TRANSPORTATION PROBLEMS EXERCISES

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1 TRANSPORTATION PROBLEMS EXERCISES Vassilis Kostoglou URL:

2 PROBLEM 1 A factory producing aluminum is supplied with bauxite from three mines (01, 02 and 03) which produce 3, 7 and 5 thousand tones of mineral per week respectively. There are 4 modes of transportation of bauxite to the factory: by ship (T1) - by trucks (T2) - by a simple railway wagon (T3) - by special railway wagons (T4). The total capacity per day is 4 thousand tones for ships, 3 thousand for cars and 4 thousands each one of the two types of rail. Transportation costs per tone are given in the following table. Vehicles Τ1 Τ2 Τ3 Τ4 Mines Identify the quantities that must be transported by any means of transportation so as to minimize the total transportation cost.

3 PROBLEM 2 One of the main products of P & T Company, a large canning industry is canned beans. There are three canneries which send products to four distribution centers. Since transportation costs are particularly increased due to large distances, the administration decided to reduce it. There have been some estimates of the quantity production and transportation to distribution centers and as for the transportation costs of each load (fully loaded truck company) it is denominated in U.S. dollars. Distribution centers Canneries Find the combination of the transportation load that minimizes the total transportation cost.

4 PROBLEM 3 A company that produces an innovative product has two branches and three main customers. The two branches will produce 60 units and 40 units respectively of the product during the next period. The company is committed to sell 50 units to the first customer and at least 20 units to the third customer. The second and the third customer also want to buy as many units from those remaining. The profit of the company (expressed in thousands of euro) depending on the transportation of its branches to customers is given in the following table. Customer Branch Branch How should we distribute the products in order to maximize the total profit?

5 PROBLEM 4 Factories X, Y and Z of a business have a monthly production capacity of a chemical product 22, 15 and 8 tons respectively. This production covers the needs of four consumer centers, which need 7, 12, 17 and 8 tons per month. The cost of transporting one tone (in ) from the factories in the centers of consumption is indicated in the following table. Factory Consumer center Ι ΙΙ ΙΙΙ ΙV Χ Y Ζ The responsible officer has formed the following program based on his experience: Χ ΙΙ: 12 tones, Χ ΙΙΙ: 1 tone, Χ ΙV: 9 tones, Y ΙΙΙ: 15 tones, C Ι : 7 tones, C ΙΙΙ : 1 tone. Consider whether the transportation program developed is the best possible. If not, then determine the optimum solution.

6 PROBLEM 5 A commercial company has three stores, let it be A, B and C, from which supplies its three main largest customers, let it be M, N and Q, with a consumer product. All warehouses have approximately the same size as with corresponding capacity of 50 tones for this product. The three customers require a certain period of time 30, 45 and 25 tones respectively. Transportation cost (in ) of each tone from each warehouse to each customer is as follows. Customer Μ Ν Q Warehouse A B C The company's management wants to know whether it would be desirable to eliminate one of the warehouses and sell the corresponding stock. What in your opinion is the most appropriate decision?

7 PROBLEM 6 A company has three branches which produce a particular product that after production is transported to distribution centers. Branches 1, 2 and 3 produce 12, 17 and 11 loads per month respectively. Each distribution center needs to collect 10 loads per month. The distance from each branch in the respective distribution centers is given (in kms) in the following table. Distribution center Branch The fixed cost of each load is 30 and the extra charge is 1.50/km. a) Design the appropriate transportation model. b) Using the method of the northwest corner find the initial basic feasible solution. c) Starting with the initial basic feasible solution determined to the question b find the optimal solution. How many loads must be transferred from each branch to each distribution center in order to minimize the total transportation cost?

8 PROBLEM 7 Let us suppose that England, France and Spain produce the whole wheat, barley and oats in the world. The demand for wheat in the world requires 125 million hectares available for its production. Similarly, 60 million hectares of land are required for the production of barley and 75 for the production of oats. The total area available for this purpose in England, France and Spain is 70, 110 and 80 million hectares respectively. The number of hours required in England, France and Spain for the production of wheat in one hectare of land is 18, 13 and 16 hours respectively. The corresponding hours in the three countries for the production of barley in a hectare of land is 15, 12 and 12 hours respectively. The number of hours required in England, France and Spain for the production of oats in one hectare of land is 12, 10 and 16 hours respectively. The cost for each working hour for the production of wheat in England, France and Spain is $ 3, $ 2.40 and $ 3.30 per hour respectively. The comparable costs for every working hour for the production of barley is $ 2.70, $ 3 and $ 2.80 respectively, while for the production of oats is $ 2.30, $ 2.50 and $ 2.10 respectively. The problem that must be addressed is the distribution of land used for each country so as to meet global needs and simultaneously minimize the total labor costs. Design and resolve the appropriate transportation model.

