transportation transshipment and assignment models

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Transportation Models and Its Variants

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Transportation problem is a special case of LPP. In a typical transportation problem, the objective is to transport various amounts of a single homogeneous commodity that are initially stored at various origins, to different destinations in such a way that the total transportation cost is minimal. In this chapter, we understand the concept of transportation problem, discuss its different types, and examine five different methods of solution including North-West Corner Rule (NWCR) Method, Row/Column Minima Method, Matrix Minima Method and Vogel’s Approximation Method. We will also discuss the transshipment and assignment problems.

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Srinivasan, R. (2014). Transportation Models and Its Variants. In: Strategic Business Decisions. Springer, New Delhi. https://doi.org/10.1007/978-81-322-1901-9_4

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Quantitative Analysis for Management, 13/e by Barry Render, Ralph M. Stair, Michael E. Hanna, Trevor S. Hale

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Chapter 9 Transportation, Assignment, and Network Models

Learning objectives.

After completing this chapter, students will be able to:

9.1 Construct LP problems for the transportation, assignment, and transshipment models.

9.2 Solve facility location and other application problems with transportation models.

9.3 Use LP to model and solve maximal-flow problems.

9.4 Use LP to model and solve shortest route problems.

9.5 Solve minimal-spanning tree problems.

Chapter 8 provided examples of a number of managerial problems that could be modeled using linear programming (LP), and this chapter will provide even more such examples. However, all of the problems in this chapter can be modeled as networks as well as linear programs. The use of networks ...

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Chapter Questions

A transportation problem involves the following costs, supply, and demand: table cant copy Solve this problem by using the computer.

A severe winter ice storm has swept across North Carolina and Virginia, followed by over a foot of snow and frigid, single-digit temperatures. These weather conditions have resulted in numerous downed power lines and power outages, causing dangerous conditions for much of the population. Local utility companies have been overwhelmed and have requested assistance from unaffected utility companies across the Southeast. The following table shows the number of utility trucks with crews available from five different companies in Georgia, South Carolina, and Florida; the demand for crews in seven different areas that local companies cannot get to; and the weekly cost (in thousands of dollars) of a crew going to a specific area (based on the visiting company's normal charges, the distance the crew has to come, and living expenses in an area):

$$ \begin{array}{lccrccccc} \text { GA-1 } & 15.2 & 14.3 & 13.9 & 13.5 & 14.7 & 16.5 & 18.7 & 12 \\ \text { GA-2 } & 12.8 & 11.3 & 10.6 & 12.0 & 12.7 & 13.2 & 15.6 & 10 \\ \text { SC-1 } & 12.4 & 10.8 & 9.4 & 11.3 & 13.1 & 12.8 & 14.5 & 14 \\ \text { FL-1 } & 18.2 & 19.4 & 18.2 & 17.9 & 20.5 & 20.7 & 22.7 & 15 \\ \text { FL-2 } & 19.3 & 20.2 & 19.5 & 20.2 & 21.2 & 21.3 & 23.5 & 12 \\ \hline \text { Crews Needed } & 9 & 7 & 6 & 8 & 10 & 9 & 7 & \\ \hline \end{array} $$

Consider the following transportation problem: table cant copy Formulate this problem as a linear programming model and solve it by using the computer.

Solve the following linear programming problem:

$$ \begin{aligned} & \text { minimize } Z=3 x_{11}+12 x_{12}+8 x_{13}+10 x_{21}+5 x_{22}+6 x_{23}+6 x_{31}+7 x_{32}+10 x_{33} \\ & \text { subject to } \\ & x_{11}+x_{12}+x_{13}=90 \\ & x_{21}+x_{22}+x_{23}=30 \\ & x_{31}+x_{32}+x_{33}=100 \\ & x_{11}+x_{21}+x_{31} \leq 70 \\ & x_{12}+x_{22}+x_{32} \leq 110 \\ & x_{13}+x_{23}+x_{33} \leq 80 \\ & x_{i j} \geq 0 \\ & \end{aligned} $$

Consider the following transportation problem: table cant copy Solve it by using the computer.

An electronics firm produces electronic components, which it supplies to various electrical manufacturers. Quality control records indicate that different employees produce different numbers of defective items. The average number of defects produced by each employee for each of six components is given in the following table: table cant copy

Steel mills in three cities produce the following amounts of steel: $$ \begin{array}{lc} \text { Location } & \text { Weekly Production (tons) } \\ \text { A. Bethlehem } & 150 \\ \text { B. Birmingham } & 210 \\ \text { C. Gary } & \underline{320} \\ & 680 \end{array} $$

These mills supply steel to four cities, where manufacturing plants have the following demand: $$ \begin{array}{lc} \text { Location } & \text { Weekly Demand (tons) } \\ \text { 1. Detroit } & 130 \\ \text { 2. St. Louis } & 70 \\ \text { 3. Chicago } & 180 \\ \text { 4. Norfolk } & \underline{240} \\ & 620 \end{array} $$ Shipping costs per ton of steel are as follows: Because of a truckers' strike, shipments are prohibited from Birmingham to Chicago. Formulate this problem as a linear programming model and solve it by using the computer.

In Problem 5, what would be the effect on the optimal solution of a reduction in production capacity at the Gary mill from 320 tons to 290 tons per week?

Coal is mined and processed at the following four mines in Kentucky, West Virginia, and Virginia:

$$ \begin{array}{lc} \text { Location } & \text { Capacity (tons) } \\ \text { A. Cabin Creek } & 90 \\ \text { B. Surry } & 50 \\ \text { C. Old Fort } & 80 \\ \text { D. McCoy } & \frac{60}{280} \end{array} $$ These mines supply the following amount of coal to utility power plants in three cities: $$ \begin{array}{lc} \text { Plant } & \text { Demand (tons) } \\ \text { 1. Richmond } & 120 \\ \text { 2. Winston-Salem } & 100 \\ \text { 3. Durham } & \underline{110} \\ & 330 \end{array} $$ The railroad shipping costs (in thousands of dollars) per ton of coal are shown in the following table. Because of railroad construction, shipments are prohibited from Cabin Creek to Richmond:

Formulate this problem as a linear programming model and solve it by using the computer.

Oranges are grown, picked, and then stored in warehouses in Tampa, Miami, and Fresno. These warehouses supply oranges to markets in New York, Philadelphia, Chicago, and Boston. The following table shows the shipping costs per truckload (in hundreds of dollars), supply, and demand. Because of an agreement between distributors, shipments are prohibited from Miami to Chicago: Formulate this problem as a linear programming model and solve it by using the computer.

