Assignment and Transportation Problem

Amit Kumar Bardhan

Faculty of Management Studies

University of Delhi

Step1. Find a bfs. Find the minimum element in each row of the mxm cost matrix. Construct a new matrix by subtracting from each cost the minimum cost in its row. For this new matrix, find the minimum cost in each column. Construct a new matrix (reduced cost matrix) by subtracting from each cost the minimum cost in its column.

Step2. Draw the minimum number of lines (horizontal and/or vertical) that are needed to cover all zeros in the reduced cost matrix. If m lines are required, an optimal solution is available among the covered zeros in the matrix. If fewer than m lines are required, proceed to step 3.

Step3. Find the smallest nonzero element (call its value k) in the reduced cost matrix that is uncovered by the lines drawn in step 2. Now subtract k from each uncovered element of the reduced cost matrix and add k to each element that is covered by two lines. Return to step2.

Assignment Problems: Hungarian method Assignment Problems: Hungarian method

Example 1: Assignment ProblemExample 1: Assignment Problem

Personnel Assignment:

Time (hours)Job 1 Job 2 Job 3 Job 4

Person 1 14 5 8 7Person 2 2 12 6 5Person 3 7 8 3 9Person 4 2 4 6 10

Example 2: Swimmer Selection

The coach of a swim team needs to assign swimmers to a 200-meters medley relay team to send to the Olympics. Since most of his best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five fastest swimmers and the best time (in seconds) they have achieved in each of the strokes (for 50 meters) are:

From: Hillier and Hillier

Swimmer Selection [contd.]

Stroke Mark Ajay Vipin Ali Vir

Backstroke 37 32 33 37 35

Breaststroke 43 33 42 34 41

Butterfly 33 28 38 30 33

Freestyle 29 26 29 28 31

The coach wishes to determine how to assign four swimmers to the four different strokes to minimize the sum of corresponding best times.

Hest the coach in team selection.

Swimmer Selection [contd.]

Swimmer Selection [Answer]

Backstroke → Vipin

Breaststroke → Ali

Butterfly → Ajay

Freestyle → Mark

Example: Flight Assignment

‘YO Air Co.’ is a small commuter airline running 6 flights per day from New Delhi to surrounding tourist destinations. Flight crews are all based in New Delhi, working flights to various locations and returning on the next flight home. Taking into account complex work rules, and pay incentives, YO Air schedulers have constructed the 8 possible work patterns detailed in the following table. Each row of the table marks the flight that could be covered in a particular pattern and the daily cost per crew (in lakhs of Rupees)

Flight Assignment [contd.]

WorkPattern

Flight Cost

1 2 3 4 5 6

1 √ √ 1.40

2 √ √ 0.96

3 √ √ √ 1.52

4 √ √ √ 1.60

5 √ √ √ 1.32

6 √ √ 1.12

7 √ √ 0.84

8 √ √ √ 1.54

Flight Assignment [contd.]

The company wants to choose a minimum total cost collection of work patterns that covers all flights exactly once.

How to find the best assignments?

Transportation Problem

Power Co: Shipping costs, Supply, and Demand for Powerco Example

From To

City 1 City 2 City 3 City 4 Supply (Million kwh)

Plant 1 8 6 10 9 35

Plant 2 9 12 13 7 50

Plant 3 14 9 16 5 40

Demand (Million kwh)

45 20 30 30

Transportation Tableau

Powerco Power Plant: Formulation

Define Variables:Let Xij = number of (million kwh) produced at plant i and sent to city j.

Objective Function:Min z = 8*X11 + 6*X12 + 10*X13 + 9*X14 + 9*X21 + 12*X22 + 13*X23 + 7*X24 + 14*X31 + 9*X32 + 16*X33 + 5*X34

Supply Constraints:X11 + X12 + X13 + X14 < = 35 (Plant 1) X21 + X22 + X23 + X24 < = 50 (Plant 2) X31 + X32 + X33 + X34 < = 40 (Plant 3)

Powerco Power Plant: Formulation (Cont’d.)

