-Hill Companies, Inc., 2003

Example: Forest industry supply chain
Wagner, H.M. (1975). Principles of Operations Research 2nd ed. Englewood Cliffs NJ: Prentice-Hall © The McGraw-Hill Companies, Inc., 2003

Assumptions 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 multiplied by the number of units distributed. © The McGraw-Hill Companies, Inc., 2003

The network for a transportation problem
Quantities - xij The Decision Variable Region 1 Region 2 Region 4 Region 3 Plant 1 Plant 2 Plant 3 Capacities - Ki Demands - Dj Variable costs - cij © The McGraw-Hill Companies, Inc., 2003

A basic transportation model – Foster Generators
2 3 1 4 7 6 5 Transportation cost per unit Plants (origin nodes) Distribution Centers Chicago 4000 St. Louis 2000 Bedford 6000 Boston 6000 Distribution Routes (arcs) Lexington 1500 York 2500 Cleveland 5000 Supplies / Plant Capacities (units) Demands (units) © The McGraw-Hill Companies, Inc., 2003

Transportation Problem: Demand Allocation Model
All demands are satisfied No capacities are exceeded Parameters cij variable costs per unit transported from Plant i to Region j Ki capacity for Plant i Dj demand for Region j m number of regions n number of plants Decision variables xij quantity transported from Plant i to Region j © The McGraw-Hill Companies, Inc., 2003

A simple Transportation/Demand Allocation problem solved

The Capacitated Plant Location Model (CPLM)
Open/Closed Fixed costs Capacities Variable costs Quantities Demands Ware- house 1 house 2 house 3 Plant 1 Plant 2 Plant 3 © The McGraw-Hill Companies, Inc., 2003

The Capacitated Plant Location Model (CPLM)
Decision variables yi binary variable indicating whether Plant i should be open (1) or closed (0) xij quantity transported from Plant i to Region j Parameters Fi fixed costs for Plant i cij variable costs per unit transported from Plant i to Region j Ki capacity for Plant i Dj demand for Region j m number of regions n number of potential plants © The McGraw-Hill Companies, Inc., 2003

The CPLM with single sourcing
Ware- house 1 house 2 house 3 Plant 1 Plant 2 Plant 3 Variable costs Open/Closed Fixed costs Capacities Assigning plants to warehouses Demands © The McGraw-Hill Companies, Inc., 2003

The CPLM with single sourcing
Parameters Fi fixed costs for Plant i cij variable costs per unit transported from Plant i to Region j Ki capacity for Plant i Dj demand for Region j m number of regions n number of potential plants Decision variables yi binary variable indicating whether Plant i should be open(1) or closed (0) xij binary variable indicating whether Plant i should supply market in Region j © The McGraw-Hill Companies, Inc., 2003

Transshipment model with sourcing
(Combines plant location, warehouse location and sourcing) Open/ Closed Open/ Closed Quantities Plant 1 Quantities Quantities Market 1 Supplier 1 Ware- house 1 Plant 2 Market 2 Supplier 2 Ware- house 2 Plant 3 Market 3 Fixed costs Variable costs Capacities Variable costs Variable costs Capacities Fixed costs Demands Capacities © The McGraw-Hill Companies, Inc., 2003

Warehouse fixed costs Supplier-Plant variable costs Plant-Warehouse variable costs Warehouse-Market variable costs Plant fixed costs subject to: for i = 1, 2, 3, …, n for e = 1, 2, 3, …, t for j = 1, 2, 3, …, m for h = 1, 2, 3, …, l Warehouse capacity constraint Warehouse flow balance Market satisfaction Plant flow balance Plant capacity constraint Source capacity constraint © The McGraw-Hill Companies, Inc., 2003