A manufacturing firm produces diesel engines in four cities—Phoenix, Seattle, St. Louis, and Detroit. The company is able to produce the following numbers of engines per month: $$ \begin{array}{lc} \text { Plant } & \text { Production } \\ \text { 1. Phoenix } & 5 \\ \text { 2. Seattle } & 25 \\ \text { 3. St. Louis } & 20 \\ \text { 4. Detroit } & 25 \end{array} $$ Three trucking firms purchase the following numbers of engines for their plants in three cities:

$$ \begin{array}{lc} \text { Firm } & \text { Demand } \\ \text { A. Greensboro } & 10 \\ \text { B. Charlotte } & 20 \\ \text { C. Louisville } & 15 \end{array} $$ The transportation costs per engine (in hundreds of dollars) from sources to destinations are shown in the following table. However, the Charlotte firm will not accept engines made in Seattle, and the Louisville firm will not accept engines from Detroit; therefore, those routes are prohibited:

The Interstate Truck Rental firm has accumulated extra trucks at three of its truck leasing outlets, as shown in the following table:

$$ \begin{array}{lc} \hline \text { Leasing Outlet } & \text { Extra Trucks } \\ \hline \text { 1. Atlanta } & 70 \\ \text { 2. St. Louis } & 115 \\ \text { 3. Greensboro } & \underline{60} \\ \quad \text { Total } & 245 \end{array} $$ The firm also has four outlets with shortages of rental trucks, as follows: $$ \begin{array}{lc} \hline \text { Leasing Outlet } & \text { Truck Shortage } \\ \hline \text { A. New Orleans } & 80 \\ \text { B. Cincinnati } & 50 \\ \text { C. Louisville } & 90 \\ \text { D. Pittsburgh } & \underline{25} \\ \quad \text { Total } & 245 \end{array} $$ The firm wants to transfer trucks from those outlets with extras to those with shortages at the minimum total cost. The following costs of transporting these trucks from city to city have been determined: Solve this problem by using the computer.

The Shotz Beer Company has breweries in two cities; the breweries can supply the following numbers of barrels of draft beer to the company's distributors each month:

$$ \begin{array}{lc} \hline \text { Brewery } & \text { Monthly Supply (bbl) } \\ \hline \text { A. Tampa } & 3,500 \\ \text { B. St. Louis } & \underline{5,000} \\ \quad \text { Total } & 8,500 \end{array} $$

In Problem 11, the Shotz Beer Company management negotiated a new shipping contract with a trucking firm between its Tampa brewery and its distributor in Kentucky. This contract reduces the shipping cost per barrel from $$\$ 0.80$$ per barrel to $$\$ 0.65$$ per barrel. How will this cost change affect the optimal solution?

Computers Unlimited sells microcomputers to universities and colleges on the East Coast and ships them from three distribution warehouses. The firm is able to supply the following numbers of microcomputers to the universities by the beginning of the academic year:

$$ \begin{array}{lc} \begin{array}{l} \text { Distribution } \\ \text { Warehouse } \end{array} & \begin{array}{c} \text { Supply } \\ \text { (microcomputers) } \end{array} \\ \hline \text { 1. Richmond } & 420 \\ \text { 2. Atlanta } & 610 \\ \text { 3. Washington, DC } & \frac{340}{2,370} \end{array} $$ Four universities have ordered microcomputers that must be delivered and installed by the beginning of the academic year: $$ \begin{array}{lc} \text { University } & \begin{array}{c} \text { Demand } \\ \text { (microcomputers) } \end{array} \\ \hline \text { A. Tech } & 520 \\ \text { B. A \& M } & 250 \\ \text { C. State } & 400 \\ \text { D. Central } & \underline{380} \\ \quad \text { Total } & 1,550 \\ \hline \end{array} $$ The shipping and installation costs per microcomputer from each distributor to each university are as follows:

In Problem 13, Computers Unlimited wants to better meet demand at the four universities it supplies. It is considering two alternatives: (1) expand its warehouse at Richmond to a capacity of 600 , at a cost equivalent to an additional $$\$ 6$$ in handling and shipping per unit; or (2) purchase a new warehouse in Charlotte that can supply 300 units with shipping costs of $$\$ 19$$ to Tech, $$\$26$$ to$A \& M$, $$\$22$$ to State, and $$\$16$$ to Central. Which alternative should management select, based solely on transportation costs (i.e., no capital costs)?

Computers Unlimited in Problem 13 has determined that when it is unable to meet the demand for microcomputers at the universities it supplies, the universities tend to purchase microcomputers elsewhere in the future. Thus, the firm has estimated a shortage cost for each microcomputer demanded but not supplied that reflects the loss of future sales and goodwill. These costs for each university are as follows:

$$ \begin{array}{lc} \hline \text { University } & \text { Cost/Microcomputer } \\ \hline \text { A. Tech } & \$ 40 \\ \text { B. A \& M } & 65 \\ \text { C. State } & 25 \\ \text { D. Central } & 50 \\ \hline \end{array} $$ Solve Problem 13 with these shortage costs included. Compute the total transportation cost and the total shortage cost.

A large manufacturing company is closing three of its existing plants and intends to transfer some of its more skilled employees to three plants that will remain open. The number of employees available for transfer from each closing plant is as follows:

$$ \begin{array}{cc} \hline \text { Closing Plant } & \text { Transferable Employees } \\ \hline 1 & 60 \\ 2 & 105 \\ 3 & \underline{70} \\ \text { Total } & 235 \\ \hline \end{array} $$ The following number of employees can be accommodated at the three plants remaining open: $$ \begin{array}{cc} \hline \text { Open Plants } & \text { Employees Demanded } \\ \hline \text { A } & 45 \\ \text { B } & 90 \\ \text { C } & \underline{35} \\ \text { Total } & 170 \\ \hline \end{array} $$

The Sav-Us Rental Car Agency has six lots in Nashville, and it wants to have a certain number of cars available at each lot at the beginning of each day for local rental. The agency would like a model it could quickly solve at the end of each day that would tell it how to redistribute the cars among the six lots in the minimum total time. The times required to travel between the six lots are as follows:

Bayville has built a new elementary school, increasing the town's total to four schools-Addison, Beeks, Canfield, and Daley. Each has a capacity of 400 students. The school board wants to assign children to schools so that their travel time by bus is as short as possible. The school board has partitioned the town into five districts conforming to population density — north, south, east, west, and central. The average bus travel time from each district to each school is shown as follows:

$$ \begin{array}{lccccc} \text { District } & \text { Addison } & \text { Beeks } & \text { Canfield } & \text { Daley } & \text { Student Population } \\ \hline \text { North } & 12 & 23 & 35 & 17 & 250 \\ \text { South } & 26 & 15 & 21 & 27 & 340 \\ \text { East } & 18 & 20 & 22 & 31 & 310 \\ \text { West } & 29 & 24 & 35 & 10 & 210 \\ \text { Central } & 15 & 10 & 23 & 16 & 290 \\ \hline \end{array} $$ Determine the number of children that should be assigned from each district to each school to minimize total student travel time.