Demand Constraints:X11 + X21 +X31 > = 45 (City 1) X12 + X22 +X32 > = 20 (City 2) X13 + X23 +X33 > = 30 (City 3) X14 + X24 +X34 > = 30 (City 4)

Nonnegativity Constraints:Xij > = 0 (i=1,..,3; j=1,..,4)

Balanced?Total Demand = 45 + 20 + 30 + 30 = 125Total Supply = 35 + 50 + 40 = 125

Yes, Balanced Transportation Problem! (If problem is unbalanced, add dummy supply or demand as required.)

Example: Cost-less Corp Ltd.

The Cost-less Corp Ltd. supplies its four retail outlets from its four plants. The shipping cost per shipment from each plant to each retail outlet is given below

Unit shipping cost

Retail outlet

A B C D

Plant

1 500 600 400 200

2 200 900 100 300

3 300 400 200 100

4 200 100 300 200

Plant 1, 2, 3 and 4 make 10, 20, 20 and 10 shipments respectively.

Retail outlets 1, 2, 3 and 4 need to receive 20, 10, 10 and 20

shipments per month respectively. The distribution manager wants to determine the best plan for how many shipments to send from each plant to respective retail outlets each month. The objective is to minimize the total shipping cost.

Example: Cost-less Corp Ltd. [contd.]

Solving transportation problem

• Can be formulated as Linear Program and solved through simplex method / solver

• Modified Simplex method applied directly on the table

Step 1: Find the Initial basic feasible solution

(North-West Corner rule, Least Cost Method, Vogel’s Approximation method)

Step 2: UV Method / Stepping stone method

Transshipment Problems

A transportation problem allows only shipments that go directly from supply points to demand points. In many situations, shipments are allowed between supply points or between demand points. Sometimes there may also be points (called transshipment points) through which goods can be transshipped on their journey from a supply point to a demand point. Fortunately, the optimal solution to a transshipment problem can be found by solving a transportation problem.

The following steps describe how the optimal solution to a transshipment problem can be found by solving a transportation problem.

Step 1: If necessary, add a dummy demand point (with a supply of 0 and a demand equal to the problem’s excess supply) to balance the problem. Shipments to the dummy and from a point to itself will be zero. Let s= total available supply.

Step 2: Construct a transportation tableau as follows: A row in the tableau will be needed for each supply point and transshipment point, and a column will be needed for each demand point and transshipment point.

Solving Transshipment Problems

Each supply point will have a supply equal to it’s original supply, and each demand point will have a demand to its original demand. Let s= total available supply. Then each transshipment point will have a supply equal to (point’s original supply)+s and a demand equal to (point’s original demand)+s. This ensures that any transshipment point that is a net supplier will have a net outflow equal to point’s original supply and a net demander will have a net inflow equal to point’s original demand. Although we don’t know how much will be shipped through each transshipment point, we can be sure that the total amount will not exceed s.

Transshipment Example

Widgetco manufactures widgets at two factories, one in Memphis and one in Denver. The Memphis factory can produce as 150 widgets, and the Denver factory can produce as many as 200 widgets per day.

Widgets are shipped by air to customers in LA and Boston. The customers in each city require 130 widgets per day. Because of the deregulation of airfares, Widgetco believes that it may be cheaper first fly some widgets to NY or Chicago and then fly them to their final destinations. The cost of flying a widget are shown next. Widgetco wants to minimize the total cost of shipping the required widgets to customers.

Transportation Tableau Associated with the Transshipment Example

NY Chicago LA Boston Dummy Supply

Memphis $8 $13 $25 $28 $0 150

Denver $15 $12 $26 $25 $0 200

NY $0 $6 $16 $17 $0 350

Chicago $6 $0 $14 $16 $0 350

Demand 350 350 130 130 90

Supply points: Memphis, Denver

Demand Points: LA Boston

Transshipment Points: NY, Chicago

The problem can be solved using the transportation simplex method

Example - 1

General Motors produces cars at L.A. and Detroit and has a warehouse in Atlanta. The company supplies cars to customers in Houston and Tampa. The cost of shipping a car between points is given in table. (‘-’ means the shipment is not allowed) L.A. can produce as many as 1100 cars and Detroit can produce as many as 2900 cars. Houston must receive 2400 cars and Tampa must receive 1500 cars.

