Assignment: Global Criterion & Goal Programming Techniques | Môn Multi-Criteria Decision Making - Trường Đại học Quốc tế, Đại học Quốc gia Thành phố Hồ Chí Minh

lOMoAR cPSD| 58504431 Student Name:… Student ID: …
MULTI-CRITERIA DECISION MAKING
METHOD OF GLOBAL CRITERION Question 1: Consider the MODM problem:
Maximize Z1=3x1+2 x2
Maximize Z2=−4 x1+x2
Maximize Z3=x1+5 x2 Subject to
5 x1−2.5 x2≤10
3 x1+5 x2≤16
−4 x1+3.5 x2≤5
2 x1+3 x2≥5
0.5 x1+4 x2≥6 x1≥0, x2≥0
a. Determine the individual optimal solutions
b. Identify the set of efficient solutions, build the payoff matrix
c. Determine the “best” efficient solution by Global Criterion technique with p=1
Question 2: Production Planning Problem
A company produces two products, P1 and P2, each with different resource requirements, demands
and profit contributions. The details are as follows: Resource Requirement Product P1 Product P2 Resource Availability Labor (hours/unit) 5 hrs 2 hrs 240 hours Material (kg/unit) 4 kg 6 kg 400 kg Demand (units) 50 30 — lOMoAR cPSD| 58504431 Student Name:… Student ID: … Profit per unit $3 $5 — Labour cost: $0.2/hour
a. Formulate the problem based on the given data, with two objectives: maximizing profit and minimizing labor cost.
b. Determine the individual optimal solutions. Identify the set of efficient solutions, build the payoff matrix
c. Formulate Global Criterion objective with p=1 and p=2.
d. Determine the “best” efficient solution by Global Criterion technique with p=1. Question 3: Consider the MODM problem:
Maximize Z1=2x1+4 x2
Maximize Z2=5x1−2x2
Minimize Z3=4 x1+3 x2 Subject to
3 x1+x2≤15 x1−x2≤4
2 x1+3 x2≥10 x1≥0, x2≥0
a. Determine the individual optimal solutions
b. Identify the set of efficient solutions, build the payoff matrix
c. Determine the “best” efficient solution by Global Criterion technique with p=1
MULTI-CRITERIA DECISION MAKING COMPROMISE PROGRAMMING Question 1: Given the following problem:
MaxZ1=2 x1+3 x2 lOMoAR cPSD| 58504431 Student Name:… Student ID: …
MaxZ2=3 x1−x2 Subject to
2 x1−x2≥0 3 x1−2x2+1≥0 x1+2x2−13≤0 x1+x2−9≤0
3 x1−x2−15≤0
(a) Solve the problem for goal Z1 (b)
Solve the problem for goal Z2
(c) Use the compromise method to solve two goals at the same time given W1=W2=1∧p=1 Question 2:
Consider the following problem:
Maximize Z1=x1+2 x2
MinimizeZ2=5 x1+x2
Subject to x1−3 x2≤14 3 x1+2 x2≤20
x1+3.5x2≤15
2 x1+5 x2≥10 x1≥0, x2≥0
a. Solve the problem separately for Z1 and Z2
b. Assume that weights of objectives are W1=1,W2=1, formulate the L1 and L∞ using Compromise Programming method
c. Define compromise solution set with L1 Question 3:
Consider the following problem
Maximize Z1=3x1+2 x2
Maximize Z2=4 x1+x2 Subject to
2 x1+x2≤100
x1+x2≤80 lOMoAR cPSD| 58504431 Student Name:… Student ID: …
x1≤40 x1, x2≥0
a. Determine the individual optimal solution using graphical method
b. Identify the set of efficient solutions. Build the payoff matrix
c. Determine the best efficient solution by Compromise Programming technique rectilinear
distance p=1 and weight w1=3 for the first objective function and w2=2 for the second objective function. Question 4:
Consider the following problem:
Maximize Z1=x1+3 x2 Maximize
Z2=2.5x1−x2
Minimize Z3=3 x1+2x2
Subject to x1+x2≤10
−x1+x2≤2
4 x1−x2≤20
2 x1+3 x2≥6 x1, x2≥0
a. Determine the individual optimal solutions.