9 PROBLEM 8 The company SAS has a chain of stores that sell equipment, hi-fi. The shareholders are thinking of ordering new midi systems, which then would sell 430 pounds. The shops are separated into three geographical areas, north, west and south and SAS believes that the demand for the system in each area will be 170, 210 and 150 units respectively. The SAS has decided to order 100 units for each geographic area stores. There are three potential suppliers for the system A, B and C. A is capable of supplying 200 units for 400 pounds the one, the B 160 units for 420 pounds the one and C 180 units for 410 pounds the one. These prices do not include the transportation costs that vary with the supplier and the supply contract according to the following table.

10 Stores Suppliers North West South Α Β Γ (Unit transportation cost in British pounds) The SAS wants to maximize the profit from the sales of the new system. Design and solve the corresponding transportation problem.

11 PROBLEM 9 An airline company buys fuel for the plane from three vendors. The company needs for each of the three airports that uses and for the next months gallons for the first, gallons for the second and gallons for the third airport. Each vendor can supply fuel to any airport in the price (dollars per gallon) given in the table below. Airport 1 Airport 2 Airport 3 Seller Seller Seller Each seller has a restriction on the total amount of fuel that can be supplied each month. The potential is gallons for the seller 1, gallons for the seller 2 and gallons for the seller 3. Find the right fuel purchasing policy so as to minimize the total cost of feeding the three airports.

12 PROBLEM 10 A farmer's association has launched public competition for the daily transportation of some cereal from the head offices of the three regional warehouses. The daily requirements of the warehouses are amounted to 18, 10 and 8 tones respectively. Overall offers were submitted by three carriers, each specifying the maximum amount of weight that can carry per day. These quantities are equal for all three companies with 12, 16 and 24 tones respectively. The costs for the transportation of one tone of grain from each company in the three regional warehouses are presented (in ) in the following table. Warehouses Companies What contracts exactly would you advise the administration of the partnership to sign, so that on the one hand to minimize the total transportation cost, on the other hand not the administration be accused for bias in favor of or against any of the carriers?

13 PROBLEM 11 A food industry produces chips at three factories, located in Birmingham, Glasgow and London. Since the customers should be supplied with fresh products, these are not stored in the factories. The monthly production ability of the factory in London is 750 tones, while of the other two are 500 tones. Every day 300 tones are given to the five warehouses in order to subsequently be transferred to customers. The profit per sale tone from the first warehouse is 0.4 pounds if produced in London, 0.6 if produced in Glasgow and 2.2 pounds if produced in Birmingham. The corresponding gains for the second warehouse are 1.1, 1.2 and 2 pounds. The profit per tone of production in London is 1.7, 1.3 and 2.5 pounds when sold by the third, fourth and fifth warehouse respectively. The corresponding gains for Birmingham are 1.6, 1 and 0.5 pounds, while for Glasgow are 1.1, 0.8 and 2.1 pounds. What is the maximum monthly profit that can be achieved by the industry?

14 PROBLEM 12 One major producer wants to buy raw material that does not exist in large stock. He needs to purchase 100 tones per week for the operation of the three factories, which have regular weekly requirements 40, 15 and 45 tons respectively. It is possible to satisfy these requirements if it obtains 40, 35 and 25 tons per week from three different suppliers. The cost of each unit of raw material is charged the same from any supplier, but the buyer must pay the costs of transportation for the three factories 6, 3.6 and 4.8 pounds for the first supplier, 2.4, 1.2 and 0.6 pounds for the second supplier and 6, 4.8 and 3.6 pounds for the third supplier, respectively. (a) How should the producer make the orders to minimize the total transportation cost and simultaneously meet requirements of the factory? (b) Assume that in the initial problem the requirement of the first factory is 50 tons per week, of the second factory 15 tons a week and of the third factory 35 tones per week. Determine the optimal solution and compare with that of the initial problem.

