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QUANTITATIVE METHODS FOR MANAGERS ASSIGNMENT MODEL
Strategic Implications of Short-Term Scheduling • By scheduling effectively, companies use assets more effectively and create greater capacity per dollar invested, which, in turn, lowers cost • This added capacity and related flexibility provides faster delivery and therefore better customer service • Good scheduling is a competitive advantage which contributes to dependable delivery
The Goals of Short-Term scheduling • Minimize completion time • Maximize utilization (make effective use of personnel and equipment) • Minimize customer wait time
Assignment Method • Assigns tasks or jobs to resources - Hungarian Method • Type of linear programming model • Objective • Minimize total cost, time etc. • Constraints • 1 job per resource (e.g., machine) • 1 resource (e.g., machine) per job
Example 1: A department head has four subordinates and four tasks to be performed. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. His estimates of the times each man would take to perform each task are given below in the matrix: How should the tasks be allocated to subordinates so as to minimize the total time?
Assignment Method • Select the smallest element in each row and subtract it from every element of that row • Then select the smallest element in each column and subtract it from every element of that column • Now make the assignment as follows: (a). Starting with row M1 of the reduced matrix given in table 2, examine all rows one by one until a row containing exactly single zero element is found. Here, row M1, M2 and M4 are the rows which have single zero. Make assignment to the respective zero by putting and strike off (X) other zeros in the column in which the assignment is made.
Assignment Method contd. (b). When rows have been examined successively, an identical procedure is applied to each and every column. Starting with column 1, examine all the columns until a column containing exactly one remaining zero is found. Here column 2 contains a single zero and therefore assignment is made to column 2 as shown below:
Since the number of assignments ( ) made are equal to the number of rows/columns, i.e., 4, the optimal solution has been reached. The complete set of assignments of subordinates to perform different tasks along with their respective hours is given below: Subordinates Tasks Hours M1 M2 M3 M4 Total =
Example 2: A car hire company has one car at each of five depots D1, D2, D3, D4 and D5. A customer in each of the five cities A, B, C, D and e requires a car. The distance (in Kms) between the depots and the cities are given in the following matrix: Depot City How should the cars be assigned to the customers so as to minimize the distance travelled?
Applying steps 1 and 2, reduced matrix • Make assignment ( ) in Row A (contain a single zero) here and strike off (X) all other zeros in column D2. Now no other row has a single zero element. Proceed column wise, column D1 has a single zero in row D. Make an assignment and strike off other zeros in row D. Similarly column D3 has a single zero in row B. Make an assignment and strike off other zeros in row B. Proceed in the same way until all zeros in rows/columns are either marked ( ) or strike off (X)
Draw the minimum number of vertical and horizontal straight lines necessary to cover all zeros in the table: (a) Tick row C and E which has no assignments (b) Tick column D2 which has zero in the ticked rows C and E (c) Tick row A which has assignment in ticked column D2 (d) Repeat steps 4(b) and 4(c) until no more rows or columns can be ticked (e) Draw the straight lines through all rows that are not ticked and marked columns
If the number of lines equals either the number of rows or the number of columns, then you can make an optimal assignment • Otherwise: • Subtract the smallest number not covered by a line from every other uncovered number. Add the same number to any number(s) lying at the intersection of any two lines. • Optimal assignments will always be at the zero locations of the table
Variation of the Assignment Problem 1. Maximization Case - objective function: maximize the total pay-off - First convert the maximization problem into a minimization problem. This involves subtracting all the elements from the highest element in the original pay-off matrix - Now use Hungarian Method to solve the problem
Multiple Optimum solution - possible to have two or more ways to strike off all zero elements in the final reduced matrix for a given problem - there will be multiple optimum solutions with the same total pay-off for assignments made. In such cases, management may exercise their judgements or preference and select that set of optimal assignments which is more suited to their requirement.
Example 3: A company has four sales territories and four salesmen available for assignment. The sales territories are not equally rich in their sales potential and the salesmen also differ in their selling ability. The following matrix gives the sales (in thousand rupees) for each salesman to be assigned to each territory. How should the territories be assigned so as to maximize the total sales?
The assignment problem has two alternative optimal solutions: Assignment set 1 Assignment set 2 Salesman Territory Sales (in Rs1000)Salesman Territory Sales A I 42 A I 42 B II 25 B III 20 C III 20 C II 25 D IV 12 D IV 12
Unbalanced Assignment Problem • if pay-off matrix of an assignment problem is not a square matrix (i.e. the no. of rows are not equal to the no. of columns), the assignment problem is called an unbalanced assignment problem • In such cases, dummy rows and/or columns are added in the matrix to make it a square matrix • Then apply the Hungarian Method to this resulting balanced assignment problem • For example, if five workers are to be assigned to six machines, a dummy row is added to transform the assignment problem into 6 x 6 square matrix. • A dummy row means that one machine is idle at a particular time and the elements corresponding to this dummy row are zero in the pay-off matrix.