Market 1 D1 X11, c11 Plant 1 K1, F1, yi X11, c11 X11, c11 Supplier 1 S1 Ware- house 1 W1, f1 , ye X12, c12 X21, c21 X21, c21 Plant 2 K2, F2 , yi Market 2 D2 Supplier l Sl Ware- house t Wt, ft , ye X2t, c2t Xl2, cl2 Xt2, ct2 Plant n Kn, Fn , yi Market m Dm Xln, cln Xnt, cnt Xtm, ctm xhi, chi xie, cie xej, cej © The McGraw-Hill Companies, Inc., 2003

Parameters m number of markets or demand points n number of potential plant or factory locations l number of suppliers t number of potential warehouse locations Dj annual demand from customer j Ki potential capacity of plant at site i Sh supply capacity at supplier h We potential warehouse capacity at site e Fi fixed cost of locating a plant at site i fe fixed cost of locating a warehouse at site e chi cost of shipping one unit from supply source h to plant i cie cost of producing and shipping one unit from plant i to warehouse e cej cost of shipping one unit from warehouse e to customer j Decision variables yi 1 if plant is located at site i, 0 otherwise ye 1 if warehouse is located at site e, 0 otherwise xei quantity shipped from warehouse e to market j xie quantity shipped from plant at site i to warehouse e xhi quantity shipped from supplier h to plant at site i © The McGraw-Hill Companies, Inc., 2003

The Assignment Problem
Arises in a variety of decision making situations: Jobs to machines Agents to tasks Sales personnel to sales territories Contracts to bidders Spacecraft to planetary missions Nuclear warheads to targets Distinguishing feature of the basic assignment problem One agent is assigned to one and only one task We seek a set of assignments that optimizes a stated objective Minimize costs Minimize time Maximize profit Maximize observation time Maximize damage …etc. © The McGraw-Hill Companies, Inc., 2003

An Assignment Problem – Fowle Marketing
Clients (destination nodes) Completion time in days Project leaders (origin nodes) 1 Client 1 - 1 10 1 Terry - 1 15 9 2 Client 2 - 1 9 18 2 Kari - 1 5 3 14 Client 3 - 1 6 3 3 Gudmund - 1 Demands Possible assignments (arcs) Supplies © The McGraw-Hill Companies, Inc., 2003

Parameters cij cost of assigning Agent i to Task j m number of Agents n number of Tasks Decision variables xij assignment of Agent i to Task j, 0 if not assigned, 1 if assigned © The McGraw-Hill Companies, Inc., 2003

A simple Assignment Problem solved

Table of Contents Chapter 6 (Transportation and Assignment Problems)
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) 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). © The McGraw-Hill Companies, Inc., 2003

P&T Company Distribution Problem
Figure 6.1 Location of the canneries and warehouses for the P&T Company problem. © The McGraw-Hill Companies, Inc., 2003

Shipping Data Cannery Output Warehouse Allocation Bellingham
75 truckloads Sacramento 80 truckloads Eugene 125 truckloads Salt Lake City 65 truckloads Albert Lea 100 truckloads Rapid City 70 truckloads Total 300 truckloads Albuquerque 85 truckloads Table 6.1 Shipping data for the P&T Company. © The McGraw-Hill Companies, Inc., 2003

Current Shipping Plan From \ To Warehouse Sacramento Salt Lake City
Rapid City Albuquerque Cannery Bellingham 75 Eugene 5 65 55 Albert Lea 15 85 Table 6.2 Current Shipping Plan for the P&T Company. © The McGraw-Hill Companies, Inc., 2003

Shipping Cost per Truckload
Warehouse From \ To Sacramento Salt Lake City Rapid City Albuquerque Cannery Bellingham $464 $513 $654 $867 Eugene 352 416 690 791 Albert Lea 995 682 388 685 Total shipping cost = 75($464) + 5($352) + 65($416) + 55($690) + 15($388) + 85($685) = $165,595 Table 6.3 Shipping Costs per Truckload for the P&T Company. © The McGraw-Hill Companies, Inc., 2003

