transportation

TRANSPORTATIONCompiled by – Sapna Bhupendra jain, 9811255704

Outline the basic idea in a Transportation Application.Transportation applications relate to a LPP where goods are to be transported from “m” production locations (factories) to “n” sales locations (warehouses). The objectives are -To meet the differing availability and requirements of these locations andTo minimize the total transportation costs.

The Transportation application can be solved in three stages – Preliminary CheckInitial Basic Feasible Solution (IBFS)Optimality Test

What are the different methods of finding the Initial Basic Feasible Solution to a transportation problem?

IBFS can be determined using any of the following methods - 1. Northwest Corner Rule 2. Least Cost Cell Method 3. Vogel’s Approximation Method (VAM)

Highlight the stages involved in determining the solution to the Transportation Problem.Stage 1: Preliminary check involves the following-

Verify Objective = Minimisation. In case of Profit Matrix, convert the same into an opportunity Loss Matrix, by subtracting each number from the highest number in the matrix.

Verify Nature of Data = Balanced. Data is said to be balanced if Total Availability = Total Requirement. In case of Unbalanced Data, a Dummy Column or Row should be introduced with Zero Transportation costs.

Stage 2: IBFS can be determined using any of the following methods_(a) Northwest corner Rule. (b) Least Cost Cell Method. (c) Vogel’s Approximation method (VAM)

Stage 3: Optimality test : It consists of the following steps -Computation of margin number, ‘U’ and ‘V’ for all rows and columns such that

U+V = Cost of allocated Cells.Computation of U+V for unallocated cells.Computation of Cost Less (U+V) for unallocated cells, i.e. Step 1 minus Step 2 above.

Procedure under North West Corner RuleEnsure Availability = Requirement, by inserting dummy row or column, if required.2. - Go to Top left-hand corner cell of the matrix

- Compare availability and requirement.- Allocated availability or requirement, whichever is less, to that cell.- Cancel the row or column where availability or requirement is exhausted.

3. Go to top left hand corner cell of the resultant matrix (after cancellation of row or column in step 2) Repeat Step 2 procedure till all the row availability and column requirements are satisfied.

Compiled by – Sapna Bhupendra jain, 9811255704

Procedure under least Cost Cell Method1. Ensure Availability = Requirement, by inserting dummy row or column , if required. 2. Identify the cell with the lowest Cost. In case of a tie, arbitrary selection may be made.

Compare availability and requirement. Allocated availability or requirement, whichever is less, to that cell.Cancel the row or column where availability or requirements exhausted.

3. Identify the next lowest cost cell in the matrix.Repeat Step 2 procedure till all the row availability and column requirements are satisfied.

Advantage: The cost associated with each route is taken into consideration. So, this method leads to a better allocation than North West Corner Rule Method.

Procedure under Vogel’s Approximation Method (VAM)Compute Cost Differences for each row and column.

Cost Difference is the difference between the least cost and the next least cost in that row\column.In case of tie in least cost, Cost Difference 0.

Ascertain the maximum of cost differences and select that row or column for allocation.Choose the least cost cell in the selected row or column for allocation.Compare availability and requirement for that cell.

Allocate availability or requirement, whichever is less, to that cell.Cancel the row or column where availability or requirement is exhausted.

Compute Cost Difference for the resultant matrix and repeat the above procedure till all row availability and column requirements are satisfied.

Steps involved in Optimality TestOptimality Test involves the following steps-

1. Table 1: Ui +Vj for allocated cells:Select the row / column with maximum number of allocations.For that row / column, Ui + Vj is equal to zero. [Ui for Rows; Vj for columns]The other set of numbers Ui /Vj are computed in such a way that Ui + Vj = Cost of allocated cells.

Note: The Ui +Vj table can be completed only if the IBFS is non-degenerate.IBFS is said to be degenerate if number of allocations <(No. of rows + No. of column-1).In case of degeneracy, a dummy allocation “e” (a number very close to Zero) is made in the least cost unallocated cell falling in non-dummy row or column.