In Problem 19, the school board determined that it does not want any of the schools to be overly crowded compared with the other schools. It would like to assign students from each district to each school so that enrollments are evenly balanced between the four schools. However, the school board is concerned that this might significantly increase travel time. Determine the number of students to be assigned from each district to each school such that school enrollments are evenly balanced. Does this new solution appear to significantly increase travel time per student?

The Easy Time Grocery chain operates in major metropolitan areas on the East Coast. The stores have a "no-frills" approach, with low overhead and high volume. They generally buy their stock in volume at low prices. However, in some cases they actually buy stock at stores in other areas and ship it in. They can do this because of high prices in the cities they operate in compared with costs in other locations. One example is baby food. Easy Time purchases baby food at stores in Albany, Binghamton, Claremont, Dover, and Edison and then trucks it to six stores in and around New York City. The stores in the outlying areas know what Easy Time is up to, so they limit the number of cases of baby food Easy Time can purchase. The following table shows the profit Easy Time makes per case of baby food, based on where the chain purchases it and at which store it is sold, plus the available baby food per week at purchase locations and the shelf space available at each Easy Time store per week: Determine where Easy Time should purchase baby food and how the food should be distributed to maximize profit.

Suppose that in Problem 21 Easy Time can purchase all the baby food it needs from a New York City distributor at a price that will result in a profit of $$\$ 9$$ per case at stores 1,3 , and 4 ; $$\$ 8$$ per case at stores 2 and 6 ; and $$\$ 7$$ per case at store 5 . Should Easy Time purchase all, none, or some of its baby food from the distributor rather than purchase it at other stores and truck it in?

In Problem 21, if Easy Time could arrange to purchase more baby food from one of the outlying locations, which should it be, how many additional cases could be purchased, and how much would this increase profit?

The Roadnet Transport Company expanded its shipping capacity by purchasing 90 trailer trucks from a competitor that went bankrupt. The company subsequently located 30 of the purchased trucks at each of its shipping warehouses in Charlotte, Memphis, and Louisville. The company makes shipments from each of these warehouses to terminals in St. Louis, Atlanta, and New York. Each truck is capable of making one shipment per week. The terminal managers have indicated their capacity of extra shipments. The manager at St. Louis can accommodate 40 additional trucks per week, the manager at Atlanta can accommodate 60 additional trucks, and the manager at New York can accommodate 50 additional trucks. The company makes the following profit per truckload shipment from each warehouse to each terminal. The profits differ as a result of differences in products shipped, shipping costs, and transport rates: Determine how many trucks to assign to each route (i.e., warehouse to terminal) in order to maximize profit.

During U.S. military action in the Middle East, large amounts of military matériel and supplies had to be shipped daily from supply depots in the United States. The critical factor in the movement of these supplies was speed. The following table shows the number of planeloads of supplies available each day from each of six supply depots and the number of daily loads demanded at each of five bases in the Middle East. (Each planeload is approximately equal in tonnage.) Also included are the transport hours per plane, including loading and fueling, actual flight time, and unloading and refueling:

Suntrek, based in China, is a global supplier of denim jeans for apparel companies around the world. They purchase raw cotton from producers in Arkansas, Mississippi, and Texas, where it is picked, ginned, and baled and then transported by flatbed trucks to ports in Houston, New Orleans, Savannah, and Charleston, where it is loaded into 80 -foot containers and shipped to factories overseas. For the coming year Suntrek has contracted with its U.S. broker for $71,000(550 \mathrm{lb}$.) bales of cotton and the transportation and handling costs from each cotton-processing facility to each port, and the container capacity (in bales) at each port are as follows:

Suntrek in Problem 26 ships the cotton it has purchased from the U.S. ports to overseas ports in Shanghai, Karachi, and Saigon, where its denim fabric factories are also located. The shipping and handling costs per bale of cotton from each U.S. port to each of Suntrek's overseas factories and the demand at these factories are as follows: Determine the optimal shipments that will result in the minimum total shipping cost.

Suntrek in Problem 27 manufactures denim fabric at its factories in Shanghai, Karachi, and Saigon and ships it to its denim jeans manufacturing facilities in China, India, Japan, Turkey, and Italy. A bale of cotton will produce approximately 325 yards of denim fabric. Following are the fabric demand at each denim jeans plant and the shipping and handling costs per yard from the fabric manufacturing facilities to the jeans plants: Determine the optimal shipments from each of the fabric plants to the denim jeans manufacturing facilities and the minimum total shipping cost.

Suntrek in Problem 28 supplies its finished denim jeans to its customers' distribution centers in the United States in New York and New Orleans, and in Europe in Bristol and Marseilles. Denim jeans require 1.5 yards of denim fabric. Following are the contracted deliveries of jeans for each of Suntrek's customer's distribution centers, and the shipping and handling costs per jeans from the jeans factories to the distribution centers: Determine the optimal shipments from each jeans factory to each distribution center and the optimal total shipping cost.

PM Computer Services produces personal computers from component parts it buys on the open market. The company can produce a maximum of 300 personal computers per month. PM wants to determine its production schedule for the first 6 months of the new year. The cost to produce a personal computer in January will be $$\$ 1,200$$. However, PM knows the cost of component parts will decline each month so that the overall cost to produce a PC will be $5 \%$ less each month. The cost of holding a computer in inventory is $$\$ 15$$ per unit per month. Following is the demand for the company's computers each month: Determine a production schedule for PM that will minimize total cost.

In Problem 30, suppose that the demand for personal computers increased each month, as follows: In addition to the regular production capacity of 300 units per month, PM Computer Services can also produce an additional 200 computers per month by using overtime. Overtime production adds $20 \%$ to the cost of a personal computer. Determine a production schedule for PM that will minimize total cost.

