Find the optimal assignment to minimize the total processing cost.
Five wagons are available at stations 1, 2, 3, 4, and 5. These are required at 5 stations I, II, III, IV, and V. The mileage between various stations are given in the table below. How should the wagons be transported so as to minimize the mileage covered?
10 | 5 | 9 | 18 | 11 | |
13 | 9 | 6 | 12 | 14 | |
3 | 2 | 4 | 4 | 5 | |
18 | 9 | 12 | 17 | 15 | |
11 | 6 | 14 | 19 | 10 |
Five different machines can do any of the five required jobs, with different profits resulting from each assignment as shown below:
30 | 37 | 40 | 28 | 40 | |
40 | 24 | 27 | 21 | 36 | |
40 | 32 | 33 | 30 | 35 | |
25 | 38 | 40 | 36 | 36 | |
29 | 62 | 41 | 34 | 39 |
Find the optimal assignment schedule.
The assignment problem is said to be balanced if ______.
Choose the correct alternative :
The assignment problem is said to be balanced if it is a ______.
In an assignment problem if number of rows is greater than number of columns then
Fill in the blank :
When an assignment problem has more than one solution, then it is _______ optimal solution.
In an assignment problem, if number of column is greater than number of rows, then a dummy column is added.
State whether the following is True or False :
In assignment problem, each facility is capable of performing each task.
Solve the following problem :
A plant manager has four subordinates, and four tasks to be performed. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. This estimate of the time each man would take to perform each task is given in the effectiveness matrix below.
7 | 25 | 26 | 10 | |
12 | 27 | 3 | 25 | |
37 | 18 | 17 | 14 | |
18 | 25 | 23 | 9 |
How should the tasks be allocated, one to a man, as to minimize the total man hours?
A dairy plant has five milk tankers, I, II, III, IV and V. These milk tankers are to be used on five delivery routes A, B, C, D and E. The distances (in kms) between the dairy plant and the delivery routes are given in the following distance matrix.
150 | 120 | 175 | 180 | 200 | |
125 | 110 | 120 | 150 | 165 | |
130 | 100 | 145 | 160 | 175 | |
40 | 40 | 70 | 70 | 100 | |
45 | 25 | 60 | 70 | 95 |
How should the milk tankers be assigned to the chilling center so as to minimize the distance travelled?
Choose the correct alternative:
The assignment problem is generally defined as a problem of ______
Choose the correct alternative:
Assignment Problem is special case of ______
When an assignment problem has more than one solution, then it is ______
The assignment problem is said to be balanced if ______
If the given matrix is ______ matrix, the assignment problem is called balanced problem
In an assignment problem if number of rows is greater than number of columns, then dummy ______ is added
State whether the following statement is True or False:
The objective of an assignment problem is to assign number of jobs to equal number of persons at maximum cost
In assignment problem, if number of columns is greater than number of rows, then a dummy row is added
State whether the following statement is True or False:
In assignment problem each worker or machine is assigned only one job
Give mathematical form of Assignment problem
Three jobs A, B and C one to be assigned to three machines U, V and W. The processing cost for each job machine combination is shown in the matrix given below. Determine the allocation that minimizes the overall processing cost.
Machine | ||||
U | V | W | ||
Jobs | A | 17 | 25 | 31 |
B | 10 | 25 | 16 | |
C | 12 | 14 | 11 |
(cost is in ₹ per unit)
A computer centre has got three expert programmers. The centre needs three application programmes to be developed. The head of the computer centre, after studying carefully the programmes to be developed, estimates the computer time in minitues required by the experts to the application programme as follows.
Programmers | ||||
P | Q | R | ||
Programmers | 1 | 120 | 100 | 80 |
2 | 80 | 90 | 110 | |
3 | 110 | 140 | 120 |
Assign the programmers to the programme in such a way that the total computer time is least.
Find the optimal solution for the assignment problem with the following cost matrix.
Area | |||||
1 | 2 | 3 | 4 | ||
P | 11 | 17 | 8 | 16 | |
Salesman | Q | 9 | 7 | 12 | 6 |
R | 13 | 16 | 15 | 12 | |
S | 14 | 10 | 12 | 11 |
Number of basic allocation in any row or column in an assignment problem can be
If number of sources is not equal to number of destinations, the assignment problem is called ______
The solution for an assignment problem is optimal if
In an assignment problem involving four workers and three jobs, total number of assignments possible are
A car hire company has one car at each of five depots a, b, c, d and e. A customer in each of the fine towers A, B, C, D and E requires a car. The distance (in miles) between the depots (origins) and the towers(destinations) where the customers are given in the following distance matrix.
a | b | c | d | e | |
A | 160 | 130 | 175 | 190 | 200 |
B | 135 | 120 | 130 | 160 | 175 |
C | 140 | 110 | 155 | 170 | 185 |
D | 50 | 50 | 80 | 80 | 110 |
E | 55 | 35 | 70 | 80 | 105 |
How should the cars be assigned to the customers so as to minimize the distance travelled?
A natural truck-rental service has a surplus of one truck in each of the cities 1, 2, 3, 4, 5 and 6 and a deficit of one truck in each of the cities 7, 8, 9, 10, 11 and 12. The distance(in kilometers) between the cities with a surplus and the cities with a deficit are displayed below:
To | |||||||
7 | 8 | 9 | 10 | 11 | 12 | ||
From | 1 | 31 | 62 | 29 | 42 | 15 | 41 |
2 | 12 | 19 | 39 | 55 | 71 | 40 | |
3 | 17 | 29 | 50 | 41 | 22 | 22 | |
4 | 35 | 40 | 38 | 42 | 27 | 33 | |
5 | 19 | 30 | 29 | 16 | 20 | 33 | |
6 | 72 | 30 | 30 | 50 | 41 | 20 |
How should the truck be dispersed so as to minimize the total distance travelled?
A dairy plant has five milk tankers, I, II, III, IV and V. Three milk tankers are to be used on five delivery routes A, B, C, D and E. The distances (in kms) between the dairy plant and the delivery routes are given in the following distance matrix.