Terminology for a Transportation Problem
P&T Company Problem Truckloads of canned peas Canneries Warehouses Output from a cannery Allocation to a warehouse Shipping cost per truckload from a cannery to a warehouse General Model Units of a commodity Sources Destinations Supply from a source Demand at a destination Cost per unit distributed from a source to a destination Table 6.4 Terminology for a transportation problem © The McGraw-Hill Companies, Inc., 2003

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

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. © The McGraw-Hill Companies, Inc., 2003

The P&T Co. Transportation Problem data
Unit Cost Destination (Warehouse): Sacrament o Salt Lake City Rapid City Albuquerque Supply Source (Cannery) Bellingham $464 $513 $654 $867 75 Eugene 352 416 690 791 125 Albert Lea 995 682 388 685 100 Demand 80 65 70 85 Table 6.5 The data for the P&T Co. problem formulated as a transportation problem. © The McGraw-Hill Companies, Inc., 2003

Network Representation
Figure 6.3 The network representation of the P&T Company transportation problem shows all the data in Table 6.5 graphically. © The McGraw-Hill Companies, Inc., 2003

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 + $685x34 subject 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 = 85 and xij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4) From Cannery 1 to all destinations From Cannery 2 to all destinations From Cannery 3 to all destinations All canneries can supply all warehouses © The McGraw-Hill Companies, Inc., 2003

Spreadsheet Formulation
Figure 6.2 A spreadsheet formulation of the P&T Co. problem as a transportation problem, including the target cell Total Cost (J17), the changing cells Shipment Quantity (D12:G14), and the optimal shipping plan obtained by the Solver. © The McGraw-Hill Companies, Inc., 2003

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

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 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 © The McGraw-Hill Companies, Inc., 2003

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. Unit Cost Product: 1 2 3 4 Capacity Available Plant $41 $27 $28 $24 75 40 29 — 23 37 30 27 21 45 Required production 20 Table 6.6 Data for the Better Products Co. problem. Question: Which plants should produce which products? © The McGraw-Hill Companies, Inc., 2003

Transportation Problem Formulation
Unit Cost Destination (Product): 1 2 3 4 Supply Source(Plant) $41 $27 $28 $24 75 40 29 — 23 37 30 27 21 45 Demand 20 Table 6.7 Data for the Better Products Co. problem formulated as a variant of a transportation problem. © The McGraw-Hill Companies, Inc., 2003

Figure 6.4 A spreadsheet formulation of the Better Products Co. problem as a variant of a transportation problem, including the target cell Total Cost (I16), the changing cells Daily Production (C11:F13), and the optimal production plan obtained by the Solver. © The McGraw-Hill Companies, Inc., 2003

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? © The McGraw-Hill Companies, Inc., 2003

Data for the Nifty Company
Unit Cost Product: 1 2 3 4 Capacity Available Plant $41 $27 $28 $24 75 40 29 — 23 37 30 27 21 45 Required production 20 Table 6.8 Data for the Nifty Company problem. Question: How many units should Nifty sell to each customer and how many units should they ship from each plant to each customer? © The McGraw-Hill Companies, Inc., 2003

Figure 6.5 A spreadsheet formulation of the Nifty Company problem as a variant of a transportation problem, including the target cell Total Profit (I17), the changing cells Shipment (C11:F13), and the optimal shipping plan obtained by the Solver. © The McGraw-Hill Companies, Inc., 2003

Metro Water (Distributing Natural Resources)
Metro Water District is an agency that administers water distribution in a large geographic 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. Cost per Acre Foot Berdoo Los Devils San Go Hollyglass Available Colombo River $160 $130 $220 $170 5 Sacron River 140 130 190 150 6 Calorie River 200 230 — Needed 2 4 1.5 (million acre feet) Table 6.9 Water resources data for Metro Water District. Question: How much water should Metro take from each river, and how much should they send from each river to each city? © The McGraw-Hill Companies, Inc., 2003