National Foods Company has five plants where it processes and packages fruits and vegetables. It has suppliers in six cities in California, Texas, Alabama, and Florida. The company owns and operates its own trucking system for transporting fruits and vegetables from its suppliers to its plants. However, it is now considering outsourcing all its shipping to outside trucking firms and getting rid of its own trucks. It currently spends $$\$ 245,000$$ per month to operate its own trucking system. It has determined monthly shipping costs (in thousands of dollars per ton) of using outside shippers from each of its suppliers to each of its plants, as shown in the following table: Should National Foods continue to operate its own shipping network or sell its trucks and outsource its shipping to independent trucking firms?

In Problem 32, National Foods would like to know what the effect would be on the optimal solution and the company's decision regarding its shipping if it negotiates with its suppliers in Sacramento, Jacksonville, and Ocala to increase their capacity to 25 tons per month. What would be the effect of negotiating instead with its suppliers at San Antonio and Montgomery to increase their capacity to 25 tons each?

Orient Express is a global distribution company that transports its clients' products to customers in Hong Kong, Singapore, and Taipei. All the products Orient Express ships are stored at three distribution centers - one in Los Angeles, one in Savannah, and one in Galveston. For the coming month the company has 450 containers of computer components available at the Los Angeles center, 600 containers available at Savannah, and 350 containers available at Galveston. The company has orders for 600 containers from Hong Kong, 500 containers from Singapore, and 500 containers from Taipei. The shipping costs per container from each U.S. port to each of the overseas ports are shown in the following table:

Orient Express, as the overseas broker for its U.S. customers, is responsible for unfulfilled orders, and it incurs stiff penalty costs from overseas customers if it does not meet an order. The Hong Kong customers charge a penalty cost of $$\$ 800$$ per container for unfulfilled demand, Singapore customers charge a penalty cost of $$\$ 920$$ per container, and Taipei customers charge $$\$ 1,100$$ per container. Formulate and solve a transportation model to determine the shipments from each U.S. distribution center to each overseas port that will minimize shipping costs. Indicate what portion of the total cost is a result of penalties.

Binford Tools manufactures garden tools. It uses inventory, overtime, and subcontracting to absorb demand fluctuations. Expected demand, regular and overtime production capacity, and subcontracting capacity are provided in the following table for the next four quarters for its basic line of steel garden tools:

The regular production cost per unit is $$\$ 20$$, the overtime cost per unit is $$\$ 25$$, the cost to subcontract a unit is $$\$ 27$$, and the inventory carrying cost is $$\$ 2$$ per unit. The company has 300 units in inventory at the beginning of the year.

Determine the optimal production schedule for the four quarters to minimize total costs.

The National Western Railroad's rail network covers most of the U.S. West and Midwest. On a daily basis it sends empty freight cars from various locations in its rail network to its customers for their use. Sometimes there are not enough freight cars to meet customer demand. The transportation costs for shipping empty freight cars, shown as follows, are directly related to distance traveled and the number of rail centers that must handle the car movement.

Determine the number of empty freight cars that should be sent from each rail network location to customers to meet demand at the minimum total cost.

Al, Barbara, Carol, and Dave have joined together to purchase two season tickets to the Giants' home football games. Because there are eight home games, each person will get tickets to two games. Each person has ranked the games they prefer from 1 to 8 , with 1 being most preferred and 8 least preferred, as follows: $$ \begin{array}{lcccc} \text { Game } & \text { Al } & \text { Barbara } & \text { Carol } & \text { Dave } \\ \hline \text { 1. Cowboys } & 1 & 2 & 1 & 4 \\ \text { 2. Redskins } & 3 & 4 & 4 & 1 \\ \text { 3. Cardinals } & 7 & 8 & 8 & 7 \\ \text { 4. Eagles } & 2 & 7 & 5 & 3 \\ \text { 5. Bengals } & 5 & 6 & 6 & 8 \\ \text { 6. Packers } & 6 & 3 & 2 & 5 \\ \text { 7. Saints } & 8 & 5 & 7 & 6 \\ \text { 8. Jets } & 4 & 1 & 3 & 2 \\ \hline \end{array} $$ Determine the two games each person should get tickets for that will result in the groups' greatest degree of satisfaction. Do you think the participants will think your allocation is fair?

World Foods, Inc., imports food products such as meats, cheese, and pastries to the United States from warehouses at ports in Hamburg, Marseilles, and Liverpool. Ships from these ports deliver the products to Norfolk, New York, and Savannah, where they are stored in company warehouses before being shipped to distribution centers in Dallas, St. Louis, and Chicago. The products are then distributed to specialty food stores and sold through catalogs. The shipping costs $$(\$ / 1,000 \mathrm{lb}$$.) from the European ports to the U.S. cities and the available supplies $$(1,000 \mathrm{lb}$$.) at the European ports are provided in the following table: The transportation costs ( $$\$ / 1,000 \mathrm{lb}$$.) from each U.S. city of the three distribution centers and the demands $$(1,000 \mathrm{lb}$$.) at the distribution centers are as follows: Determine the optimal shipments between the European ports and the warehouses and the distribution centers to minimize total transportation costs.

A sports apparel company has received an order for a college basketball team's national championship T-shirt. The company can purchase the T-shirts from textile factories in Mexico, Puerto Rico, and Haiti. The shirts are shipped from the factories to companies in the United States that silk-screen the shirts before they are shipped to distribution centers. Following are the production and transportation costs ( $$\$ /$$ shirt) from the T-shirt factories to the silk-screen companies to the distribution centers, plus the supply of T-shirts at the factories and demand for the shirts at the distribution centers: Determine the optimal shipments to minimize total production and transportation costs for the apparel company.

Walsh's Fruit Company contracts with growers in Ohio, Pennsylvania, and New York to purchase grapes. The grapes are processed into juice at the farms and stored in refrigerated vats. Then the juice is shipped to two plants, where it is processed into bottled grape juice and frozen concentrate. The juice and concentrate are then transported to three food warehouses/distribution centers. The transportation costs per ton from the farms to the plants and from the plants to the distributors, and the supply at the farms and demand at the distribution centers are summarized in the following tables:

a. Determine the optimal shipments from farms to plants to distribution centers to minimize total transportation costs. b. What would be the effect on the solution if the capacity at each plant were 140,000 tons?

A national catalog and Internet retailer has three warehouses and three major distribution centers located around the country. Normally, items are shipped directly from the warehouses to the distribution centers; however, each of the distribution centers can also be used as an intermediate transshipment point. The transportation costs (\$/unit) between warehouses and distribution centers, the supply at the warehouses (100 units), and the demand at the distribution centers (100 units) for a specific week are shown in the following table: The transportation costs ($/unit) between the distribution centers are Determine the optimal shipments between warehouses and distribution centers to minimize total transportation costs.