150 | 120 | 175 | 180 | 200 | |
125 | 110 | 120 | 150 | 165 | |
130 | 100 | 145 | 160 | 170 | |
40 | 40 | 70 | 70 | 100 | |
45 | 25 | 60 | 70 | 95 |
A job production unit has four jobs P, Q, R, and S which can be manufactured on each of the four machines I, II, III, and IV. The processing cost of each job for each machine is given in the following table:
| ||||
P | 31 | 25 | 33 | 29 |
Q | 25 | 24 | 23 | 21 |
R | 19 | 21 | 23 | 24 |
S | 38 | 36 | 34 | 40 |
A department store has four workers to pack goods. The times (in minutes) required for each worker to complete the packings per item sold is given below. How should the manager of the store assign the jobs to the workers, so as to minimize the total time of packing?
Books | Toys | Crockery | Cutlery | |
3 | 11 | 10 | 8 | |
13 | 2 | 12 | 12 | |
3 | 4 | 6 | 1 | |
4 | 15 | 4 | 9 |
Five wagons are available at stations 1, 2, 3, 4 and 5. These are required at 5 stations I, II, III, IV and V. The mileage between various stations are given in the table below. How should the wagons be transported so as to minimize the mileage covered?
10 | 5 | 9 | 18 | 11 | |
13 | 9 | 6 | 12 | 14 | |
7 | 2 | 4 | 4 | 5 | |
18 | 9 | 12 | 17 | 15 | |
11 | 6 | 14 | 19 | 10 |
A job production unit has four jobs P, Q, R, S which can be manufactured on each of the four machines I, II, III and IV. The processing cost of each job for each machine is given in the following table :
| ||||
31 | 25 | 33 | 29 | |
25 | 24 | 23 | 21 | |
19 | 21 | 23 | 24 | |
38 | 36 | 34 | 40 |
Complete the following activity to find the optimal assignment to minimize the total processing cost.
Step 1: Subtract the smallest element in each row from every element of it. New assignment matrix is obtained as follows :
| ||||
6 | 0 | 8 | 4 | |
4 | 3 | 2 | 0 | |
0 | 2 | 4 | 5 | |
4 | 2 | 0 | 6 |
Step 2: Subtract the smallest element in each column from every element of it. New assignment matrix is obtained as above, because each column in it contains one zero.
Step 3: Draw minimum number of vertical and horizontal lines to cover all zeros:
Step 4: From step 3, as the minimum number of straight lines required to cover all zeros in the assignment matrix equals the number of rows/columns. Optimal solution has reached.
Examine the rows one by one starting with the first row with exactly one zero is found. Mark the zero by enclosing it in (`square`), indicating assignment of the job. Cross all the zeros in the same column. This step is shown in the following table :
| ||||
6 | 8 | 4 | ||
4 | 3 | 2 | ||
2 | 4 | 5 | ||
4 | 2 | 6 |
Step 5: It is observed that all the zeros are assigned and each row and each column contains exactly one assignment. Hence, the optimal (minimum) assignment schedule is :
P | II | `square` |
Q | `square` | 21 |
R | I | `square` |
S | III | 34 |
Hence, total (minimum) processing cost = 25 + 21 + 19 + 34 = ₹`square`
A plant manager has four subordinates and four tasks to perform. The subordinates differ in efficiency and task differ in their intrinsic difficulty. Estimates of the time subordinate would take to perform tasks are given in the following table:
3 | 11 | 10 | 8 | |
13 | 2 | 12 | 2 | |
3 | 4 | 6 | 1 | |
4 | 15 | 4 | 9 |
Complete the following activity to allocate tasks to subordinates to minimize total time.
Step I: Subtract the smallest element of each row from every element of that row:
0 | 8 | 7 | 5 | |
11 | 0 | 10 | 0 | |
2 | 3 | 5 | 0 | |
0 | 11 | 0 | 5 |
Step II: Since all column minimums are zero, no need to subtract anything from columns.
Step III : Draw the minimum number of lines to cover all zeros.
Since minimum number of lines = order of matrix, optimal solution has been reached
Optimal assignment is A →`square` B →`square`
C →IV D →`square`
Total minimum time = `square` hours.
Table of Contents
This blog will explain how to solve assignment problems using optimization techniques to achieve maximum results. Learn the basics, step-by-step approach, and real-world applications of maximizing assignment problems.
In an assignment problem, the goal is to assign n tasks to n agents in such a way that the overall cost or benefit is minimized or maximized. The maximization problem arises when the objective is to maximize the overall benefit rather than minimize the cost.
The maximization problem can be solved using the Hungarian algorithm, which is a special case of the transportation problem. The algorithm involves converting the assignment problem into a matrix, finding the minimum value in each row, and subtracting it from all the elements in that row. Similarly, we find the minimum value in each column and subtract it from all the elements in that column. This is known as matrix reduction.
Next, we cover all zeros in the matrix using the minimum number of lines. A line covers a row or column that contains a zero. If the minimum number of lines is n, an optimal solution has been found. Otherwise, we continue with the next step.
In step three, we add the minimum uncovered value to each element covered by two lines, and subtract it from each uncovered element. Then, we return to step two and repeat until an optimal solution is found.
The above approach provides a step-by-step process to maximize an assignment problem. Here are the steps in summary:
Maximization in an Assignment Problem has numerous real-world applications. For example, it can be used to optimize employee allocation to projects, to allocate tasks in a manufacturing process, or to assign jobs to machines for maximum productivity. By using optimization techniques to maximize the benefits of an assignment problem, businesses can save time, money, and resources.
This blog has provided an overview of Maximisation in an Assignment Problem, explained how to solve it using the Hungarian algorithm, and discussed real-world applications. By following the step-by-step approach, you can optimize your assignments for maximum benefit.
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1 Operations Research-An Overview
2 Linear Programming: Formulation and Graphical Method
3 Linear Programming-Simplex Method
4 Transportation Problem
5 Assignment Problem
6 Application of Excel Solver to Solve LPP
7 Goal Programming
8 Integer Programming
9 Dynamic Programming
10 Non-Linear Programming
11 Introduction to game theory and its Applications
12 Monte Carlo Simulation
13 Queueing Models
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Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons.
The assignment problem in the general form can be stated as follows:
“Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to assign each facility to one and only one job in such a way that the measure of effectiveness is optimised (Maximised or Minimised).”
Several problems of management has a structure identical with the assignment problem.