Figure 6.6 A spreadsheet formulation of the Metro Water District problem as a variant of a transportation problem, including the target cell of Total Cost (I17), the changing cells Water Distribution (C11:F13), and the optimal solution obtained by the Solver. © The McGraw-Hill Companies, Inc., 2003

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. Maximum Production Unit Cost of Production ($million) Unit Cost of Storage ($thousand) Month Scheduled Installations Regular Time Overtime 1 10 20 1.08 1.10 15 2 30 1.11 1.12 3 25 4 5 1.13 1.15 Table Production scheduling data for the Northern Airplane Company problem. 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? © The McGraw-Hill Companies, Inc., 2003

Optimal Production at Northern Airplane
Month 1 (RT) 2 (RT) 3 (RT) 3 (OT) 4 (RT) Production 20 10 25 5 Installations 15 Stored Table Optimal production schedule for the Northern Airplane Company. © The McGraw-Hill Companies, Inc., 2003

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

Data for the Middletown School District
Distance (Miles) to School Tract 1 2 3 Number of High School Students 2.2 1.9 2.5 500 1.4 1.3 1.7 400 0.5 1.8 1.1 450 4 1.2 0.3 2.0 5 0.9 0.7 1.0 6 1.6 0.6 7 2.7 1.5 8 0.8 9 Minimum enrollment 1,200 1,100 1,000 Maximum enrollment 1,800 1,700 1,500 Table Data for the Middletown School District problem. © The McGraw-Hill Companies, Inc., 2003

Figure 6.8 A spreadsheet formulation of the Middletown School District problem as a variant of a transportation problem, including the target cell Total Distance (H30), the changing cells Number of Students (C17:E25), and the optimal zoning plan obtained by the Solver. © The McGraw-Hill Companies, Inc., 2003

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? © The McGraw-Hill Companies, Inc., 2003

Cost Data for Energetic
Unit Cost Energy Need: Electricity Water Heating Space Heating Source of Energy $400 $500 $600 Natural gas — 600 500 Solar heater 300 400 Table Cost data for the Energetic Co. problem. © The McGraw-Hill Companies, Inc., 2003

Figure 6.9 A spreadsheet formulation of the Energetic Co. problem as a variant of a transportation problem, including the target cell Total Cost (I18), the changing cells Daily Energy Use (D12:F14), and the optimal energy-sourcing plan obtained by the Solver. © The McGraw-Hill Companies, Inc., 2003

Location of Texago’s Facilities
Type of Facility Locations Oil fields 1. Several in Texas 2. Several in California 3. Several in Alaska Refineries 1. Near New Orleans, Louisiana 2. Near Charleston, South Carolina 3. Near Seattle, Washington Distribution Centers 1. Pittsburgh, Pennsylvania 2. Atlanta, Georgia 3. Kansas City, Missouri 4. San Francisco, California Table Location of Texago’s current facilities. © The McGraw-Hill Companies, Inc., 2003

Potential Sites for Texago’s New Refinery
Main Advantages Near Los Angeles, California 1. Near California oil fields. 2. Ready access from Alaska oil fields. 3. Fairly near San Francisco distribution center. Near Galveston, Texas 1. Near Texas oil fields. 2. Ready access from Middle East imports. 3. Near corporate headquarters. Near St. Louis, Missouri 1. Low operating costs. 2. Centrally located for distribution centers. 3. Ready access to crude oil via the Mississippi River. Table Potential sites for Texago’s new refinery and their main advantages. © The McGraw-Hill Companies, Inc., 2003

Production Data for Texago
Refinery Crude Oil Needed Annually (Million Barrels) Oil Fields Crude Oil Produced Annually (Million Barrels) New Orleans 100 Texas 80 Charleston 60 California Seattle Alaska New site 120 Total 240 360 Needed imports = 360 – 240 = 120 Table Production data for Texago Corp. © The McGraw-Hill Companies, Inc., 2003