b. Identifying the set of efficient solutions, build the payoff matrix
c. Determine the “best” efficient solution by Compromise Programming technique with
W1=2,W2=W3=1, and p=1 lOMoAR cPSD| 58504431 Student Name:… Student ID: …
MULTI-CRITERIA DECISION MAKING GOAL PROGRAMMING Question 1
The Bay City Parks and Recreation Department has received a federal grant of $600,000 to expand
its public recreation facilities. City council representatives have demanded four different types of
facilities – gymnasiums, athletic fields, tennis courts, and swimming pools. In fact, the demand by
various communities in the city has been for 7 gyms, 10 athletic fields, 8 tennis courts, and 12
swimming pools. Each facility costs a certain amount, requires a certain number of acres, and is
expected to be used a certain amount, as follows: Facility Cost Required Acres Expected Usage (people/ week) Gymnasium $80,000 4 1,500 Athletic field 24,000 8 3,000 Tennis court 15,000 3 500 Swimming pool 40,000 5 1,000
The Parks and Recreation Department has located 50 acres of land for construction (although more
land could be located, if necessary). The department has established the following goals, listed in order of their priority:
(1) The department wants to spend the total grant because any amount not spent must be returned to the government.
(2) The department wants the facilities to be used by a total of at least 20,000 people each week.
(3) The department wants to avoid having to secure more than the 50 acres of land already located.
(4) The department would like to meet the demands of the city council for new facilities.
However, this goal should be weighted according to the number of people expected to use each facility. Question 2:
The Wearever Carpet Company manufactures two brands of carpet – shag and sculptured – in 100-
yard lots. It requires 8 hours to produce one lot of shag carpet and 6 hours to produce one lot of
sculptured carpet. The company has the following production goals, in prioritized order:
(1) Do not underutilize the production capacity, which is 480 hours. lOMoAR cPSD| 58504431 Student Name:… Student ID: …
(2) Achieve product demand of 40 (100-yard) lots for shag and 50 (100-yard) lots for
sculptured carpet. Meeting demand for shag is more important than meeting demand for
sculpture, by a ratio of 5 to 2.
(3) Limit production overtime to 20 hours.
Formulate a goal programming model to determine the amount of shag and sculptured carpet to
produce to best meet the company’s goal. Question 3:
Computers Unlimited sells microcomputers and distributes them from three warehouses to four universities.
The available supply at the three warehouses, demand at the four universities, and shipping costs
are shown in the following table: Warehouse University Tech A&M State Central Supply Richmond $22 17 30 18 420 Atlanta 15 35 20 25 610 Washington 28 21 16 14 340 Demand 520 250 400 380
Instead of its original objective of cost minimization, Computers Unlimited has indicated the
following goals, arranged in order of their importance:
(1) A&M has been one of its better long-term customers, so Computers Unlimited wants to
meet all of A&M’s demands.
(2) Because of recent problems with the trucking union, it wants to ship at least 80 units from
the Washington warehouse to Central University.
(3) To maintain the best possible relations with all its customers, Computers Unlimited would
like to meet no less than 80% of each customer’s demand.
(4) It would like to keep total transportation costs to no more than 110% of the $22,470 total
cost achieved with the optimal allocation, using the transportation solution method.
(5) Because of dissatisfaction with the trucking firm it uses for the Atlanta-to-State deliveries,
it would like to minimize the number of units shipped over this route. Question 4:
The Oregon Atlantic Company produces two kinds of paper – newsprint and white wrapping paper
(butcher paper). It requires 5 minutes to produce a yard of newsprint and 8 minutes to produce a
yard of wrapping paper. The company has 4,800 minutes of normal production capacity available
each week. The profit of $0.20 for a yard of newsprint and $0.25 for a yard of wrapping paper.
The weekly demand is for 500 yards of newsprint and 400 yards of wrapping paper. lOMoAR cPSD| 58504431 Student Name:… Student ID: …
The company has established the following goals, in order of priority:
(1) Limit overtime to 480 minutes.