2. Table 2: Ui + Vj for unallocated Cells:Draw a matrix for the given rows and columns Block out the allocated cells.Compute Ui + Vj (total of margin numbers) for all unallocated cells.

3. Table 3: Net Evaluation Table = ∆ijDraw a matrix for the given rows and columns.Block out the allocated cells.Compute Cost Difference for all unallocated cells. Cost Difference = umber in Table (1) Less Number in

Table (2).DECISION:If all numbers in the Net Evaluation Table are non - negative (i.e.>0), IBFS is optimal and unique.If all numbers are positive and one of the cells contains zero, IBFS is optimal but not unique. Alternative solution

exists. Total number of solutions = Number of Zeroes + 1.


If any one number in Table 3 is negative, solution is not optimal. Reallocation should be made to determine alternative Basic Feasible Solution (ABFS).

ABFS is tested for optimality by adopting the same procedure.

“Loop” Diagram in TransportationIdentify the worst negative in Table 3. If there is a tie, choose the least cost cell.Draw a loop with the following principle-

Loop should commence from and end in the selected worst negative cell.It should have only allocated cells as its other corners.

Only horizontal and vertical lines (not diagonal) shall be permitted.Loop should result in an even sided figure.

Identify the selected cell as having scope for allocation. It is marked with plus (+) sign.Other corners of the loop are identified with (-) and (+) signs alternatively. ABFS is determined by reference to reallocation as specified by the corners of the loop.Reallocation is done as under-

Identify the allocations to the negative cells (Corners marked with (-) sign.Select the minimum out of the above allocations.Add this minimum to (+) corners and subtract this minimum from (-) corners. Other allocations remain

unaffected.This ABFS is tested for optimality using the procedure outlined in the earlier question. In case negatives still arise in the Net Evaluation Table, the reallocation procedure should be continued further..

QuestionsQ.1- Profit Maximisation – Unbalanced Data A Company has four factories F1, F2, F3 and F4 manufacturing the same product. Production and raw material costs differ from factory to factory and are given in the following table. The transportation costs from the factories to sales depots S1, S2 and S3 are also given. The Sales price ad the total requirement at each depot as also the product capacity at each factory is also stated. Determine the most profitable production and distribution schedule and the corresponding profit. The Surplus production should be taken to yield zero profit.

Particulars F1 F2 F3 F4 Sale prices Per unit

Requirements Per unit.

Production Cost per unitRaw Material Cost per unit

1510

18 9

1412

13 9

Transportation Cost per unit to S1 S2 S3

3 1 5

9 7 8

5 4 3

4 5 6

34 32 31

80 120 150

Production capacity (units) 10 150 50 100


Q. 2- Unbalanced Maximisation-Degeneracy A Company has three factories and four customers. It furnishes the following schedule of profit per unit on transportation of goods to customers in rupees. You are required to solve the transportation problem to maximize the profit. Determine the resultant optimal profit.

Factory / Customer A B C D Supply

P Q R

40 44 38

25 35 38

22 30 28

33 30 30

100 30 70

Demand 40 20 60 30

Q. 3- Balanced MinimisationThe information on the available supply to each warehouse, requirement of each market and the unit transportation cost from each warehouse to each market is given below: Warehouse Market Supply

M1 M2 M3 M4

A 5 2 4 3 22 B 4 8 1 6 15 C 4 6 7 5 8 Demand 7 12 17 9

The shipping clerk has worked out the following schedule from his experience.

Unit 12 1 9 15 7 1

From-Warehouse A A A B C C

To-Market M2 M3 M4 M3 M1 M3

You required to-Check and see if the clerk has the optimal schedule.Find the optimal schedule and minimum total shipping cost and If the check is approach by a carrier of route C to M2, who offers to reduce his rate in the hope of getting some business, by how much should the rate be reduced before the clerk considers giving him an order.