15 PROBLEM 13 A pastry company produces a special type of bread in two branches with the following production data. Branch Production capacity (kg) Production cost ($/kg) Α Β Four restaurant chains want to purchase this special type of bread. Their requirements and the amount they offer are given in the table below.

16 Chain Maximum demand (in kg) Offering price ($/kg) Transportation costs (in $) of one kg of bread from each branch of the chain restaurants are: Chain 1 Chain 2 Chain 3 Chain 4 Branch Α Branch Β Design the delivery plan, which maximizes the net profit.

17 PROBLEM 14 A wholesaler sells some food in four major markets, let it be A, B, C and D. The weekly demand has been accurately predicted and is 40 bags for the market A, 30 for B and C and 20 bags for the market D. The wholesaler always orders the goods he markets from two local producers, P & R. The first of them charges 1 for the transportation of one bag in market A, 4 for B, 5 for C and 6 for D. As expected the cost of food is charged separately. The producer P charges 1 for the transportation bags to the market A, 2 for B, 7 for C and 9 for D, without including the cost of the product. One particular week both producers can supply from 65 bags, but the producer P sells food for 1 more expensive per bag than P. Which are the orders which must be placed this week by the wholesalers? Next week the producers can still supply the same quantities, but now P sells 1 per bag more expensive than P, even though transport costs have not changed at all. Should the wholesaler change the orders? And if so, how exactly?

18 PROBLEM 15 The air company "Greek Air-transportations" uses four types of aircrafts (Β 727, Β 737, Β 707, Α 300) and aims to use them in four new routes of transportation of the fresh agricultural products: ΕΑ 101: Athens - Bucharest ΕΑ 108: Athens - Glasgow ΕΑ 205: Thessaloniki - Laussane ΕΑ 207: Thessaloniki - Hamburg The weekly demand of the products is calculated to 400 tons for Bucharest, to 530 for Glasgow, to 450 for Laussane and to 480 for Hamburg. The offering transportation ability for the same time period is 420 tones with Β 727, 390 tones with Β 737, 480 tones with Β 707 and 570 tones with Α 300 (without the number of the aircrafts or the routes getting examined).

19 The Β 737 cannot be used to the route Athens - Glasgow due to restricted action radius (a medium supply is disadvantageous) while the Α 300 can t fly to Bucharest, because there is not specialized technical land potential. The profit per tone (in ) for each flighttype aircraft combination is: Flight ΕΑ 101 ΕΑ 108 ΕΑ 205 ΕΑ 207 Aircrafts Α Β Β Β Design the routes so as to maximize the profit.

20 PROBLEM 16 An agricultural partnership has the following orders (in tones) for fresh and canned peaches. Product Fresh Can Month May June July August The collection of one peach tone requires four working hours, the sorting and their package for direct disposition (fresh) six working hours, while the canning of the same quantity requires five working hours. The partnership employs totally 10 employees, that work in average 25 days per month by full daily 8-hour. 5 of them should be moved to other cultivation for 13 days every month in May and for 16 days in July and in August.

21 Monthly wages are formed per four-month as follows: Labor - Month May June July August Collection Package Canning Late delivery of an order is not possible. Production that exceeds demand of the same period reflects a charge against salary of 50 during the month that production exceeds demand. Design the most advantageous production to the partnership.

22 PROBLEM 17 A company has decided to produce three new products. Its five branches are currently over production capacity. The construction cost of one unit of the first product will be $ 31, $ 29, $ 32, $ 28 and $ 29 in 1 st, 2 nd, 3 rd, 4 th and 5 th branch respectively. Construction cost of a unit of the second product will be $ 45, $ 41, $ 46, $ 42 and $ 43 in 1 st, 2 nd, 3 rd, 4 th and 5 th branch respectively. Construction cost of a unit of the third product will be $ 38, $ 35 and $ 40 in 1 st, 2 nd and 3 rd branch, respectively, while the 4 th and 5 th branch do not have the ability to produce this product. The sales division is that they can be produced 6000, and 8000 units of products 1, 2 and 3 respectively per day. Branches 1, 2, 3, 4 and 5 have the potential to produce 4000, 6000, 4000, 6000 and units per day, respectively, without including the combination of products available. It is assumed that whichever branch has the ability and capacity to produce these products it can also produce combinations of these in any quantity.

