WOLLO UNIVERSITY KOMBOLCHA INSTITUTE OF TECHNOLOGY Production Engineering and Management – M.Sc Program
Operations Research – Assignment 1. A manufacturing company is engaged in producing three types of products: A, B, and C. The production department produces, each day, components sufficient to make 50 units of A, 25 units of B, and 30 units of C. The management is confronted with the problem of optimizing the daily production of products in assembly department where only 100manhours are available daily to assemble the products. The following additional information is available.
Type of Product A
Profit contribution per unit of products (Birr) 12
Assembly time time per product 0.8
B
20
1.7
C
45
2.5
The company has a daily order commitment for 20 units of product A and a total of 15 units of products B and C. formulate this problem as an LP model so as to maximize the total profit.
2. A company has two grades of inspectors 1 and 2, who are to be assigned for a quality control inspection. It is required that at least 2000 pieces be inspected per 8 hour day. Grade 1 inspector cab check pieces at the rate of 40 per hour, with an accuracy of 97 percent. Grade 2 inspector checks at the t he rate of 30 pieces per hour with an accuracy of 95 percent. The wage rate of a grade 1 inspector is birr 5 per hour while that of a grade 2 inspector is birr 4 per hour. An error made by an inspector costs c osts birr 3 to the company. c ompany. There are only 1
nine grade 1 inspectors and eleven grade 2 inspectors available in the company. The company wishes to assign work to the available inspectors so as to minimize the total cost of the inspection. Formulate this problem as an LP models so as to minimize daily inspection cost. 3. An electronic company is engaged in the production of two components C1 and C2 used in radio sets. Each unit of C1 costs the company Birr 5 in wages and Birr 5 in material, while each of C2 costs the company Birr 25 in wages and Birr 15 in material. The company sells both products on one-period credit terms, but the company's labour and material expenses must be paid in cash. The selling price of C1 is Birr 30 per unit and of C2 it is Birr 70 per unit. Because of the strong monopoly of the company for these components, it is assumed that the company can sell at the prevailing prices as many units as it produces. The company's production capacity is, however, limited by two considerations. First, at the beginning of period 1, the company has an initial balance of Birr 4,000 (cash plus bank credit plus collections from past credit sales). Second, the company has available in each period 2,000 hours of machine time and 1,400 hours of assembly time. The production of each C1 requires 3 hours of machine time and 2 hours of assembly time, whereas the production of each C2 requires 2 hours of machine time and 3 hours of assembly time. Formulate this problem as an LP model so as to maximize the total profit to the company. 4. A firm makes two products X and Y, and has a total production capacity of 9 tones per day, X and Y requiring the same production capacity. The f irm has a permanent contract to supply at least 2 tones of X and at least 3 tones of Y per day to another company. Each tone of X requires 20 machine hours of production time and each tone of Y requires 50 machine hours of production time. The daily maximum possible number of machine hours is 360. All the firm's output can be sold, and the profit made is Birr 80 per tone of X and Birr 120 per tone of Y. It is required to determine the production schedule for maximum profit and to calculate this profit. (Hint: Use graphical method) 5. An electronic company produces-three types of parts for automatic washing machines. It purchases casting of the parts from a local foundry and then finishes the part on drilling, shaping and polishing machines. The selling prices of parts A, B and C, are Birr 8, Birr 10 and Birr 14 respectively. All parts made can be sold. Castings for parts A, B and C, cost Birr 5, Birr 6 and Birr 10 respectively . The shop possesses only one of each type of machine. The costs per hour to run each of the three machines are Birr 20 for drilling, Birr
2
30 for shaping and Birr 30 for polishing . The capacities (parts per hour) for each part on each machine are shown in the following table: Machine
Capacity per hour Part A
Part B
Part C
Drilling
25
40
25
Shaping
25
20
20
Polishing
40
30
40
The management of the shop wants to know how many parts of each type it should produce per hour in order to maximize profit for an hour's run. Formulate this problem as an LP model so as to maximize total profit to the company.