Cost Data for Shipping to Refineries
Cost per Unit Shipped to Refinery or Potential Refinery (Millions of Dollars per Million Barrels) New Orleans Charleston Seattle Los Angeles Galveston St. Louis Source Texas 2 4 5 3 1 California Alaska 7 Middle East Table Cost data for shipping crude oil to a Texago refinery. © The McGraw-Hill Companies, Inc., 2003

Cost Data for Shipping to Distribution Centers
Cost per Unit Shipped to Distribution Center (Millions of Dollars) Pittsburgh Atlanta Kansas City San Francisco Refinery New Orleans 6.5 5.5 6 8 Charleston 7 5 4 Seattle 3 Potential Refinery Los Angeles 2 Galveston St. Louis 1 Number of units needed 100 80 Table Cost data for shipping finished product to a distribution center. © The McGraw-Hill Companies, Inc., 2003

Estimated Operating Costs for Refineries
Site Annual Operating Cost (Millions of Dollars) Los Angeles Galveston St. Louis 620 570 530 Table Estimated operating costs for a Texago refinery at each potential site. © The McGraw-Hill Companies, Inc., 2003

Basic Spreadsheet for Shipping to Refineries
Figure The basic spreadsheet formulation for the Texago transportation problem for shipping crude oil from the oil fields to the refineries, including the new refinery at a site still to be selected. The target cell is Total Cost (J20). Before entering the data for a new site and then clicking on the Solve button, a trial solution of 0 has been entered into each of the changing cells Shipment Quantity (D13:G16). © The McGraw-Hill Companies, Inc., 2003

Shipping to Refineries, Including Los Angeles
Figure The changing cells Shipment Quantity (D13:G16) give Texago management an optimal plan for shipping crude oil if Los Angeles is selected as the new site for the refinery in column G of Figure 6.10. © The McGraw-Hill Companies, Inc., 2003

Shipping to Refineries, Including Galveston
Figure The changing cells Shipment Quantity (D13:G16) give Texago management an optimal plan for shipping crude oil if Galveston is selected as the new site for the refinery in column G of Figure 6.10. © The McGraw-Hill Companies, Inc., 2003

Shipping to Refineries, Including St. Louis
Figure The changing cells Shipment Quantity (D13:G16) give Texago management an optimal plan for shipping crude oil if St. Louis is selected as the new site for the refinery in column G of Figure 6.10. © The McGraw-Hill Companies, Inc., 2003

Basic Spreadsheet for Shipping to D.C.’s
Figure The basic spreadsheet formulation for the Texago transportation problem for shipping finished product from the refineries (including the new one at a site still to be selected) to the distribution centers. The target cell is Total Cost (J20). Before entering the data for a new site and then clicking on the Solve button, a trial solution of 0 has been entered into each of the changing cells Shipment Quantity (D13:G16). © The McGraw-Hill Companies, Inc., 2003

Shipping to D.C.’s When Choose Los Angeles
Figure The changing cells Shipment Quantity (D13:G16) give Texago management an optimal plan for shipping finished product if Los Angeles is selected as the new site for a refinery in rows 8 and 16 of Figure 6.14. © The McGraw-Hill Companies, Inc., 2003

Shipping to D.C.’s When Choose Galveston
Figure The changing cells Shipment Quantity (D13:G16) give Texago management an optimal plan for shipping finished product if Galveston is selected as the new site for a refinery in rows 8 and 16 of Figure 6.14. © The McGraw-Hill Companies, Inc., 2003

Shipping to D.C.’s When Choose St. Louis
Figure The changing cells Shipment Quantity (D13:G16) give Texago management an optimal plan for shipping finished product if St. Louis is selected as the new site for a refinery in rows 8 and 16 of Figure 6.14. © The McGraw-Hill Companies, Inc., 2003