23 The administration wants to know how to allocate new products to the branches in order to minimize construction costs. a) Design the problem as a transportation model. b) Starting with the Vogel method for finding the initial possible solution use the Simplex method of transposition problems to determine the optimal solution.

24 PROBLEM 18 A company which produces a unique product has three branches and four main customers. The three branches will produce 6, 8 and 4 units respectively during the following period. The company has pledged to sell 4 units to the first client, 6 units to the second client and at least 2 units to the third client. The third and fourth clients want to buy as much as possible of what will be left over. The net profit from the transportation of a unit from the branch i to customer j is given in the table below. Branches Clients The management wants to know how many units to sell to its interest to the third and

25 fourth clients and how many units should be transferred from the branches to each of the four clients in order to maximize the total profit. PROBLEM 19 New designs must be done for the energy systems of a new building. The three possible sources of energy are electrical, gas and solar energy. The building needs the energy for electricity, water heating and heating of the interior spaces. The respective daily requirements are: Electricity Water heating Heating : 20 units : 10 units : 30 units The size of the roof reduces solar modules to 30 points, while there is no restriction for the rest. The electricity needs can be satisfied by the electricity market ($ 200 unit). The needs of the two other sources can be met from some source or combination of sources. Prices of units are: Electricity Natural Solar

26 gas energy Water heating $ 450 $ 300 $ 150 Heating $ 400 $ 250 $ 200 (a) Design the problem as a transportation model. (b) Use the northwest corner method for finding the initial basic possible solution to the problem as designed at (a). (c) Starting with the basic feasible solution of (b), use the Simplex transportation method to determine the best solution. (d) Use Vogel method for finding the initial feasible solution of the problem as designed at (b). e) Starting with the initial basic feasible solution of the question (d), use the methodology of transposition problems to find the optimal solution. Compare the numbers of steps required to determine the optimal solution using the above two methods.

27 PROBLEM 20 A company has two branches that produce a specific product distributed in three reception centers. The production of each unit has the same cost in both branches and the transport cost (in hundreds of U.S. dollars) per unit for the product is presented for each combination of branch and reception center in the table below. Branch Reception center Α Β A total of 60 units of product are to be produced and transported per week. Each branch can produce and send any number of units with a maximum of 50 units per week, i.e. there is flexibility in how they will share the total production between

28 the two branches, in order to reduce transportation costs. The objective of the management is to determine how many should be produced at each branch and then what will be the total mode of transportation to minimize the transportation costs. Answers wanted to the following questions: (a) Suppose that each reception center must accept 20 units per week. Design the problem as a transportation model. (b) Use the northwest corner method to find an initial basic feasible solution of the problem, as designed in question (a). Then determine the optimal solution. (c) Suppose now that each distribution center can receive an amount between 10 and 30 units per week to reduce transportation costs, but the total cargo transported remains at 60 points. Design the problem as a transportation model.

29 (d) Use the Vogel method to find the initial basic feasible solution for the problem as designed at (c) and then determine the optimal solution. PROBLEM 21 One of the most important products of a multinational IT company is produced in two of its factories and is mostly available in three main clients of the firm. The two factories will produce over the next period 600 and 400 units of the product, respectively. The company is committed with contracts for the sale of 500 units to the first client and at least 200 units to the second. Also the second and third clients both want to buy as many of the remaining units of the product. The net profit from the sale of each unit depends on its origin (factory) and the destination (client) and ranges according to the following table. Client Factory

30 How exactly should the 1000 units of the product be allocate in order to maximize the total profit? PROBLEM 22 A student who is studying abroad, decided that he needs for the next four years a car for his movements. Since his costs are much he wants to carry out his wish by the most inexpensive way. What he can t decide is whether to buy an old car or a newer one. Also he does not know whether he should sell it through these four years. The following data are given. Purchase price Cost of car use per year ($) Car purchase price per year ($) ($) 1 st 2 nd 3 rd 4 th 1 st 2 nd 3 rd 4 th Old car New car If the student changes his car within the next four years, he will do it at the end of that year and will take a car of one of two types. However, he plans to get in the

31 future a newer of that what he gets now. He wants to find the most appropriate solution. a) Describe how the problem can be expressed as a transportation model b) Find the initial solution using any relevant method. c) Determine the optimal solution by the solving transportation problems methodology PROBLEM 23 A large construction company undertook computerization items to equip a ministry with the PC terminal units. It was agreed to supply 150 units next October and 225 units in November. Working an eight hour shift, the manufacturer can only produce 160 terminal units per month. Extending working hours with two hours overtime, it is possible to construct 30 additional units per month, with an additional unit cost of 20. The terminal units can be stored at a monthly cost of 3 per unit. The cost of producing each unit of PC is constant, regardless of the month of construction. Formulate the model (the initial table) in order to find out the production schedule, which minimizes the total cost.