Q. 4: Balanced Minimization -Multiple optimal solution.Home Build Construction Company is interested in taking loans from banks for its projects - P,Q,R,S,T. The rates of interest and the lending capacity differ from bank to bank. All these projects are to be completed. The relevant details are provided below. Assuming the role of a consultant, advise the Company as to how it should take the loans so that the total interest payable is least. Find our alternate optimum solution, if any.

Sources Bank Interest rate in % for projects Max credit (in 000s) P Q R S T Private Bank 20 18 18 17 17 Any amount Nationalized Bank 16 16 16 15 16 400


Co-operative Bank 15 15 15 13 14 250 Amount required (in 000) 200 150 200 125 75

Q. 5:Unablanced Minimisation – Big M Cost-Production PlanningTimely and Co, a manufacturer must produce a product in sufficient quantity to meet contractual sales in next four months. The production capacity and unit Cost of production vary from month to month. The production produced in one month may be held for sale in later months but at an estimated storage cost of Re.1 per unit per month. No storage cost is incurred for goods sold in the same month in which they are produced. There is no opening inventory and none is desired at the end of four months. The necessary details are given below. Month Contracted Sales Maximum production Unit cost of Production 1 2 3 4

20 30 50 40

40 50 30 50

14 16 15 17

How much should the manufacturer produce each month to minimise total Cost?

Q.6- Unbalanced Minimisation –production and supply schedulingAlpha Co. has 3 plants and 3 warehouses. The cost of sending a unit from different plants to the warehouses, production at different plants, and demand at different warehouses are shown in the following matrix: Pant Warehouses Production

A B C X 8 16 16 152 Y 32 48 32 164 Z 16 32 48 154 Demand 144 204 82Determine the transportation schedule, so that the cost is minimised. Assume that cost in the cost matrix is given in thousands of rupees.

Q.7 - Unbalanced MinimisationConsider the following transportation cost table. The costs are given in rupees, supply and demand are in units Determine an optimal solutions.Source / Destination 1 2 3 4 5 Supply

I II III

40 38 36

36 28 38

26 34 24

38 34 28

30198 30

160 280 240

Demand 160 160 200 120 240

Q. 8- Unbalanced Maximisation- Multiple Optimal SolutionsSomehow Achieve Profits (SAP) Ltd. has four production plants and four wholesale warehouse outlets. The warehouses are situated away from the production plants. The production and transportation costs, the selling prices, production capacities, and sales quantities are given below:Production Plants Warehouses

1 2 3 4Production Capacity (Units)

Cost of per unit(Rs.)Material Labour &OH


A 10 14 7 10 140 4 6

B 8 12 5 10 100 5 8C 3 7 11 8 150 4 9

D 9 12 6 13 160 3 8

Warehouse requirement in units 80 120 130 110

Selling price(ex warehouse)per unit 26 32 30 25

The cost of transporting a unit from a given plant to a warehouse is shown in the body of the matrix in rupees per unit. Compute a plan for production and distribution that will achieve maximum profit for the company. Also, state the profit achieved by such a plan.

Q. 9- Unbalanced MaximisationBishop Company has four terminals A, B, C and D. At the start of a particular day, 10,4,6 and 5 trailers respectively are available at these terminals. During the previous night 13, 10,6and 6 trailers respectively were loaded at plant P, Q, R and S. The Company dispatcher has come up with the following costs:Terminals Plants

P Q R S

ABCD

20 36 10 2840 20 45 2075 35 45 5030 35 30 25

Find the allocation of loaded trailers from plants to terminals in order to minimise transportation costs.

Q. 10- Unbalanced Minimisation A Company produces a small component for all industrial products and distributes it to five wholesalers at a fixed price of Rs. 2.50 per unit. Sales forecasts indicate that monthly deliveries will be 3000,3000,10000,5000and 4000units to wholesalers W1, W2, W3, W4and W5 respectively. The monthly production capabilities are 5000,10000,12500,at plants P1, P2, and P3 respectively. The direct costs of production of each unit are Rs. 1.00,Rs.0.90, and Rs.0.80 at plants P1, P2 and P3 respectively. The transportation costs of shipping a unit from a plant to a wholesaler is given below:

Plant WarehousesW1 W2 W3 W4 W5

P1 P2 P3

0.05 0.07 0.10 0.15 0.150.08 0.06 0.09 0.12 0.141.10 0.09 0.08 0.10 0.15

Find how many components each plant supplies to each wholesaler in order to maximize profit.


