1. A manufacturer produces two different models: X and Y, of the same products. Model X makes a contribution of Birr 50 per unit and modelY, birr 30 per unit towards total profit. Raw materials R1 and R2 are required for production. At least 18 kg of R1 and 12 of if R2 must be used daily. Also, at most 34 hours of labor are to be utilized. A quantity of 2 kg of R1 is needed for model X and 1 kg of R1 for model Y. For each of X and Y, 1 kg of R2 is required. It takes 3 hours to manufacture model X and 2 hours to manufacture model Y. How many units of each model should be produced to maximize the profit?

Required:

 

1. Formulate a linearprogramming model for this problem
2. Solve this model graphically and interpret the results
3. Analysis the sensitivity of the objective functions and right-hand side values?
4. A manufacturing company produces products 1, 2, and 3. The three products have the following resource requirements and produce the following profit

 

Product	Labour (hr/unit)	Material (IB/unit)	Profit ($/Unit)
1	5	4	3
2	2	6	5
3	4	3	2

 

At present, the firm has a daily labor capacityof 240 available hours and a daily supply of 400 pounds of material. The general linear programming formulation for this problem is as follows:

Max Z = 3X1 + 5X2 + 2X3

s.t. 5X1 + 2X2 + 4X3 ≤ 240 4X1+6X2+3X3 ≤ 400 X1, X2. X3 ≥ 0

Management has developed the following set of goals,arranged in order of their importance to the firm:

1. Because of recent labor relations difficulties, management wants to avoid underutilization of normal production capacity.
2. Management has established a satisfactory profitlevel of $500 per day.

 

1. Overtime is to be minimized as much as possible.
2. Management wants to minimize the purchase of additional materials to avoid handling and storage problems.

Required: formulate a goal programming model to determinethe number of each productto produce to best satisfy the goals and solve using the Lingo.

 

1. ABC a manufacturer of large-size pressure cookers, produces and sells three models of cookers. While market demands pose no restriction, the capacity to produce is currently constrained by the limited suppliesof special gradealuminum limited to 1500kg per week and machine processing time limited to 1200 hours per week. To determine the optimal product mix to maximize weekly contribution, a linear programming model was formulated as below.

 

Maximize  Z=60X1+ 40X2+ 80X3

Subject to

6X1+ 3X2+ 5X3 ≤1500 (Aluminum) 3X1+ 4X2+ 5X3 ≤1200 (Machine Hours) X1, X2, X3 ≥ 0

Using the simplex method,the following tableauwas obtained

 

 

 

Basic V	Cj	60 X1	40 X2	80 X3	0 S1	0 S2	RHS
X1	_	1	-1/3	0	1/3	-1/3	100
X3	_	0	1	1	-1/5	2/5	180
Zi		_	_	_	_	_	_
Cj-Zi		_	_	_	_	_	_

1. Fill in all the numerical valuesin the above table
2. Is the current solutionoptimal? If not, carry out the iterations until an optimalsolution is reached
3. Analyze the sensitivity of the optimalsolution to the following changes
4. Due to a machinebreakdown, the machinehours available get reduced to 1050 hours
5. An additional quantity of 150 kg of aluminumcan be obtained
6. Conduct sensitivity analysis and determinethe sensitivity for the right-hand side values and the coefficients of the objective function value?

 

1. Formulate the dual of the problem and determine the dual solution from the primal optimal tableau

 

1. Two competing firms must simultaneously determine how much of a product to produce. The total profit earned by the two firms is always $1,000. If firm 1's production level is low and firm 2's is also low, then firm 1 earns a profit of $500; if firm 1's level is low and 2's is high, then firm 1's profit is $400. If firm 1's production level is high and so is firm 2's, then firm 1's profit is $600; but if firm 1's level is high while firm 2's level is low, then firm 1's profit is only $300. Find the value and optimal strategies for this constant-sum game.
2. Jhon has left an estate of Birr 200,000to support his three ex-wives. Unfortunately, Jhon's attorney has determined that each ex-wife needs the following amount of money to take care of Jhon's children: wife 1 100,000birr; wife 2 200,000 birr; wife 3 300,000 birr. Jhon's attorney must determine how to divide the money among the three wives. He defines the value of a coalition S of ex-wives to be the maximum amount of money left for the ex- wives in S after all ex-wivesnot in S receive what they need. Using this definition, construct

 

a characteristic function for this problem. Then determine the core and Shapley value for this game.

1. Azaria Healthcare operatesthree hospitals and a numberof clinics in its citywidenetwork. It is planning to open a new wellness center and clinic facility that focuses on geriatric clients in one of four suburbs. The following table shows the weighted criteria for each location:

Score (0 to 100)
Location factors	Weight	Ashcroft	Brainerd	Crabtree	Dowling
Elderly population	0.55	75	80	65	75
Income level	0.15	65	75	90	85
Land availability	0.10	90	70	90	80
Average age	0.10	80	70	80	75
Public transportation	0.05	95	55	75	95
Crime rate	0.05	95	70	85	90

 

Recommend a site for the new Balston Healthcare facility, basedon these weightedlocation factors and scores.