(2) Achieve a profit of $300 each week
(3) Fulfill the demand for the products in order of magnitude of their profits
(4) Avoid underutilization of production capacity
Formulate a goal programming model to determine the number of yards of each type of paper. Question 5:
Mac’s Warehouse is a large discount store that operates 7 days per week. The store needs the
following number of full-time employees working each day of the week: Day Number of Employees Sunday 47 Monday 22 Tuesday 28 Wednesday 35 Thursday 34 Friday 43 Saturday 53
Each employee must work 5 consecutive days each week and then have 2 days off. For example,
any employee who works Sunday through Thursday has Friday and Saturday off. The store
currently has a total of 60 employees available to work. Mac’s has developed the following set of
prioritized goals for employee scheduling:
(1) The store would like to avoid hiring any additional employees.
(2) The most important days for the store to be fully staffed are Saturday and Sunday.
(3) The next most important day to be fully staffed is Friday
(4) The store would like to be fully staffed for the remaining 4 days of the week.
Formulate a goal programming model to determine the number of employees who should begin
their 5-day workweek each day of the week to achieve the store’s objectives. Question 6:
A company produces two products P1, P2. The two products have resource requirements as follows: Product P1 Product P2 Resource availability lOMoAR cPSD| 58504431 Student Name:… Student ID: … Labor (hr/unit) 5 2 240 Material (kg/unit) 4 6 400 Profit 3 5
The management board wants to decide the production plan for P1 and P2 and sets up the following prioritized goals:
1. To avoid underutilization of normal production capacity (i.e. no layoff)
2. A profit level of at least 500
3. Overtime is to be minimized as much as possible
4. Management wants to minimize the purchase of additional materials. Question 7:
A small firm produces washers and dryers. Production of either product requires 1 hour of
production time. The plant has a normal production capacity of 40 hours per week. A maximum
of 24 washers and 30 dryers can be stored per week. The profit is $80 for a washer and $40 for a
dryer. The manager has established the following goals:
● P1: Avoid underutilization of normal production capacity
● P2: Produce as many products as possible. However, since the profit of a washer is twice as
much as that for a dryer, the manager has twice as much desire to achieve the production of
washers to achieve the production of dryers.
● P3: Minimize overtime as much as possible.
a. Formulate the goal programming problem
b. The firm adds one new priority: Overtime should not exceed 10 hours per week if possible.
The priority of this new goal places it between the old P1 and P2 levels. Reformulate the goal programming problem.
MULTI-CRITERIA DECISION MAKING DE NOVO PROGRAMMING Question 1: lOMoAR cPSD| 58504431 Student Name:… Student ID: …
Let x1, x2, x3 be the number of item types 1, 2, and 3 respectively. The production plan aims to
maximize 3 goals which are calculated as:
MaxZ1=2 x1+2 x2+3x3
MaxZ2=1.3 x1+3x2+2 x3
MaxZ3=x1+1.5 x2+2 x3
Each item requires 2 resources R1 and R2. It is known that the constraints for resources are given as:
3x1+2x2+2x3≤24 (resource1)
x1+x2+2x3≤25 (resource2)
The prices of resources are given as p=[1;1.2] and the budget B is 60.
a. Solve separately Z1,Z2,Z3
b. Use the De Novo method to solve this problem. Question 2:
Consider the following problems:
Maximize Z1=3x1+2 x2
Maximize Z2=4 x2 Subject to x (1) 1+x2≤15 2 x (2)
1+2 x2≤20 x1+4 x2≤35 (3)
x1, x2≥0
a. Formulate the De Novo Programming model given unit cost of resource (1) is $3, resource
(2) is $2 and resource (3) is $2, budget B = $45.
b. Use De Novo method to solve this problem
c. Given the total maximum budget is $100. Solve the model if we want to get maximum total achievements. Question 3:
Consider the following problem: lOMoAR cPSD| 58504431 Student Name:… Student ID: …
Maxf1=400 x1+300 x2
Max f2=6 x1+8 x2 Subject to 4 x1≤10
2 x1+6 x2≤12
12 x1+4 x2≤30 3 x2≤5.25
4 x1+4 x2≤13 x1, x2≥0
Where p1=30, p2=40, p3=9.5, p4=20,and p5=10are market price ($ per unit) a. Find the ideal solution
b. Find the minimum budget to get the ideal solution
c. If given B=$2,700, find f1, f2
d. Redesign the mode with B=$2,700