Horizon Computers manufactures laptops in Germany, Belgium, and Italy. Because of high tariffs between international trade groups, it is sometimes cheaper to ship partially completed laptops to factories in Puerto Rico, Mexico, and Panama and have them completed before final shipment to U.S. distributors in Texas, Virginia, and Ohio. The cost (\$/unit) of the completed laptops plus tariffs and shipment costs from the European plants directly to the United States and supply and demand are as follows:

The Midlands Field Produce Company contracts with potato farmers in Colorado, Minnesota, North Dakota, and Wisconsin for monthly potato shipments. Midlands picks up the potatoes at the farms and ships mostly by truck (and sometimes by rail) to its sorting and distribution centers in Ohio, Missouri, and Iowa. At these centers the potatoes are cleaned, rejects are discarded, and the potatoes are sorted according to size and quality. They are then shipped to combination plants and distribution centers in Virginia, Pennsylvania, Georgia, and Texas, where the company produces a variety of potato products and distributes bags of potatoes to stores. Exceptions are the Ohio distribution center, which will accept potatoes only from farms in Minnesota, North Dakota, and Wisconsin, and the Texas plant, which won't accept shipments from Ohio because of disagreements over delivery schedules and quality issues. Following are summaries of the shipping costs from the farms to the distribution centers and the processing and shipping costs from the distribution centers to the plants, as well as the available monthly supply at each farm, the processing capacity at the distribution centers, and the final demand at the plants (in bushels): table cant copy Formulate and solve a linear programming model to determine the optimal monthly shipments from the farms to the distribution centers and from the distribution centers to the plants to minimize total shipping and processing costs.

KanTech Corporation is a global distributor of electrical parts and components. Its customers are electronics companies in the United States, including computer manufacturers and audio/ visual product manufacturers. The company contracts to purchase components and parts from manufacturers in Russia, Eastern and Western Europe, and the Mediterranean, and it has them delivered to warehouses in three European ports, Gdansk, Hamburg, and Lisbon. The various components and parts are loaded into containers based on demand from U.S. customers. Each port has a limited fixed number of containers available each month. The containers are then shipped overseas by container ships to the ports of Norfolk, Jacksonville, New Orleans, and Galveston. From these seaports, the containers are typically coupled with trucks and hauled to inland ports in Front Royal (Virginia), Kansas City, and Dallas. There are a fixed number of freight haulers available at each port each month. These inland ports are sometimes called "freight villages," or intermodal junctions, where the containers are collected and transferred from one transport mode to another (i.e., from truck to rail or vice versa). From the inland ports, the containers are transported to KanTech's distribution centers in Tucson, Pittsburgh, Denver, Nashville, and Cleveland. Following are the handling and shipping costs (\$/container) between each of the embarkation and destination points along this overseas supply chain and the available containers at each port:

In Problem 44, KanTech Corporation is just as concerned that its U.S. distributors receive shipments in the minimum amount of time as they are about minimizing their shipping costs. Suppose that each U.S. distributor receives one major container shipment each month. Following are summaries of the shipping times (in days) between each of the embarkation and destination points along KanTech's overseas supply chain. These times encompass not only travel time but also processing, loading, and unloading times at each port: table cant copy

Blue Mountain Coffee Company produces various blends of Free Trade, organic specialty coffees that it sells to wholesale customers. The company imports 25 million pounds of coffee beans annually from coffee plantations in Brazil, Indonesia, Kenya, Colombia, Côte d'Ivoire, and Guatemala. The beans are shipped from these countries to U.S. ports in Galveston, New Orleans, Savannah, and Jacksonville, where they are loaded onto container trucks and shipped to the company's plant in Vermont. The shipping costs (in dollars per million pounds) from the countries to the U.S. ports, the amount of beans (in millions of pounds) contracted from the growers in each country, and the port capacities are shown in the following table: table cant copy

The shipping costs from each port to the plant in Vermont are shown in the following table:

$$ \begin{array}{lr} \hline \text { U.S. Port } & \text { Vermont } \\ \hline \text { 7. Galveston } & \$ 61,000 \\ \text { 8. New Orleans } & 55,000 \\ \text { 9. Savannah } & 38,000 \\ \text { 10. Jacksonville } & 43,000 \\ \hline \end{array} $$ Determine the optimal shipments from the grower countries to the plant in Vermont that will minimize shipping costs.

The Pinnacle Company is a U.S.-based manufacturer of furniture and appliances that offshored all of its actual manufacturing operations to Asia about a decade ago. It then set up distribution centers at various locations on the East Coast, near ports where its items were imported on container ships. In many cases, Pinnacle's appliances and furniture arrive partially assembled, and the company completes the assembly at its distribution centers before sending the finished products to retailers. For example, appliance motors, electric controls, housings, and furniture pieces might arrive from different Asian manufacturers in separate containers. Recently Pinnacle began exporting its products to various locations in Europe, and demand steadily increased. As a result, the company determined that shipping items to the United States, assembling the products, and then turning around and shipping them to Europe was inefficient and not cost effective. The company now plans to open three new distribution centers near ports in Europe, and it will ship its items from Asian ports to distribution centers at the European ports, offload some of the items for final product assembly, and then ship the partially filled containers on to the U.S. distribution centers. The following table shows the seven possible locations near container ports in Europe, and their container capacity that Pinnacle has identified to construct its proposed three distribution centers; the container shipments from each of its Asian ports; and the container shipping cost from each of the Asian ports to each possible distribution center location.

In Problem 47, reformulate the model so that Pinnacle minimizes its shipping costs while selecting its three new distribution centers within a budget of $$\$ 45$$ million. What is the difference in this solution, if any?

Formulate and solve a transshipment model for the shipments of cottons from the cotton processing facilities to the U.S. ports and to the overseas ports and Suntrek's fabric factories as described in Problems 26 and 27.

A plant has four operators to be assigned to four machines. The time (minutes) required by each worker to produce a product on each machine is shown in the following table: $$ \begin{array}{crrrr} \text { Operator } & \text { A } & \text { B } & \text { C } & \text { D } \\ \hline 1 & 10 & 12 & 9 & 11 \\ 2 & 5 & 10 & 7 & 8 \\ 3 & 12 & 14 & 13 & 11 \\ 4 & 8 & 15 & 11 & 9 \\ \hline \end{array} $$ Determine the optimal assignment and compute total minimum time.