Annual Variable Costs Site Total Cost of Shipping Crude Oil
Total Cost of Shipping Finished Product Operating Cost for New Refinery Total Variable Cost Los Angeles $880 million $1.57 billion $620 million $3.07 billion Galveston 920 million 1.63 billion 570 million 3.12 billion St. Louis 960 million 1.43 billion 530 million 2.92 billion Table Annual variable costs resulting from the choice of each site for the new Texago refinery. © The McGraw-Hill Companies, Inc., 2003

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

Data for the Sellmore Problem
Required Time per Task (Hours) Temporary Employee Word Processing Graphics Packets Registrations Hourly Wage Ann 35 41 27 40 $14 Ian 47 45 32 51 12 Joan 39 56 36 43 13 Sean 25 46 15 Table Data for the Sellmore Company problem. © The McGraw-Hill Companies, Inc., 2003

Figure A spreadsheet formulation of the Sellmore Co. problem as an assignment problem, including the target cell Total Cost (J30). The values of 1 in the changing cells Assignment (D24:G27) show the optimal plan obtained by the Solver for assigning the people to the tasks. © The McGraw-Hill Companies, Inc., 2003

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

The Network Representation
Figure The network representation of the Sellmore Co. assignment problem shows all the possible assignments and their costs graphically. © The McGraw-Hill Companies, Inc., 2003

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? © The McGraw-Hill Companies, Inc., 2003

Materials-Handling Cost Data
Cost per Hour Location: 1 2 3 4 5 Machine $13 $16 $12 $14 $15 15 — 13 20 16 7 10 6 Table Materials-handling cost data for the Job Shop Co. problem. © The McGraw-Hill Companies, Inc., 2003

Figure A spreadsheet formulation of the Job Shop Co. problem as a variant of an assignment problem, including the target cell Total Cost (J17). The values of 1 in the changing cells Assignment (C11:G13) show the optimal plan obtained by the Solver for assigning the machines to the locations. © The McGraw-Hill Companies, Inc., 2003

Better Products (No Product Splitting)
Figure In contrast to Figure 6.4, product splitting is not allowed, so the Better Products Co. problem becomes a variant of an assignment problem. The target cell is Total Cost (I24). The values of 1 in the changing cells Assignment (C19:F21) display the optimal production plan obtained by the Solver. © The McGraw-Hill Companies, Inc., 2003

Middletown School District (No Tract Splitting)
Figure In contrast to Figure 6.8, tract splitting is no longer allowed, so the Middletown School District problem becomes a variant of an assignment problem. The target cell is Total Distance (H30). The values of 1 in the changing cells Assignment (C18:E26) show the optimal zoning plan found by the Solver. © The McGraw-Hill Companies, Inc., 2003

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. Slides 6.57–6.75 are based upon a lecture from a 2nd-year MBA elective “Modeling with Spreadsheets” at the University of Washington (as taught by one of the authors). The lecture covers transportation and assignment problems. © The McGraw-Hill Companies, Inc., 2003

The Network Representation

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. Unit Shipping Cost Distribution Center Sacramento St. Louis Pittsburg h Plant Seattle $2 $6 $8 Atlanta $7 $5 $3 Question: How many units should be shipped from each plant to each distribution center? © The McGraw-Hill Companies, Inc., 2003

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. Unit Shipping Cost Customer 1 2 3 4 Distributio n Center Sacramento $8 $10 $7 $11 St. Louis $12 $9 $6 Pittsburgh $15 Customer Demand 40 30 25 This formulation assumes that each distribution center has received 40 units of product (as was required in the previously solved transportation problem). Question: How many units should be shipped from each distribution center to each customer? © The McGraw-Hill Companies, Inc., 2003

Spreadsheet Solution The total weekly cost from Plants to Distribution Centers is $420. The total weekly cost from Distribution Centers to Customers is $965. The combined total weekly cost is $1,385. However, we so far have considered the two problems (Plants to Distribution Centers, Distribution Centers to Customers) separately. By combining the two, we may be able to come up with a better solution. For example, it may not be best to ship 40 units through each distribution center. The next spreadsheet combines the whole problem into a single model. © The McGraw-Hill Companies, Inc., 2003