Example I A manager has four persons (i.e. facilities) available for four separate jobs (i.e. jobs) and the cost of assigning (i.e. effectiveness) each job to each ...
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The assignment problem is a special type of transportation problem , where the objective is to minimize the cost or time of completing a number of jobs by a number of persons.
In other words, when the problem involves the allocation of n different facilities to n different tasks, it is often termed as an assignment problem.
The model's primary usefulness is for planning. The assignment problem also encompasses an important sub-class of so-called shortest- (or longest-) route models. The assignment model is useful in solving problems such as, assignment of machines to jobs, assignment of salesmen to sales territories, travelling salesman problem, etc.
It may be noted that with n facilities and n jobs, there are n! possible assignments. One way of finding an optimal assignment is to write all the n! possible arrangements, evaluate their total cost, and select the assignment with minimum cost. But, due to heavy computational burden this method is not suitable. This chapter concentrates on an efficient method for solving assignment problems that was developed by a Hungarian mathematician D.Konig.
"A mathematician is a device for turning coffee into theorems." -Paul Erdos
Suppose a company has n persons of different capacities available for performing each different job in the concern, and there are the same number of jobs of different types. One person can be given one and only one job. The objective of this assignment problem is to assign n persons to n jobs, so as to minimize the total assignment cost. The cost matrix for this problem is given below:
The structure of an assignment problem is identical to that of a transportation problem.
To formulate the assignment problem in mathematical programming terms , we define the activity variables as
x = | 1 if job j is performed by worker i | |
0 otherwise |
for i = 1, 2, ..., n and j = 1, 2, ..., n
In the above table, c ij is the cost of performing jth job by ith worker.
The optimization model is
Minimize c 11 x 11 + c 12 x 12 + ------- + c nn x nn
subject to x i1 + x i2 +..........+ x in = 1 i = 1, 2,......., n x 1j + x 2j +..........+ x nj = 1 j = 1, 2,......., n
x ij = 0 or 1
x ij = 0 or 1 for all i and j
An assignment problem can be solved by transportation methods, but due to high degree of degeneracy the usual computational techniques of a transportation problem become very inefficient. Therefore, a special method is available for solving such type of problems in a more efficient way.
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In this paper, we consider a variant of the classic Assignment Problem (AP), called the Balanced Assignment Problem (BAP) [ 2 ]. The BAP seeks to find an assignment solution which has the smallest value of max-min distance : the difference between the maximum assignment cost and the minimum one. However, by minimizing only the max-min distance, the total cost of the BAP solution is neglected and it may lead to an inefficient solution in terms of total cost. Hence, we propose a fair way based on Nash equilibrium [ 1 , 3 , 4 ] to inject the total cost into the objective function of the BAP for finding assignment solutions having a better trade-off between the two objectives: the first aims at minimizing the total cost and the second aims at minimizing the max-min distance. For this purpose, we introduce the concept of Nash Fairness (NF) solutions based on the definition of proportional-fair scheduling adapted in the context of the AP: a transfer of utilities between the total cost and the max-min distance is considered to be fair if the percentage increase in the total cost is smaller than the percentage decrease in the max-min distance and vice versa.
We first show the existence of a NF solution for the AP which is exactly the optimal solution minimizing the product of the total cost and the max-min distance. However, finding such a solution may be difficult as it requires to minimize a concave function. The main result of this paper is to show that finding all NF solutions can be done in polynomial time. For that, we propose a Newton-based iterative algorithm converging to NF solutions in polynomial time. It consists in optimizing a sequence of linear combinations of the two objective based on Weighted Sum Method [ 5 ]. Computational results on various instances of the AP are presented and commented.
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Authors and affiliations.
INP Clermont Auvergne, Univ Clermont Auvergne, Mines Saint-Etienne, CNRS, UMR 6158 LIMOS, 1 Rue de la Chebarde, Aubiere Cedex, France
Minh Hieu Nguyen, Mourad Baiou & Viet Hung Nguyen
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Correspondence to Viet Hung Nguyen .
Editors and affiliations.
ESSEC Business School of Paris, Cergy Pontoise Cedex, France
Ivana Ljubić
IBM TJ Watson Research Center, Yorktown Heights, NY, USA
Francisco Barahona
Georgia Institute of Technology, Atlanta, GA, USA
Santanu S. Dey
Université Paris-Dauphine, Paris, France
A. Ridha Mahjoub
Proposition 1 . There may be more than one NF solution for the AP.
Let us illustrate this by an instance of the AP having the following cost matrix
By verifying all feasible assignment solutions in this instance, we obtain easily three assignment solutions \((1-1, 2-2, 3-3), (1-2, 2-3, 3-1)\) , \((1-3, 2-2, 3-1)\) and \((1-3, 2-1, 3-2)\) corresponding to 4 NF solutions (280, 36), (320, 32), (340, 30) and (364, 28). Note that \(i-j\) where \(1 \le i,j \le 3\) represents the assignment between worker i and job j in the solution of this instance. \(\square \)
We recall below the proofs of some recent results that we have published in [ 10 ]. They are needed to prove the new results presented in this paper.
Theorem 2 [ 10 ] . \((P^{*},Q^{*}) = {{\,\mathrm{arg\,min}\,}}_{(P,Q) \in \mathcal {S}} PQ\) is a NF solution.
Obviously, there always exists a solution \((P^{*},Q^{*}) \in \mathcal {S}\) such that
Now \(\forall (P',Q') \in \mathcal {S}\) we have \(P'Q' \ge P^{*}Q^{*}\) . Then
The first inequality holds by the Cauchy-Schwarz inequality.
Hence, \((P^{*},Q^{*})\) is a NF solution. \(\square \)
Theorem 3 [ 10 ] . \((P^{*},Q^{*}) \in \mathcal {S}\) is a NF solution if and only if \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P(\alpha ^{*})}\) where \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) .
Firstly, let \((P^{*},Q^{*})\) be a NF solution and \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) . We will show that \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P(\alpha ^{*})}\) .
Since \((P^{*},Q^{*})\) is a NF solution, we have
Since \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) , we have \(\alpha ^{*}P^{*}+Q^{*} = 2Q^{*}\) .
Dividing two sides of ( 6 ) by \(P^{*} > 0\) we obtain
So we deduce from ( 7 )
Hence, \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P}(\alpha ^{*})\) .