A shop has four machinists to be assigned to four machines. The hourly cost of having each machine operated by each machinist is as follows: table cant copy However, because he does not have enough experience, machinist 3 cannot operate machine B. a. Determine the optimal assignment and compute total minimum cost. b. Formulate this problem as a general linear programming model.

The Omega pharmaceutical firm has five salespersons, whom the firm wants to assign to five sales regions. Given their various previous contacts, the salespersons are able to cover the regions in different amounts of time. The amount of time (days) required by each salesperson to cover each city is shown in the following table: table cant copy Which salesperson should be assigned to each region to minimize total time? Identify the optimal assignments and compute total minimum time.

The Bunker Manufacturing firm has five employees and six machines and wants to assign the employees to the machines to minimize cost. A cost table showing the cost incurred by each employee on each machine follows: table cant copy Because of union rules regarding departmental transfers, employee 3 cannot be assigned to machine $\mathrm{E}$, and employee 4 cannot be assigned to machine B. Solve this problem, indicate the optimal assignment, and compute total minimum cost.

Given the following cost table for an assignment problem, determine the optimal assignment and compute total minimum cost: table cant copy

A dispatcher for Citywide Taxi Company has six taxicabs at different locations and five customers who have called for service. The mileage from each taxi's present location to each customer is shown in the following table: table cant copy Determine the optimal assignment(s) that will minimize the total mileage traveled.

The Southeastern Conference has nine basketball officials who must be assigned to three conference games, three to each game. The conference office wants to assign the officials so that the total distance they travel will be minimized. The distance (in miles) each official would travel to each game is given in the following table: table cant copy Determine the optimal assignment(s) to minimize the total distance traveled by the officials.

In Problem 57, officials 2 and 8 recently had a confrontation with one of the coaches in the game in Athens. They were forced to eject the coach after several technical fouls. The conference office decided that it would not be a good idea to have these two officials work the Athens game so soon after this confrontation, so they decided that officials 2 and 8 will not be assigned to the Athens game. How will this affect the optimal solution to this problem?

State University has planned six special catered events for the Saturday of its homecoming football game. The events include an alumni brunch, a parents' brunch, a booster club luncheon, a postgame party for season ticket holders, a lettermen's dinner, and a fund-raising dinner for major contributors. The university wants to use local catering firms as well as the university catering service to cater these events, and it has asked the caterers to bid on each event. The bids (in thousands of dollars) based on menu guidelines for the events prepared by the university are shown in the following table: table cant copy The Bon Apetít, Custom, and University caterers can handle two events, whereas each of the other four caterers can handle only one. The university is confident that all the caterers will do a highquality job, so it wants to select the caterers for the events that will result in the lowest total cost.

Determine the optimal selection of caterers to minimize total cost.

A university department head has five instructors to be assigned to four different courses. All the instructors have taught the courses in the past and have been evaluated by the students. The rating for each instructor for each course is given in the following table (a perfect score is 100 ): table cant copy The department head wants to know the optimal assignment of instructors to courses to maximize the overall average evaluation. The instructor who is not assigned to teach a course will be assigned to grade exams.

The coach of the women's swim team at State University is preparing for the conference swim meet and must choose the four swimmers she will assign to the 800-meter medley relay team. The medley relay consists of four strokes - backstroke, breaststroke, butterfly, and freestyle. The coach has computed the average times (in minutes) each of her top six swimmers has achieved in each of the four strokes for 200 meters in previous swim meets during the season, as follows: table cant copy Determine for the coach the medley relay team and its total expected relay time.

Biggio's Department Store has six employees available to assign to four departments in the store-home furnishings, china, appliances, and jewelry. Most of the six employees have worked in each of the four departments on several occasions in the past and have demonstrated that they perform better in some departments than in others. The average daily sales for each of the six employees in each of the four departments are shown in the following table: table cant copy Employee 3 has not worked in the china department before, so the manager does not want to assign this employee to china.

Determine which employee to assign to each department and indicate the total expected daily sales.

The Vanguard Publishing Company wants to hire seven of the eight college students who have applied as salespeople to sell encyclopedias during the summer. The company desires to allocate them to three sales territories. Territory 1 requires three salespeople, and territories 2 and 3 require two salespeople each. It is estimated that each salesperson will be able to generate the amounts of dollar sales per day in each of the three territories as given in the following table: table cant copy Help the company allocate the salespeople to the three territories so that sales will be maximized.

Carolina Airlines, a small commuter airline in North Carolina, has six flight attendants that it wants to assign to six monthly flight schedules in a way that will minimize the number of nights they will be away from their homes. The numbers of nights each attendant must be away from home with each schedule are given in the following table:

$$ \begin{array}{crrrrrr} \text { Attendant } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { F } \\ \hline 1 & 7 & 4 & 6 & 10 & 5 & 8 \\ 2 & 4 & 5 & 5 & 12 & 7 & 6 \\ 3 & 9 & 9 & 11 & 7 & 10 & 8 \\ 4 & 11 & 6 & 8 & 5 & 9 & 10 \\ 5 & 5 & 8 & 6 & 10 & 7 & 6 \\ 6 & 10 & 12 & 11 & 9 & 9 & 10 \\ \hline \end{array} $$ Identify the optimal assignments that will minimize the total number of nights the attendants will be away from home.

The football coaching staff at Tech focuses its recruiting on several key states, including Georgia, Florida, Virginia, Pennsylvania, New York, and New Jersey. The staff includes seven assistant coaches, two of whom are responsible for Florida, a high school talent-rich state, whereas one coach is assigned to each of the other five states. The staff has been together for a long time, and at one time or another all the coaches have recruited in all the states. The head coach has accumulated some data on the past success rate (i.e., percentage of targeted recruits signed) for each coach in each state, as shown in the following table:

$$ \begin{array}{lllllll} \text { Coach } & \text { GA } & \text { FL } & \text { VA } & \text { PA } & \text { NY } & \text { NJ } \\ \hline \text { Allen } & 62 & 56 & 65 & 71 & 55 & 63 \\ \text { Bush } & 65 & 70 & 63 & 81 & 75 & 72 \\ \text { Crumb } & 46 & 53 & 62 & 55 & 64 & 50 \\ \text { Doyle } & 58 & 66 & 70 & 67 & 71 & 49 \\ \text { Evans } & 77 & 73 & 69 & 80 & 80 & 74 \\ \text { Fouch } & 68 & 73 & 72 & 80 & 78 & 57 \\ \text { Goins } & 72 & 60 & 74 & 72 & 62 & 61 \\ \hline \end{array} $$

Determine the optimal assignment of coaches to recruiting regions that will maximize the overall success rate and indicate the average percentage success rate for the staff with this assignment.