Managing the Whole Supply Chain (Plant to D.C. to Customer)
This spreadsheet combines the previous two transportation problems into a single problem that encompasses the whole supply chain. The two original transportation models can be simply copied and pasted into a new spreadsheet. However, the link between the two models needs to be established. The cells G28:I30 establish that link. What gets shipped out of the distribution centers (G28:G30) must be less than or equal to what gets shipping into the distribution centers (carried down from C13:E13 to I28:I30). The weekly cost of each of the two components, in H13 and I31, are then totalled to give the overall cost in I35. © The McGraw-Hill Companies, Inc., 2003

Question: Should they move their distribution center to Omaha?
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

Spreadsheet Solution to Site Selection
The shipping costs with Omaha are $1,325. This is $50 more than with St. Louis ($1,275). However, the leasing costs of the two sites have not been considered in the model. If Omaha’s lease is more than $50 less than St. Louis, it may be worthwhile to move the distribution center. Other factors should be considered as well. Other possible extensions to the model that warrant discussion: distribution center capacity constraints fixed cost and/or variable cost at the distribution centers. multiple products (1 transportation model for each, possibly linked with distribution center capacity constraints). multiple time periods (can include lead-time for shipping) © The McGraw-Hill Companies, Inc., 2003

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 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 © The McGraw-Hill Companies, Inc., 2003

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

Network Representation

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. Backstroke Breaststroke Butterfly Freestyle Carl 37.7 43.4 33.3 29.2 Chris 32.9 33.1 28.5 26.4 David 33.8 42.2 38.9 29.6 Tony 37.0 34.7 30.4 Ken 35.4 41.8 33.6 31.1 Question: How should the swimmers be assigned to make the fastest relay team? © The McGraw-Hill Companies, Inc., 2003

Algebraic Formulation
Let xij = 1 if swimmer i swims stroke j; 0 otherwise tij = best time of swimmer i in stroke j Minimize Time = ∑ i ∑ j tij xij subject to: each stroke swum: ∑ i xij = 1 for each stroke j each swimmer swims 1: ∑ j xij ≤ 1 for each swimmer i and xij ≥ 0 for all i and j. © The McGraw-Hill Companies, Inc., 2003

Other applications of assignment problems : assigning tasks to workers assigning machines to locations assigning products to plants assigning jobs to machines Major League baseball uses a version of this model to assign umpiring crews to games. © The McGraw-Hill Companies, Inc., 2003

Question: How should students be assigned to the classes?
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? Discuss: what is the right objective? © The McGraw-Hill Companies, Inc., 2003

Points Bid for Electives
Student Quantitative Methods Finance Operations Management Accounting George 60 10 20 Fred 40 Ann 45 5 Eric 50 25 Susan 30 Liz Ed 70 David 35 15 Tony Jennifer © The McGraw-Hill Companies, Inc., 2003

Spreadsheet Solution (Maximizing Total Points)
This solution maximizes the total bid points achieved (or, equivalently, the average bid points per student achieved). However, is this fair? For example, Eric bid more than Ann for QMETH, but Ann got in while Eric didn’t. Liz got 100 bid points worth of her classes, while Eric got only 45. © The McGraw-Hill Companies, Inc., 2003

Spreadsheet Solution (Maximizing the Minimum Student Point Total)
This formulation tries to make the least happy student as happy as possible by maximizing the student with the minimum total bid points assigned. This is no longer a true assignment problem because of the side constraints in K17:M26. It therefore does not have the integer solutions property, so binary constraints must be added. The minimum point total in K30 is both the target cell (to maximize), and also a changing cell. Each student point total is constrained to be greater-than-or-equal-to the minimum point total by K17:M26 (or, equivalently, the minimum point total is constrained to be less-than-or-equal-to every student’s bid point total). © The McGraw-Hill Companies, Inc., 2003

Transportation and Assignment Models

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