Now suppose \(\alpha ^{*} = \frac{Q^{*}}{P^{*}}\) and \((P^{*},Q^{*})\) is an optimal solution of \(\mathcal {P}(\alpha ^{*})\) , we show that \((P^{*},Q^{*})\) is a NF solution.
If \((P^{*},Q^{*})\) is not a NF solution, there exists a solution \((P',Q') \in \mathcal {S}\) such that
We have then
which contradicts the optimality of \((P^{*},Q^{*})\) . \(\square \)
Lemma 3 [ 10 ] . Let \(\alpha , \alpha ' \in \mathbb {R}_+\) and \((P_{\alpha }, Q_{\alpha })\) , \((P_{\alpha '}, Q_{\alpha '})\) be the optimal solutions of \(\mathcal {P(\alpha )}\) and \(\mathcal {P(\alpha ')}\) respectively, if \(\alpha \le \alpha '\) then \(P_{\alpha } \ge P_{\alpha '}\) and \(Q_{\alpha } \le Q_{\alpha '}\) .
The optimality of \((P_{\alpha }, Q_{\alpha })\) and \((P_{\alpha '}, Q_{\alpha '})\) gives
By adding both sides of ( 8a ) and ( 8b ), we obtain \((\alpha - \alpha ') (P_{\alpha } - P_{\alpha '}) \le 0\) . Since \(\alpha \le \alpha '\) , it follows that \(P_{\alpha } \ge P_{\alpha '}\) .
On the other hand, inequality ( 8a ) implies \(Q_{\alpha '} - Q_{\alpha } \ge \alpha (P_{\alpha } - P_{\alpha '}) \ge 0\) that leads to \(Q_{\alpha } \le Q_{\alpha '}\) . \(\square \)
Lemma 4 [ 10 ] . During the execution of Procedure Find ( \(\alpha _{0})\) in Algorithm 1 , \(\alpha _{i} \in [0,1], \, \forall i \ge 1\) . Moreover, if \(T_{0} \ge 0\) then the sequence \(\{\alpha _i\}\) is non-increasing and \(T_{i} \ge 0, \, \forall i \ge 0\) . Otherwise, if \(T_{0} \le 0\) then the sequence \(\{\alpha _i\}\) is non-decreasing and \(T_{i} \le 0, \, \forall i \ge 0\) .
Since \(P \ge Q \ge 0, \, \forall (P, Q) \in \mathcal {S}\) , it follows that \(\alpha _{i+1} = \frac{Q_i}{P_i} \in [0,1], \, \forall i \ge 0\) .
We first consider \(T_{0} \ge 0\) . We proof \(\alpha _i \ge \alpha _{i+1}, \, \forall i \ge 0\) by induction on i . For \(i = 0\) , we have \(T_{0} = \alpha _{0} P_{0} - Q_{0} = P_{0}(\alpha _{0}-\alpha _{1}) \ge 0\) , it follows that \(\alpha _{0} \ge \alpha _{1}\) . Suppose that our hypothesis is true until \(i = k \ge 0\) , we will prove that it is also true with \(i = k+1\) .
Indeed, we have
The inductive hypothesis gives \(\alpha _k \ge \alpha _{k+1}\) that implies \(P_{k+1} \ge P_k > 0\) and \(Q_{k} \ge Q_{k+1} \ge 0\) according to Lemma 3 . It leads to \(Q_{k}P_{k+1} - P_{k}Q_{k+1} \ge 0\) and then \(\alpha _{k+1} - \alpha _{k+2} \ge 0\) .
Hence, we have \(\alpha _{i} \ge \alpha _{i+1}, \, \forall i \ge 0\) .
Consequently, \(T_{i} = \alpha _{i}P_{i} - Q_{i} = P_{i}(\alpha _{i}-\alpha _{i+1}) \ge 0, \, \forall i \ge 0\) .
Similarly, if \(T_{0} \le 0\) we obtain that the sequence \(\{\alpha _i\}\) is non-decreasing and \(T_{i} \le 0, \, \forall i \ge 0\) . That concludes the proof. \(\square \)
Lemma 5 [ 10 ] . From each \(\alpha _{0} \in [0,1]\) , Procedure Find \((\alpha _{0})\) converges to a coefficient \(\alpha _{k} \in \mathcal {C}_{0}\) satisfying \(\alpha _{k}\) is the unique element \(\in \mathcal {C}_{0}\) between \(\alpha _{0}\) and \(\alpha _{k}\) .
As a consequence of Lemma 4 , Procedure \(\textit{Find}(\alpha _{0})\) converges to a coefficient \(\alpha _{k} \in [0,1], \forall \alpha _{0} \in [0,1]\) .
By the stopping criteria of Procedure Find \((\alpha _{0})\) , when \(T_{k} = \alpha _{k} P_{k} - Q_{k} = 0\) we obtain \(\alpha _{k} \in C_{0}\) and \((P_{k},Q_{k})\) is a NF solution. (Theorem 3 )
If \(T_{0} = 0\) then obviously \(\alpha _{k} = \alpha _{0}\) . We consider \(T_{0} > 0\) and the sequence \(\{\alpha _i\}\) is now non-negative, non-increasing. We will show that \([\alpha _{k},\alpha _{0}] \cap \mathcal {C}_{0} = \alpha _{k}\) .
Suppose that we have \(\alpha \in (\alpha _{k},\alpha _{0}]\) and \(\alpha \in \mathcal {C}_{0}\) corresponding to a NF solution ( P , Q ). Then there exists \(1 \le i \le k\) such that \(\alpha \in (\alpha _{i}, \alpha _{i-1}]\) . Since \(\alpha \le \alpha _{i-1}\) , \(P \ge P_{i-1}\) and \(Q \le Q_{i-1}\) due to Lemma 3 . Thus, we get
By the definitions of \(\alpha \) and \(\alpha _{i}\) , inequality ( 9 ) is equivalent to \(\alpha \le \alpha _{i}\) which leads to a contradiction.
By repeating the same argument for \(T_{0} < 0\) , we also have a contradiction. \(\square \)
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Cite this paper.