32 PROBLEM 24 A company is expected to face in the next four months monthly demands of 95, 120, 110 and 100 units of products. The production capacity is 90 units per month at a cost of 20 per unit in regular employment. The overtime, which can reach 20% of normal, costs 30per unit. The storage cost is 1 per unit and per month, while the cost for late delivery of order is 3. The company is worried about the high cost of overtime expenses and thinks to make a limited expansion of its facilities so that the normal capacity to be increased to 99 units in order to reduce overtime, which seem to overburden the operating costs. The total production capacity will not be changed. If the company is getting expanded at the beginning of the four months, the costs will remain the same. If it extends at the beginning of the 3rd month, then the cost will be increased by 1, but according to latest legislative incentives for the extension will receive a subsidy by the government with 400.

33 What exactly would you suggest the company to do? PROBLEM 25 The Build-Em-Fast Company has agreed to supply the best customers with three products every week for three weeks, though the production will require some overtime. The data involved are the following: Week Maximum production (normal period) Maximum production (overtime) Production cost per unit (normal period) $ 6000 $ $ 8000 The cost per unit produced in overtime for each week is $ 2000 more than in the regular season. The storage cost is $ 1000 per unit for each week of storage. There are already

34 two products in construction, but the administration does not want to have to manufacture products after the end of the three weeks. The administration wants to know how many units it should produce per week to maximize its profit. (a) Design the problem as a transportation model creating cost tables and requirement tables. (b) Use the Simplex transportation method to determine the optimal solution.

35 PROBLEM 26 The Metro Water District is a consulting company of environmental projects which is responsible for the management of the transportation of water in a large geographical area. The area is relatively dry so the water must be transported from other areas (from rivers Colombo, Sacron and Calorie) and then the company distributes it in its own turn throughout the area. The main customers are in the areas of Berdoo, Los Devils, San Goand, Hollyglass. Water can be transferred from any river in any city other than the river Calorie which does not communicate with the area Hollyglass. Due to the geographical specificity of local transportation costs depends to a large extent on the location of rivers and towns. The transportation cost per thousand cubic meters of water in dollars from all the rivers to all the cities is given in the following table.

36 Area Berdoo Los Devils San Go Hollyglass Capacity supply River Colombo Sacron Calorie Minimum requirement Normal requirement The charge of consumers by the company is the same for every cubic meter of water irrespectively of location and distance. The company is worried about how to distribute the water in the summer months, during which the amount of water available is limited (last column in table). The company must supply every city with a minimum amount of water to meet the minimum requirement of each city (except San Go area that has its own water source). The line of the normal requirement shows that Berdoo wants cubic meters of water above the minimum requirement, the San Go wants cubic

37 meters water and Hollyglass wants as much as possible. The administration wants to distribute water supplies in such a way as to minimize the total transportation cost. Formulate and solve an appropriate transportation model. PROBLEM 27 The Northern Airplane Company manufactures commercial airplanes for several airlines in the world. The last stage in the manufacturing process is the construction of the engine and its installation to the spindle of the airplane. The company works only with contracts with main term the delivery of a number of airplanes at certain time. There must be an engine production plan for the next four months. In order the company to prevent delivery days should produce engines in the way given in the third column of the table below. In addition all engines which must be produced per month are at least 10, 15, 25 and 20 respectively. But it may need some engines to be constructed one or more months after the initial programming due to production conditions. These engines however should be stored and the storage cost reaches the $ per month.

38 Scheduled facilities Maximum Unit production Unit store cost Month production cost (The cost elements are expressed in millions of dollars) The company wants to decide which is the best way of engines production per month in order to minimize the total cost. Formulate and solve the appropriate model.