Kathleen Taylor is a freshman at Roanoke College, and she wants to develop her schedule for the spring semester. Courses are offered with class periods either on Monday and Wednesday or Tuesday and Thursday for 1 hour and 15 minutes duration, with 15 minutes between class periods. For example, a course designated as $8 \mathrm{M}$ meets on Monday and Wednesday from 8:00 A.M. to 9:15 A.M.; the next class on Monday and Wednesday (9M) meets from 9:30 A.M. to 10:45 A.M.; the next class $(11 \mathrm{M})$ is from 11:00 A.M. to 12:15 P.M.; and so on. Kathleen wants to take the following six freshman courses, with the available sections shown in order of her preference, based on the professor who's teaching the course and the time: table cant copy

For example, there are eight sections of math offered, and Kathleen's first preference is the 11T section, her second choice is the $12 \mathrm{~T}$ section, and so forth. a. Determine a class schedule for Kathleen that most closely meets her preferences. b. Determine a class schedule for Kathleen if she wants to leave 11:00 A.M. to noon open for lunch every day. c. Suppose Kathleen wants all her classes on two days, either Monday and Wednesday or Tuesday and Thursday. Determine schedules for each and indicate which most closely matches her preferences.

CareMed, an HMO health care provider, operates a 24-hour outpatient clinic in Draperton, near the Tech campus. The facility has a medical staff with doctors and nurses who see regular local patients according to a daily appointment schedule. However, the clinic sees a number of Tech students who visit the clinic each day and evening without appointments because their families are part of the CareMed network. The clinic has 12 nurses who work according to three 8-hour shifts. Five nurses are needed from 8:00 A.M. to 4:00 P.M., four nurses work from 4:00 P.M. to midnight, and 3 nurses work overnight from midnight to 8:00 A.M. The clinic administrator wants to assign nurses to a shift according to their preferences and seniority (i.e., when the number of nurses who prefer a shift exceeds the shift demand, the nurses are assigned according to seniority). While the majority of nurses prefer the day shift, some prefer other shifts because of the work and school schedules of their spouses and families. Following are the nurses' shift preferences (where 1 is most preferred) and their years working at the clinic:

$$ \begin{array}{lcccc} \text { Nurse } & 8 \text { A.M. to 4 P.M. } & \text { 4 P.M. to Midnight } & \text { Midnight to 8 A.M. } & \text { Years' Experience } \\ \hline \text { Adams } & 1 & 2 & 3 & 2 \\ \text { Baxter } & 1 & 3 & 2 & 5 \\ \text { Collins } & 1 & 2 & 3 & 7 \\ \text { Davis } & 3 & 1 & 2 & 1 \\ \text { Evans } & 1 & 3 & 2 & 3 \\ \text { Forrest } & 1 & 2 & 3 & 4 \\ \text { Gomez } & 2 & 1 & 3 & 1 \\ \text { Huang } & 3 & 2 & 1 & 1 \\ \text { Inchio } & 1 & 3 & 2 & 2 \\ \text { Jones } & 2 & 1 & 3 & 3 \\ \text { King } & 1 & 3 & 2 & 5 \\ \text { Lopez } & 2 & 3 & 1 & 2 \\ \hline \end{array} $$

Formulate and solve a linear programming model to assign the nurses to shifts according to their preferences and seniority.

The MidSouth Trucking Company based in Nashville has eight trucks located throughout the East and Midwest that have delivered their loads and are available for shipments. Through their Internet logistics site MidSouth has received shipping requests from 12 customers. The following table shows the mileage for a truck to travel to a customer location, pick up the load, and deliver it.

$$ \begin{array}{ccccccccccccc} \hline \text { Truck } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { F } & \text { G } & \text { H } & \text { I } & \text { J } & \text { K } & \text { L } \\ \hline 1 & 500 & 730 & 620 & 410 & 550 & 600 & 390 & 480 & 670 & 710 & 440 & 590 \\ 2 & 900 & 570 & 820 & 770 & 910 & 660 & 650 & 780 & 840 & 950 & 590 & 670 \\ 3 & 630 & 660 & 750 & 540 & 680 & 750 & 810 & 560 & 710 & 1,200 & 490 & 650 \\ 4 & 870 & 1,200 & 810 & 670 & 710 & 820 & 1,200 & 630 & 700 & 900 & 540 & 620 \\ 5 & 950 & 910 & 740 & 810 & 630 & 590 & 930 & 650 & 840 & 930 & 460 & 560 \\ 6 & 1,100 & 860 & 800 & 590 & 570 & 550 & 780 & 610 & 1,300 & 840 & 550 & 790 \\ 7 & 610 & 710 & 910 & 550 & 810 & 730 & 910 & 720 & 850 & 760 & 580 & 630 \\ 8 & 560 & 690 & 660 & 640 & 720 & 670 & 830 & 690 & 880 & 1,000 & 710 & 680 \\ \hline \end{array} $$ Determine the optimal assignment of trucks to customers that will minimize the total mileage.

In Problem 68, assume that the customers have the following truck capacity percentage loads:

$$ \begin{array}{ccccccccccccc} & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { F } & \text { G } & \text { H } & \text { I } & \text { J } & \text { K } & \text { L } \\ \hline \text { Capacity } & 89 & 78 & 94 & 82 & 90 & 83 & 88 & 79 & 71 & 96 & 78 & 85 \\ \hline \end{array} $$ Determine the optimal assignment of trucks to customers that will minimize total mileage while also achieving at least an average truck load capacity of $85 \%$. Does this load capacity requirement significantly increase the total mileage?

The Hilton Island Tennis Club is hosting its annual professional tennis tournament. One of the tournament committee's scheduling activities is to assign chair umpires to the various matches. The tennis association rates chair umpires from 1 (best) to worst (4) based on experience, consistency, and player and coach evaluations. On a particular afternoon 12 matches start at 1:00 P.M. Fifteen chair umpires are available to officiate the matches. The matches are prioritized from 1 to 12 in order of importance, based on the rankings and seedings of the players involved, and whether it is a singles or a doubles match. The committee wants to assign the best umpires to the highest priority matches. The following tables show the match priorities and the umpire ratings.

$$ \begin{array}{llllllllllllllll} \hline \text { Umpire } & 1 & 2 & 3 & 4 & 5 & 6 & \mathbf{7} & \mathbf{8} & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \text { Rating } & \mathbf{3} & \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{2} & \mathbf{3} & \mathbf{2} & \mathbf{4} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{3} \\ \hline \end{array} $$

$$ \begin{array}{lcccccccccccc} \hline \text { Match } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \text { Priority } & 2 & 11 & 6 & 4 & 9 & 10 & 12 & 3 & 1 & 5 & 7 & 8 \\ \hline \end{array} $$ Determine the optimal assignment of umpires to matches.