Nguyen, M.H., Baiou, M., Nguyen, V.H. (2022). Nash Balanced Assignment Problem. In: Ljubić, I., Barahona, F., Dey, S.S., Mahjoub, A.R. (eds) Combinatorial Optimization. ISCO 2022. Lecture Notes in Computer Science, vol 13526. Springer, Cham. https://doi.org/10.1007/978-3-031-18530-4_13
DOI : https://doi.org/10.1007/978-3-031-18530-4_13
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What is assignment model.
→ Assignment model is a special application of Linear Programming Problem (LPP) , in which the main objective is to assign the work or task to a group of individuals such that;
i) There is only one assignment.
ii) All the assignments should be done in such a way that the overall cost is minimized (or profit is maximized, incase of maximization).
→ In assignment problem, the cost of performing each task by each individual is known. → It is desired to find out the best assignments, such that overall cost of assigning the work is minimized.
Suppose there are 'n' tasks, which are required to be performed using 'n' resources.
The cost of performing each task by each resource is also known (shown in cells of matrix)
→ Assignment Model is a special application of Linear Programming (LP).
→ The mathematical formulation for Assignment Model is given below:
→ Let, C i j \text {C}_{ij} C ij denotes the cost of resources 'i' to the task 'j' ; such that
→ Now assignment problems are of the Minimization type. So, our objective function is to minimize the overall cost.
→ Subjected to constraint;
(i) For all j t h j^{th} j t h task, only one i t h i^{th} i t h resource is possible:
(ii) For all i t h i^{th} i t h resource, there is only one j t h j^{th} j t h task possible;
(iii) x i j x_{ij} x ij is '0' or '1'.
(i) balanced assignment problem.
(i) integer programming method:.
In assignment problem, either allocation is done to the cell or not.
So this can be formulated using 0 or 1 integer.
While using this method, we will have n x n decision varables, and n+n equalities.
So even for 4 x 4 matrix problem, it will have 16 decision variables and 8 equalities.
So this method becomes very lengthy and difficult to solve.
As assignment problem is a special case of transportation problem, it can also be solved using transportation methods.
In transportation methods ( NWCM , LCM & VAM), the total number of allocations will be (m+n-1) and the solution is known as non-degenerated. (For eg: for 3 x 3 matrix, there will be 3+3-1 = 5 allocations)
But, here in assignment problems, the matrix is a square matrix (m=n).
So total allocations should be (n+n-1), i.e. for 3 x 3 matrix, it should be (3+3-1) = 5
But, we know that in 3 x 3 assignment problem, maximum possible possible assignments are 3 only.
So, if are we will use transportation methods, then the solution will be degenerated as it does not satisfy the condition of (m+n-1) allocations.
So, the method becomes lengthy and time consuming.
It is a simple trail and error type method.
Consider a 3 x 3 assignment problem. Here the assignments are done randomly and the total cost is found out.
For 3 x 3 matrix, the total possible trails are 3! So total 3! = 3 x 2 x 1 = 6 trails are possible.
The assignments which gives minimum cost is selected as optimal solution.
But, such trail and error becomes very difficult and lengthy.
If there are more number of rows and columns, ( For eg: For 6 x 6 matrix, there will be 6! trails. So 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 trails possible) then such methods can't be applied for solving assignments problems.
It was developed by two mathematicians of Hungary. So, it is known as Hungarian Method.
It is also know as Reduced matrix method or Flood's technique.
There are two main conditions for applying Hungarian Method:
(1) Square Matrix (n x n). (2) Problem should be of minimization type.
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The Quadratic Assignment Problem (QAP) is an optimization problem that deals with assigning a set of facilities to a set of locations, considering the pairwise distances and flows between them.
The problem is to find the assignment that minimizes the total cost or distance, taking into account both the distances and the flows.
The distance matrix and flow matrix, as well as restrictions to ensure each facility is assigned to exactly one location and each location is assigned to exactly one facility, can be used to formulate the QAP as a quadratic objective function.
The QAP is a well-known example of an NP-hard problem , which means that for larger cases, computing the best solution might be difficult. As a result, many algorithms and heuristics have been created to quickly identify approximations of answers.
There are various types of algorithms for different problem structures, such as:
Example: Given four facilities (F1, F2, F3, F4) and four locations (L1, L2, L3, L4). We have a cost matrix that represents the pairwise distances or costs between facilities. Additionally, we have a flow matrix that represents the interaction or flow between locations. Find the assignment that minimizes the total cost based on the interactions between facilities and locations. Each facility must be assigned to exactly one location, and each location can only accommodate one facility.
Facilities cost matrix:
L1 | L2 | L3 | L4 | |
---|---|---|---|---|
F1 | 0 | 2 | 3 | 1 |
F2 | 2 | 0 | 1 | 4 |
F3 | 3 | 1 | 0 | 2 |
F4 | 1 | 4 | 2 | 0 |
Flow matrix:
F1 | F2 | F3 | F4 | |
---|---|---|---|---|
L1 | 0 | 1 | 2 | 3 |
L2 | 1 | 0 | 4 | 2 |
L3 | 2 | 4 | 0 | 1 |
L4 | 3 | 2 | 1 | 0 |
To solve the QAP, various optimization techniques can be used, such as mathematical programming, heuristics, or metaheuristics. These techniques aim to explore the search space and find the optimal or near-optimal solution.
The solution to the QAP will provide an assignment of facilities to locations that minimizes the overall cost.
The solution generates all possible permutations of the assignment and calculates the total cost for each assignment. The optimal assignment is the one that results in the minimum total cost.
To calculate the total cost, we look at each pair of facilities in (i, j) and their respective locations (location1, location2). We then multiply the cost of assigning facility1 to facility2 (facilities[facility1][facility2]) with the flow from location1 to location2 (locations[location1][location2]). This process is done for all pairs of facilities in the assignment, and the costs are summed up.
Overall, the output tells us that assigning facilities to locations as F1->L1, F3->L2, F2->L3, and F4->L4 results in the minimum total cost of 44. This means that Facility 1 is assigned to Location 1, Facility 3 is assigned to Location 2, Facility 2 is assigned to Location 3, and Facility 4 is assigned to Location 4, yielding the lowest cost based on the given cost and flow matrices.This example demonstrates the process of finding the optimal assignment by considering the costs and flows associated with each facility and location. The objective is to find the assignment that minimizes the total cost, taking into account the interactions between facilities and locations.