1. Labran Jones has played for the Cleveland professional basketball team for the past eight seasons and has established himselfas one of the top players in the league.He has recently become a free agent, meaninghe can sign a new contract with Cleveland or with any other team in the league. While he has enjoyed playing for Cleveland, which is near his hometown, the team has never seriously contended for a championship, so Labran is strongly considering moving to one of three other teams that he thinks have more championship potential. Other factors he is considering are salary (although Cleveland is offering him more money than the other teams), the possible media attention and endorsements he might receive by playing in another city, and the city itself and the lifestyle where he would be playing. Following are the pairwise comparisons for the four teams for these four criteria and the pairwise comparisons for criteria:

CITY		SLARY (IN$)
CLEVELAND	Cleveland	Miami	New York	Chicago
	1	5	4	3
MIAMI	1/5	1	1/3	½
NEW YORK	1/4	3	1	2
CHICAGO	1/3	2	½	1

 

 

MEDIA EXPOSURE/ENDORSEMENTS
CITY	Cleveland	Miami	New York	Chicago
CLEVELAND	1	1	¼	½
MIAMI	1	1	¼	½
NEW YORK	4	4	1	1
CHICAGO	2	2	½	½
	City/Lifestyle
CITY	Cleveland	Miami	New York	Chicago
CLEVELAND	1	¼	3	2
MIAMI	4	1	5	4
NEW YORK	1/3	1/5	1	½
CHICAGO	½	¼	2	1

 

 

HAMPIONSHIP POTENTIAL
CITY	Cleveland	Miami	New York	Chicago
CLEVELAND	1	1/7	½	¼
MIAMI	7	1	5	3
NEW YORK	2	1/5	1	½
CHICAGO	4	1/3	2	1

 

 

CRITERION
CITY	Cleveland	Miami	New York	Chicago
CLEVELAND	1	¼	1/3	1/5
MIAMI	4	1	½	1/3
NEW YORK	3	2	1	½
CHICAGO	5	3	2	1

Required:

1. Using AHP, determinewhich team Labranshould select to sign a new contractwith.
2. check the consistency of the pairwisecomparison matrices for all four criteria and the criteria.

 

1. TEAM Ltd. Co. wishes to purchasea maximum of 3600 units of two types of products, A & B which are available in the market.Product A occupiesa space of 3 cubicfeet & cost Birr 9 whereas B occupies a space of 1 cubic feet & cost Birr 13 per unit. The budgetary constraints of the company do not allow spending more than Birr 39,000. The total availability of space in the company is 6000 cubic feet. The profit marginof both products A & B is Birr 3 & Birr 4 respectively. Formulate the above problem as a linearprogramming model and solve it by using a graphical method. You are required to ascertain the best possible combination of purchases of A & B so that the total profit is maximized.

Required: Analyse the sensitivity of RHS and objective function value using the graphical approach.

1. Machinco producesfour products, requiring time on two machines and two types (skilled and unskilled) of labor. The amount of machine time and labor (in hours) used by each product and the sales prices are given in the following table. Each month, 700 hours are available on machine1 and 500 hours on machine 2. Each month,Machinco can purchase up to 600 hours of skilledlabor at $8 per hour and up to 650 hours of unskilled labor at $6 perhour. Formulate an LPthat will enableMachinco to maximizeits monthly profit.Solve this LP and use the output to answer the following questions:

PRODUCT	MACHINE	MACHINE	SKILLED	UNSKILLED	SALES (IN$)
1	11	4	8	7	300
2	7	6	5	8	260
3	6	5	4	7	220
4	5	4	6	4	180

1. By how much does the priceof product 3 have to in- creasebefore it becomesoptimal to produce it?
2. If product1 sold for $290, then what would be the new optimalsolution to the problem?
3. What is the most Machinco would be willing to pay for an extra hour of time on each machine?
4. What is the most Machinco would be willing to pay for an extra hour of each type of labor?
5. If up to 700 hours of skilled labor could be purchased each month, then what would

be Machinco's monthly profits?

 

1. Patients suffering from kidney failure can either get a transplant or undergo periodic dialysis. During any one year, 30% undergo cadaveric transplants, and 10% receive living- donor kidneys. In the year following a transplant, 30% ofthose who receive the cadaveric transplants and 15% of living-donor recipients go back to dialysis. Death percentages among the two groups are 20% and 10%, respectively. Of those in the dialysis pool, 10% die, and of those who survive more than one year after a transplant, 5% die and 5% go back to dialysis. Represent the situation as a Markov chain.

Required:

1. For a patient who is currentlyon dialysis, what is the probability of receiving a transplant in two years?
2. For a patientwho is currently a more-than-one-year survivor, what is the probability of surviving four more years?

Answer rating: 100% (QA)

I can assist with these linear programming problems However its best to addres... View the full answer