Q. 11- Unbalanced Maximisation-Optimal Sales DistributionA household product is manufactured in factories A, B, C and D and is sold at centres1,2and 3. The cost in Rupees of the product per unit and capacity in kilograms per unit time of each plant as also the sale price in Rs. per unit and the demand in Kg. Per unit time are given in the following table. You are required to find the optimal sales distribution.

Factory Cost per unit(Rs.) Capacity per unit (Kg.)

A 12 100

B 15 20

C 11 60

D 13 80

Sales Centre Sale price per unit(Rs.) Demand per unit (Kg)

1 15 120

2 14 140

3 16 60

Q. 12- Unbalanced Maximisation –Optimum Investment StrategyHigh Yield Ltd. has provided the following data seeking your advice on the optimum investment strategy:

Investment made at the Beginning of the year.

Net Return Data (in paise)of Selected investments P Q R S

Amount Available (Lacs)

1 95 80 70 60 70

2 75 65 60 50 40

3 70 45 50 40 90

4 60 40 40 30 30

Maximum investment(Lacs) 40 50 60 60

-

The following additional information also provided:P, Q, R and S represent the selected investment.The company has decided to have four years investment plan.The policy of the company is that amount invested in any year will remain so until the end of the fourth year horizon

(for e.g., a rupee invested in investment P at the beginning of year 1 will grow to Rs. 1.95 by the end of the fourth year yielding a return of 95 paise).

Using the above information, determine the optimum investment strategy.Q.13- Product Distribution –Multiple Optimal SolutionSolve the following transportation problem and state whether the solution derived by you is unique.Godown 1 2 3 4 5 6 Stock Available


Factory 1 7 5 7 7 5 3 60

Factory 2 9 11 6 11 - 5 20

Factory 3 11 10 6 2 2 8 90

Factory 4 9 10 9 6 9 12 50

Demand 60 20 40 20 40 40

Note: It is not possible to transport any quantity from Factory 2 to Godown 5.

Q.14- Advertisement Campaign decision.The manufacturer of jeans is interested in developing an advertisement campaign that will reach four different age groups. Advertising campaigns can be conducted through TV, Radio and Magazines. The following table gives the estimated cost in paise per exposure for each age group according to the medium employed. In addition, maximum exposure levels possible in each of the media, namely TV, Radio and Magazine are 40,30 and 20 million respectively. Also the minimum desired exposures within each age group, namely 13-18,19-25, 26-35 and 36 and older are 30,25,15and 10 millions. The objective is to maximize the cost of attaining the minimum exposure level in each group. Age group 13-18 19-25 26-35 36 and older

TV 12 7 10 10

Radio 10 9 12 10

Magazines 14 12 9 12

(a) Formulate the above as a transportation problem, and find the optimum solution.(b) Solve this problem if the policy is to provide at least 4 million exposures through TV in the 13-18 age group, and at least 8 million exposures through TV in the age group 19-25.

Q. 15- Investment StrategyA Company wishes to determine an investment strategy for each of the next four years. Five investment types have been selected, investment capital has bee allocated for each of the coming four years, and maximum investment levels have been established for each investment type. An assumption is that amounts invested in any year will remain invested until the end of the planning horizon of four years. The following table summarizes the data for this problem. The values in the body of the table represent net return on investment of one rupee upto the end of the planning horizon. For example, a rupee invested in investment Type B at the beginning of Year 1 will grow to Rs. 1.90 by the end of the fourth year, yielding a net return of Rs. 0.90.