Applications of the QAP include facility location, logistics, scheduling, and network architecture, all of which require effective resource allocation and arrangement.
Similar reads, improve your coding skills with practice.
Bloom’s taxonomy.
Armstrong, P. (2010). Bloom’s Taxonomy. Vanderbilt University Center for Teaching. Retrieved [todaysdate] from https://cft.vanderbilt.edu/guides-sub-pages/blooms-taxonomy/. |
Background Information | The Original Taxonomy | The Revised Taxonomy | Why Use Bloom’s Taxonomy? | Further Information
The above graphic is released under a Creative Commons Attribution license. You’re free to share, reproduce, or otherwise use it, as long as you attribute it to the Vanderbilt University Center for Teaching. For a higher resolution version, visit our Flickr account and look for the “Download this photo” icon.
In 1956, Benjamin Bloom with collaborators Max Englehart, Edward Furst, Walter Hill, and David Krathwohl published a framework for categorizing educational goals: Taxonomy of Educational Objectives . Familiarly known as Bloom’s Taxonomy , this framework has been applied by generations of K-12 teachers and college instructors in their teaching.
The framework elaborated by Bloom and his collaborators consisted of six major categories: Knowledge, Comprehension, Application, Analysis, Synthesis, and Evaluation. The categories after Knowledge were presented as “skills and abilities,” with the understanding that knowledge was the necessary precondition for putting these skills and abilities into practice.
While each category contained subcategories, all lying along a continuum from simple to complex and concrete to abstract, the taxonomy is popularly remembered according to the six main categories.
Here are the authors’ brief explanations of these main categories in from the appendix of Taxonomy of Educational Objectives ( Handbook One , pp. 201-207):
The 1984 edition of Handbook One is available in the CFT Library in Calhoun 116. See its ACORN record for call number and availability.
Barbara Gross Davis, in the “Asking Questions” chapter of Tools for Teaching , also provides examples of questions corresponding to the six categories. This chapter is not available in the online version of the book, but Tools for Teaching is available in the CFT Library. See its ACORN record for call number and availability.
A group of cognitive psychologists, curriculum theorists and instructional researchers, and testing and assessment specialists published in 2001 a revision of Bloom’s Taxonomy with the title A Taxonomy for Teaching, Learning, and Assessment . This title draws attention away from the somewhat static notion of “educational objectives” (in Bloom’s original title) and points to a more dynamic conception of classification.
The authors of the revised taxonomy underscore this dynamism, using verbs and gerunds to label their categories and subcategories (rather than the nouns of the original taxonomy). These “action words” describe the cognitive processes by which thinkers encounter and work with knowledge:
In the revised taxonomy, knowledge is at the basis of these six cognitive processes, but its authors created a separate taxonomy of the types of knowledge used in cognition:
Mary Forehand from the University of Georgia provides a guide to the revised version giving a brief summary of the revised taxonomy and a helpful table of the six cognitive processes and four types of knowledge.
The authors of the revised taxonomy suggest a multi-layered answer to this question, to which the author of this teaching guide has added some clarifying points:
Citations are from A Taxonomy for Learning, Teaching, and Assessing: A Revision of Bloom’s Taxonomy of Educational Objectives .
Section III of A Taxonomy for Learning, Teaching, and Assessing: A Revision of Bloom’s Taxonomy of Educational Objectives , entitled “The Taxonomy in Use,” provides over 150 pages of examples of applications of the taxonomy. Although these examples are from the K-12 setting, they are easily adaptable to the university setting.
Section IV, “The Taxonomy in Perspective,” provides information about 19 alternative frameworks to Bloom’s Taxonomy, and discusses the relationship of these alternative frameworks to the revised Bloom’s Taxonomy.
Resource library.
Problem-based learning (PBL) is a student-centered approach in which students learn about a subject by working in groups to solve an open-ended problem. This problem is what drives the motivation and the learning.
Nilson (2010) lists the following learning outcomes that are associated with PBL. A well-designed PBL project provides students with the opportunity to develop skills related to:
Rather than teaching relevant material and subsequently having students apply the knowledge to solve problems, the problem is presented first. PBL assignments can be short, or they can be more involved and take a whole semester. PBL is often group-oriented, so it is beneficial to set aside classroom time to prepare students to work in groups and to allow them to engage in their PBL project.
Students generally must:
Nilson, L. B. (2010). Teaching at its best: A research-based resource for college instructors (2nd ed.). San Francisco, CA: Jossey-Bass.
The tutorial explains how to add and where to find Solver in different Excel versions, from 2016 to 2003. Step-by-step examples show how to use Excel Solver to find optimal solutions for linear programming and other kinds of problems.
Everyone knows that Microsoft Excel contains a lot of useful functions and powerful tools that can save you hours of calculations. But did you know that it also has a tool that can help you find optimal solutions for decision problems?
In this tutorial, we are going to cover all essential aspects of the Excel Solver add-in and provide a step-by-step guide on how to use it most effectively.
Excel Solver belongs to a special set of commands often referred to as What-if Analysis Tools. It is primarily purposed for simulation and optimization of various business and engineering models.
The Excel Solver add-in is especially useful for solving linear programming problems, aka linear optimization problems, and therefore is sometimes called a linear programming solver . Apart from that, it can handle smooth nonlinear and non-smooth problems. Please see Excel Solver algorithms for more details.
The Solver add-in is included with all versions of Microsoft Excel beginning with 2003, but it is not enabled by default.
To add Solver to your Excel, perform the following steps:
To get Solver on Excel 2003 , go to the Tools menu, and click Add-Ins . In the Add-Ins available list, check the Solver Add-in box, and click OK .
Now that you know where to find Solver in Excel, open a new worksheet and let's get started!
Before running the Excel Solver add-in, formulate the model you want to solve in a worksheet. In this example, let's find a solution for the following simple optimization problem.
Problem . Supposing, you are the owner of a beauty salon and you are planning on providing a new service to your clients. For this, you need to buy a new equipment that costs $40,000, which should be paid by instalments within 12 months.
Goal : Calculate the minimal cost per service that will let you pay for the new equipment within the specified timeframe.
And now, let's see how Excel Solver can find a solution for this problem.