(a) Formulate a balanced transportation problem that can be used to minimize the shipping cost incurred in meeting demands at Houston and Tampa.

(b) Modify the answer to (a) If shipments between LA and Detroit are not allowed

(c) Modify the answer to (a) if shipments between Houston and Tampa are allowed at a cost of $5.

Example - 1

To ($)

From L.A. Detroit Atlanta Houston Tampa

L.A. 0 140 100 90 225

Detroit 145 0 111 110 119

Atlanta 105 115 0 113 78

Houston 89 109 121 0 -

Tampa 210 117 82 - 0

Example – 1a (answer)

L.A. Detroit Atlanta Houston Tampa Dummy

L.A 0 140 100 90 225 0 5,100

Detroit 145 0 111 110 119 0 6,900

Atlanta 105 115 0 113 78 0 4,000

Houston 89 109 121 0 M 0 4,000

Tampa 210 117 82 M 0 0 4,000

4,000 4,000 4,000 6,400 5,500 100

Example – 1b (answer)

L.A. Detroit Atlanta Houston Tampa Dummy

L.A 0 M 100 90 225 0 5,100

Detroit M 0 111 110 119 0 6,900

Atlanta 105 115 0 113 78 0 4,000

Houston 89 109 121 0 M 0 4,000

Tampa 210 117 82 M 0 0 4,000

4,000 4,000 4,000 6,400 5,500 100

Example – 1c (answer)

L.A. Detroit Atlanta Houston Tampa Dummy

L.A 0 140 100 90 225 0 5,100

Detroit 145 0 111 110 119 0 6,900

Atlanta 105 115 0 113 78 0 4,000

Houston 89 109 121 0 5 0 4,000

Tampa 210 117 82 5 0 0 4,000

4,000 4,000 4,000 6,400 5,500 100

Limitations of Transportation Problem

• One commodity ONLY: any one product supplied and demanded at multiple locations – Merchandise– Electricity, water

• Invalid for multiple commodities: (UNLESS transporting any one of the multiple commodities is completely independent of transporting any other commodity and hence can be treated by itself alone)– Example: transporting product 1 and product 2 from the supply

points to the demand points where the total amount (of the two products) transported on a link is subject to a capacity constraint

– Example: where economy of scale can be achieved by transporting the two products on the same link at a larger total volume and at a lower unit cost of transportation

Limitations of Transportation Problem

– Difficult to generalize the technique to accommodate (these are generic difficulty for “mathematical programming,” including linear and non-linear programming

• Economy of scale the per-unit cost of transportation on a link decreasing with the volume (nonlinear and concave; there is a trick to convert a “non-linear program with a piecewise linear but convex objective function to a linear program; no such tricks exists for a piecewise linear but concave objective function)

• Fixed-cost: transportation usually involves fixed charges. For example, the cost of truck rental (or cost of trucking in general) consists of a fixed charge that is independent of the mileage and a mileage charge that is proportional to the total mileage driven. Such fixed charges render the objective function NON-LINEAR and CONCAVE and make the problem much more difficult to solve

Assignment 1

A company must meet the following demands for a product: January 30 units; February, 30 units; March 20 units. Demand may be backlogged at a cost of Rs. 5 per unit per month. All demand must be met by the end of March. Thus, if one unit of January demand is met during March, a backlogging cost of 5 (2) = 10 is incurred. Monthly production capacity and unit production cost during each month are given in the table. A holding cost of Rs. 20 per unit is assessed on the inventory at the end of each month.

(a)Formulate a balanced transportation problem that could be used to determine, how to minimize the total cost (including backlogging, holding and production costs) of meeting demand.

(b)Use the transportation simplex to determine how to meet each month’s demand.

Question 1…

Month Production capacity

Unit production cost

January 35 Rs. 400

February 30 Rs. 420

March 35 Rs. 410

Assignment - 2

• Identify the limitations of transportation problem as discussed in the class. How can we overcome these limitations. Give one specific example.

(within 350 words)