The National Collegiate Lacrosse Association is planning its annual national championship tournament. It selects 16 teams from conference champions and the highest ranked at-large teams

to play in the single-elimination tournament. The teams are ranked from 1 (best) to 16 (worst), and in the first round of the tournament, the association wants to pair the teams so that highranked teams play low-ranked teams (i.e., seed them so that 1 plays 16,2 plays 15 , etc.). The eight first-round game sites are predetermined and have been selected based on stadium size and conditions, as well as historical local fan interest in lacrosse. Because of limited school budgets for lacrosse and a desire to boost game attendance, the association wants to assign teams to game sites so that all schools will have to travel the least amount possible. The following table shows the 16 teams in order of their ranking and the distance (in miles) for each of the teams to each of the eight game sites.

Formulate and solve a linear programming model that will assign the teams to the game sites according to the association's guidelines.

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Transportation Problem Set 8 | Transshipment Model-1

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• job | a thon dec 21 | Question 9

Transshipment Model is a model which comes under the transportation problem. There are two types of transshipment problem:   

• With sources and destination acting as transient (i.e. intermediary) nodes
• With some transient nodes between sources and destination

Look at the given transportation problem diagram.   

In a transportation problem, the shipment moves from one particular source to another particular destination, maybe from S1 to D1, S1 to D2, S1 to D3, S2 to D1, S2 to D2 and so on. 

This article will discuss the Transshipment problem with sources and destination acting as transient (i.e. intermediary) nodes. The next set of this article will discuss the Transshipment problem with some transient (i.e. intermediary) nodes between sources and destination. 

Transshipment problem with sources and destination acting as transient nodes    

In this method, the shipment passes through one or more intermediary node before it reaches its desired destination. This method allows the shipment to pass from one source to another source and from one destination to another destination before it reaches the desired destination. 

Note: The intermediary nodes can be sources and destinations themselves. 

Consider the following transshipment problem involving 4 sources and 2 destinations. The supply value of the sources S1, S2, S3 and S4 are 200 units, 250 units, 300 units and 450 units respectively. The demand value for destinations D1 and D2 are 600 units and 600 units respectively. The transportation cost per unit between different sources and destinations are summarized in the following table.  Solve the transshipment problem.   

Steps to solve the transshipment problem:   

• Check whether it is balanced or unbalanced. Balanced, if Total sum of supply = Total sum of demand = B so the B value here is 1200.  In this case the problem is balanced see the table below. In case the problem were not balanced we could add dummy row or column to make it balance. Refer this article    

• Add the value of B to all rows and columns. See the table below:   

• Find out the total transportation cost using Vogel’s Approximation Method as it gives the optimized solution than Least Cost Cell Method and North West Corner Method .  After solving the transportation problem using Vogel’s Approximation Method , we get the following solution,   

• Just ignore the zero cost cells and calculate the transportation cost.  Total transportation cost is : (200 * 6) + (450 * 5) + (300 * 7) + (450 * 6) + (150 * 4) = 8850
• Draw the shipping pattern. Note: The allocations in the main diagonal cells are to be ignored.  To draw the shipping diagram first draw the four sources and two destinations as shown below:   

• Look the table above the first allocated cell is (S1, S2). The first shipment starts from S1 to S2. The allocated value is 200.   

• The second allocated cell is (S2, D1). The shipment moved from S2 to D1. The allocated value is 450.   

• The next allocated cell is (S3, D2) and after that we have (S4, D2). The allocated value for these two cells are 300 and 450 respectively.   

• The next and last allocated cell is (D2, D1). The shipment moved from D2 to D1. The allocated value for this cell is 150.   

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Home » Management Science » Transportation and Assignment Models in Operations Research

Transportation and Assignment Models in Operations Research

Transportation and assignment models are special purpose algorithms of the linear programming.   The simplex method of Linear Programming Problems(LPP)   proves to be inefficient is certain situations like determining optimum assignment of jobs to persons, supply of materials from several supply points to several destinations and the like. More effective solution models have been evolved and these are called assignment and transportation models.

The transportation model is concerned with selecting the routes between supply and demand points in order to minimize costs of transportation subject to constraints of supply at any supply point and demand at any demand point.   Assume a company has 4 manufacturing plants with different capacity levels, and 5 regional distribution centres.     4 x 5 = 20 routes are possible.   Given the transportation costs per load of each of 20 routes between the manufacturing (supply) plants and the regional distribution (demand) centres, and supply and demand constraints, how many loads can be transported through different routes so as to minimize transportation costs?   The answer to this question is obtained easily through the transportation algorithm.

Similarly, how are we to assign different jobs to different persons/machines, given cost of job completion for each pair of job machine/person?   The objective is minimizing total cost.   This is best solved through assignment algorithm.

Uses of Transportation and Assignment Models in Decision Making

The broad purposes of Transportation and Assignment models in LPP are just mentioned above.   Now we have just enumerated the different situations where we can make use of these models.

Transportation model is used in the following:

• To decide the transportation of new materials from various centres to different manufacturing plants.   In the case of multi-plant company this is highly useful.
• To decide the transportation of finished goods from different manufacturing plants to the different distribution centres.   For a multi-plant-multi-market company this is useful.
• To decide the transportation of finished goods from different manufacturing plants to the different distribution centres.   For a multi-plant-multi-market company this is useful.   These two are the uses of transportation model.   The objective is minimizing transportation cost.

Assignment model is used in the following:

• To decide the assignment of jobs to persons/machines, the assignment model is used.
• To decide the route a traveling executive has to adopt (dealing with the order inn which he/she has to visit different places).
• To decide the order in which different activities performed on one and the same facility be taken up.

In the case of transportation model, the supply quantity may be less or more than the demand.   Similarly the assignment model, the number of jobs may be equal to, less or more than the number of machines/persons available.   In all these cases the simplex method of LPP can be adopted, but transportation and assignment models are more effective, less time consuming and easier than the LPP.

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Exclussive dff. And easy understude

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