2. define the problem.
The Solver Parameters window will open where you have to set up the 3 primary components:
Constraints.
Exactly what does Excel Solver do with the above parameters? It finds the optimal value (maximum, minimum or specified) for the formula in the Objective cell by changing the values in the Variable cells, and subject to limitations in the Constraints cells.
The Objective cell ( Target cell in earlier Excel versions) is the cell containing a formula that represents the objective, or goal, of the problem. The objective can be to maximize, minimize, or achieve some target value.
Variable cells ( Changing cells or Adjustable cells in earlier versions) are cells that contain variable data that can be changed to achieve the objective. Excel Solver allows specifying up to 200 variable cells.
In this example, we have a couple of cells whose values can be changed:
The Excel Solver Constrains are restrictions or limits of the possible solutions to the problem. To put it differently, constraints are the conditions that must be met.
To add a constraint(s), do the following:
Excel Solver allows specifying the following relationships between the referenced cell and the constraint.
To edit or delete an existing constraint do the following:
In this example, the constraints are:
After you've configured all the parameters, click the Solve button at the bottom of the Solver Parameters window (see the screenshot above) and let the Excel Solver add-in find the optimal solution for your problem.
Depending on the model complexity, computer memory and processor speed, it may take a few seconds, a few minutes, or even a few hours.
The Solver Result window will close and the solution will appear on the worksheet right away.
Below you will find two more examples of using the Excel Solver addin. First, we will find a solution for a well-known puzzle, and then solve a real-life linear programming problem.
I believe everyone is familiar with "magic square" puzzles where you have to put a set of numbers in a square so that all rows, columns and diagonals add up to a certain number.
For instance, do you know a solution for the 3x3 square containing numbers from 1 to 9 where each row, column and diagonal adds up to 15?
It's probably no big deal to solve this puzzle by trial and error, but I bet the Solver will find the solution faster. Our part of the job is to properly define the problem.
With all the formulas in place, run Solver and set up the following parameters:
This is an example of a simple transportation optimization problem with a linear objective. More complex optimization models of this kind are used by many companies to save thousands of dollars each year.
Problem : You want to minimize the cost of shipping goods from 2 different warehouses to 4 different customers. Each warehouse has a limited supply and each customer has a certain demand.
Goal : Minimize the total shipping cost, not exceeding the quantity available at each warehouse, and meeting the demand of each customer.
To define our linear programming problem for the Excel Solver, let's answer the 3 main questions:
To make our transportation optimization model easier to understand, create the following named ranges:
Range name | Cells | Solver parameter |
Products_shipped | B7:E8 | Variable cells |
Available | I7:I8 | Constraint |
Total_shipped | G7:G8 | Constraint |
Ordered | B10:E10 | Constraint |
Total_received | B9:E9 | Constraint |
Shipping_cost | C12 | Objective |
The last thing left for you to do is configure the Excel Solver parameters:
When solving a certain model, you may want to save your Variable cell values as a scenario that you can view or re-use later.
For example, when calculating the minimal service cost in the very first example discussed in this tutorial, you may want to try different numbers of projected clients per month and see how that affects the service cost. At that, you may want to save the most probable scenario you've already calculated and restore it at any moment.
Saving an Excel Solver scenario boils down to selecting a range of cells to save the data in. Loading a Solver model is just a matter of providing Excel with the range of cells where your model is saved. The detailed steps follow below.
To save the Excel Solver scenario, perform the following steps:
When you decide to restore the saved scenario, do the following:
When defining a problem for the Excel Solver, you can choose one of the following methods in the Select a Solving Method dropdown box:
This is how you can use Solver in Excel to find the best solutions for your decision problems. At the end of this post, you can download the sample workbook with all the examples discussed in this tutorial and reverse-engineer them for better understanding. I thank you for reading and hope to see you on our blog next week.
You may also be interested in.
Table of contents
Schedule a meeting in Teams
After you've invited people to your meeting, you can add up to 10 co-organizers to help manage your meeting. Co-organizers are displayed as additional organizers in the meeting participant list and have most of the capabilities of the meeting organizer.
|
|
---|---|
Access and change meeting options | Manage the meeting recording |
Manage breakout rooms | Remove or change the meeting organizer's role |
Bypass the lobby | |
Admit people from the lobby during a meeting | |
Lock the meeting | |
Present content | |
Change another participant’s meeting role | |
Change meeting options during a channel meeting* | |
|
To allow co-organizers to change meeting options in a channel meeting, they must be directly invited in the channel meeting invitation.
External users can't be made co-organizers.
To allow co-organizers to manage breakout rooms, they must be from the same organization as the meeting organizer.
To add co-organizers to a meeting:
Open a meeting in your Teams Calendar.
Make sure the people you want to add as co-organizers have already been added as required attendees.
In Choose co-organizers , select their names from the dropdown menu.
Select Save .
Note: Co-organizers must be in the same organization as the meeting organizer, or be using a guest account in the same org.
Want to learn more? See Overview of meetings in Teams .
Invite people to a meeting in Teams
Want more options.
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IMAGES
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The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment.
After reading this article you will learn about:- 1. Meaning of Assignment Problem 2. Definition of Assignment Problem 3. Mathematical Formulation 4. Hungarian Method 5. Variations. Meaning of Assignment Problem: An assignment problem is a particular case of transportation problem where the objective is to assign a number of resources to an equal number of activities so as to minimise total ...
The assignment problem is said to be unbalance if _____ The assignment problem is said to be balanced if _____. Choose the correct alternative : The assignment problem is said to be balanced if it is a _____. Fill in the blank : When an assignment problem has more than one solution, then it is _____ optimal solution.
ASSIGNMENT PROBLEM Introduction: Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons. The assignment problem in the general form can be stated as follows:
The assignment problem is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics. In an assignment problem , we must find a maximum matching that has the minimum weight in a weighted bipartite graph .
An assignment problem is a particular case of transportation problem in which a number of operations are to be assigned to an equal number of operators, where each operator performs only one operation. The objective is to minimize overall cost or to maximize the overall profit for a given assignment schedule.
Understanding the Assignment Problem. Before we dive into the solution, it is important to understand the problem itself. In the assignment problem, we have a matrix of costs, where each row represents a resource and each column represents a task. The objective is to assign each resource to a task in such a way that the total cost of ...