Investment made at beginning Of year

Net Return Data from Investment Type A B C D E

Rupee Available (000’s)

1 0.80 0.90 0.60 0.75 1.00 500

2 0.55 0.65 0.40 0.60 0.50 600

3 0.30 0.25 0.30 0.50 0.20 750


4 0.15 0.12 0.25 0.35 0.10 800

Max. Rupee invt. (000’s) 750 600 500 800 1000

The objective in this problem is to determine the amount to be invested at the beginning of each year in an investment type, so as to maximize the net rupee return for the four-year period. Solve the above transportation problem and get an optimal solution. Also calculate the net return on investment for the 4 year planning period.

Q. 16- 2001-Dec MEC Ltd. has received an order for supply of100 units of a particular machine. Four types of components (C1to C4) one each required per machine have to be fabricated. The firm has five workshops ( W1to W5) each of which is capable of fabricating any component. However , for efficiency and cost savings, a workshop will be assigned to fabricate at most one type of component. The costs( rupees per unit) of fabrication vary and data are given below:Find the optimum assignment of the components to the workshops and the overall costs.If prior to executing the above plane, there is a breakdown in W1. How will this affect the above optimum

plan?

C1 C2 C3 C4W1 21 23 29 17W2 22 21 31 18W3 24 20 30 19W4 17 18 26 16W5 23 21 29 18

Q. 17- 2002 Dec A Company has 4 Factories F1,F2, F3and F4 manufacturing the same product. Production and raw material costs differ from factory to factory and are given in the table below in the first two rows. The transportation costs from the factories to the sales depots S1, S2and S3 are also given. The last two columns in the table below give the sales price and total requirement at each depot and the production capacity of each factory is given in the last row.

F1 F2 F3 F4 Sales price /Unit(Rs.) Requirement Production Cost / Unit (Rs) !5 18 14 13Raw Material Cost / unit (Rs.) 10 9 12 9Transportation Cost / Unit (Rs.)

S1 3 9 5 4 34 80S2 1 7 4 5 32 120S3 5 8 3 6 31 150

Production capacity 10 150 50 100Determine the optimal solution and the associated profit by using the Vogel’s Approximation

Method(VAM).

Q. 18- 2003 June The Bombay Transport Company has trucks available at four different sites in the following numbers:

Site A - 5 trucksSite B – 10 trucks

Site C - 7 trucks


Site D – 3 trucksCustomers – W, X and Y require trucks as shown below.Customer W- 5 trucksCustomer X- 8 trucksCustomer Y- 10 trucks

Variable Costs of getting trucks to the Customers are given below.From A to W- Rs. 7, to X- Rs. 3, to Y – Rs. 6From B to W – Rs. 4, to X- Rs. 6, to Y- Rs. 8From C to W- Rs .5, to X- Rs. 8 to Y- Rs. 4From D to W – Rs. 8, to X- Rs. 4, to Y- Rs. 3

Solve the above transportation problem.Q. 19- 2004 June; The following table shows all the necessary information on the availability of supply to each factory of Best Industries Ltd., the requirement of each destination and the unit transport cost( in Rs.) from each factory to each destination:

Factory Destination SupplyI II III

A 5 1 7 10B 6 4 6 80C 3 2 5 15Demand 75 20 50Since there is not enough supply, some of the demands at the three destinations may not be satisfied.

For the unsatisfied demands, let the penalty costs be Re. 1, Rs. 2 and Rs. 3 respectively.Find the optimal allocation that minimize the transportation and penalty costs by using the Vogel’s

Approximation Method (VAM).

Q20. Unbalanced Maximaisation-Maximum ProfitsFashion Store wishes to purchase following types of dresses.Dress Type SS S M L XL

Quantity 150 100 75 250 200

Tenders are submitted by four different manufacturers who undertake to supply not more than the quantities below (all dress types combined).Manufacture W X Y Z

Quantity 300 250 150 200

The store estimates that its profit per dress will vary with the manufacturer as show below:Manufacturer / Dress Type SS S M L XL

W 11 14 17 9 6

X 12 13 18 7 4

Y 10 14 19 8 5

Z 13 11 16 10 7


Use transportation techniques to determine how the orders should be placed. What is the maximum profit?


transportation

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