The Hungarian Method can also solve such assignment problems, as it is easy to obtain an equivalent minimization problem by converting every number in the matrix to an opportunity loss. The conversion is accomplished by subtracting all the elements of the given matrix from the highest element. It turns out that minimizing opportunity loss ...
In an assignment problem, the goal is to assign n tasks to n agents in such a way that the overall cost or benefit is minimized or maximized. The maximization problem arises when the objective is to maximize the overall benefit rather than minimize the cost. Understanding Maximisation in an Assignment Problem. The maximization problem can be ...
Assignment Problem is a special type of linear programming problem where the objective is to minimise the cost or time of completing a number of jobs by a number of persons. The assignment problem in the general form can be stated as follows: "Given n facilities, n jobs and the effectiveness of each facility for each job, the problem is to ...
Assignment Problem: Linear Programming. The assignment problem is a special type of transportation problem, where the objective is to minimize the cost or time of completing a number of jobs by a number of persons. In other words, when the problem involves the allocation of n different facilities to n different tasks, it is often termed as an ...
Session 2.3: Solution of Maximization Assignment Problem OBJECTIVES By the end of this unit you should be able to: 1. Identify and explain an Assignment Problem. 2. Solve Minimization and Maximization Assignment Problems. Note: In order to achieve these objectives, you need to spend a maximum of two (2) hours working through the sessions.
Problem 4. Job shop needs to assign 4 jobs to 4 workers. The cost of performing a job is a function of the skills of the workers. Table summarizes the cost of the assignments. Worker1 cannot do job3, and worker 3 cannot do job 4. Determine the optimal assignment using the Hungarian method. Job.
An assignment problem is a special type of linear programming problem where the objective is to minimize the cost or time of completing a number of jobs by a number of persons. Furthermore, the structure of an assignment problem is identical to that of a transportation problem. Application Areas of Assignment Problem.
An assignment problem is a particular case of transportation problem. The objective is to assign a number of resources to an equal number of activities . So as to minimize total cost or maximize total pro t of allocation. The problem of assignment arises because available resources such as
The Assignment Problem (AP) is a fundamental combinatorial optimization problem. It can be formally defined as follows. Given a set n workers, a set of n jobs and a \(n \times n\) cost matrix whose elements are positive representing the assignment of any worker to any job, the AP aims at finding an one-to-one worker-job assignment (i.e., a bipartite perfect matching) that minimizes certain ...
1) Costs appear in the objective function only. 2) All decision variable values are either 0 or 1. 3) All constraint left-hand-side coefficient values are 1. False. In an assignment problem, one agent can be assigned to several tasks. False. A dummy origin in a transportation problem is used when supply exceeds demand. Nodes.
The balanced assignment problem, described in Martello et al. [47], attempts to recognize both objectives by minimizing the difference between the maximum and minimum assignment values. One example given is an American travel agency planning a program of trips to Europe with all the travelers, each of whom will take one of the trips, going and ...
There are two main conditions for applying Hungarian Method: (1) Square Matrix (n x n). (2) Problem should be of minimization type. Assignment model is a special application of Linear Programming Problem (LPP), in which the main objective is to assign the work or task to a group of individuals such that; i) There is only one assignment.
The Quadratic Assignment Problem (QAP) is an optimization problem that deals with assigning a set of facilities to a set of locations, ... can be used to formulate the QAP as a quadratic objective function. The QAP is a well-known example of an NP-hard problem, which means that for larger cases, computing the best solution might be difficult ...
d. All decision variable values are either 0 or 1. b. All constraints are of the ≥ form. The assignment problem constraint x<sub>31 + x<sub>32 + x<sub>33 + x<sub>34 ≤ 2 means. a. agent 3 can be assigned to 2 tasks. b. agent 2 can be assigned to 3 tasks. c. a mixture of agents 1, 2, 3, and 4 will be assigned to tasks.
Study with Quizlet and memorize flashcards containing terms like If a basic transportation has four origins and five destinations, the LP formulation of the problem will have _____ regular constraints., The assignment problem constraint X21 + X22 + X23 + X24 + X25 <= 2 means, Arcs in a transshipment problem and more.
Objectives (learning goals) are important to establish in a pedagogical interchange so that teachers and students alike understand the purpose of that interchange. Organizing objectives helps to clarify objectives for themselves and for students. Having an organized set of objectives helps teachers to: "plan and deliver appropriate ...
The objective of this assignment is to apply Fishbone Diagram (Cause-and-Effect Diagram) in identifying and analyzing potential causes of a specific quality problem at the Toronto Transit Commission (TTC). Background The Fishbone Diagram, also known as the Ishikawa Diagram or Cause-and-Effect Diagram, is a visual tool used to systematically ...
Problem-based learning (PBL) is a student-centered approach in which students learn about a subject by working in groups to solve an open-ended problem. This problem is what drives the motivation and the learning. ... Consider making the self and peer assessments a part of the assignment grade. References. Nilson, L. B. (2010).
Constraints. The Excel Solver Constrains are restrictions or limits of the possible solutions to the problem. To put it differently, constraints are the conditions that must be met. To add a constraint(s), do the following: Click the Add button right to the "Subject to the Constraints" box.; In the Constraint window, enter a constraint.; Click the Add button to add the constraint to the list.
c) Assignment Title: The objective of this assignment is to train and directly assist farmers and host staffs on integrated pest management (IPM). d) Dates of Assignment: 20/8/2016-11/9/2016 inclusive of travel to and from the United States. e) Number of days worked: 23 days 1.2.1 Objective 1 Train and Assist Farmers on integrated pest management.
The objective of the transportation problem is to a. identify one origin that can satisfy total demand at the destinations and at the same time minimize total ... origins must equal the number of destinations in the transportation problem c. each supply and demand value is 1 in the assignment problem d. there are many differences between the ...
Note: This material is based upon work supported by funding under an award with the U.S. Department of Housing and Urban Development. The substance and findings of the work are dedicated to the public. Neither the United States Government, nor any of its employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or ...
To add co-organizers to a meeting: Open a meeting in your Teams Calendar. Make sure the people you want to add as co-organizers have already been added as required attendees. Select Options > More options. Select Roles .