6. A phar maceutical company produces two phar maceutical products: A and B. Pr oduction of both products r equires the same process, I and II. The production of B results also in a by product C at no extr a cost. The product A can be sold at a profit of Birr 3 per u nit and B at a pr ofit of Birr 8 per unit. Some of this by-product can be sold at a unit profit of Birr 2; the r emainder has to be destroyed and the destr uction cost is Birr 1 per unit. Forecasts show that only up to 5 units of C can be sold . The company gets 3 units of C for each unit of B produced. The manuf acturing times are 3 hour s per unit for A on process I and II, r espectively, and 4 hours and 5 hours per unit for B on process I and II, res pectively. Because the product C r esults f ro m producing B, no time is used in producing C. The available times are 18 and 21 hours of process I and II, respectively. For mulate this pro blem as an LP model to determine the quantity of A and B which should be produced, keeping C in mind, to make the highest total profit to the com pany.
7. An investor has three investment o p portunities available at the beginning of each for the next 5 years, and also has a total of Birr 500,000 available for investment at the beginning of the first year . A summary of the financial characteristics of the three investment alter natives is presented in the following table.
3
Investment Alternative
Allowable size of initial investment
Return (%)
Timing of Return
Immediate Reinvestment possible?
1
100,000
13
1 Year later
Yes
2
unlimited
20
2 Year later
Yes
3
50,000
25
3 Year later
Yes
The investor wishes to determine the investment plan that will maximize the amount of th
money can be accumulated by the beginning of the 6 year in the future. Formulate this problem as an LP model to maximize total return.
8. Use the graphical method to solve the following LP pr oblem. Maximize Z = 15x1 + l0x2 Subject to the constr aints: 4x1 + 6x2 ≤ 360 3x1 + 0x2 ≤ 180 0x1 + 5x2 ≤ 200 x1, x2 ≥ 0. 9. Use the graphical method to solve the following LP pr oblem. Maximize Z = 2x1 + x2 Subject to the constr aints: x1 + 2x2 ≤ 10 x1 + x2 ≤ 6 x1 - x2 ≤ 2 x1 - 2x2 ≤ 1 x1, x2 ≥ O. 10. Use the graphical method to solve the following LP pr oblem. Maximize Z = -x1 + 2x2 Subject to the constr aints: -x1 + 3x2 ≤ 10 x1 + x2 ≤ 6 x1 - x2 ≤ 2 x1, x2 ≥ O.
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11. A manufacturer produces two different models: X and Y, of the same product. Model X makes a contribution of Birr 50 per unit and model Y, Birr 30 per unit towards total profit. Raw materials R 1 and R 2 are required for production. At least 18 kg of R 1 and 12 kg of R 2 must be used daily. Also at most 34 hours of labor are to be utilized. A quantity of 2 kg of R 1 is needed for model X and 1 Kg of R 1 for model Y. For each of X and Y, 1 kg of R 2 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? (Hint: use graphical method). 12. An advertising agency wishes to reach two types of audiences: customers with annual income greater than one 1000 Birr (target audience A) and customers with annual income of less than 1000 Birr (target audience B). The total advertising budget is Birr 200,000. One program of TV advertising costs Birr 50,000; one program of radio advertising costs Birr 20,000. For contract reasons, at least three programs ought to be on TV and the number of radio programs must be limited to 5. Surveys indicate that a single TV program reaches 450,000 prospective customers in target audience A and 50,000 in target audience B. One radio program reaches 20,000 prospective customers in target audience A and 80,000 in target audience B. Determine the media mix to maximize the total reach. (Hint: use Graphical method). 13. A company makes two kinds of leather belts. Belt A is a high quality belt and belt B is of lower quality. The respective profits are 4 USD and 3 USD per belt. The production of each of type A requires twice as much time as a belt of type B, and if all belts were of type B, the company could make 1000 per day. The supply of leather is sufficient for only 800 belts per day (both A and B combined). Belt A requires a fancy buckle and only 400 per day are available. There are only 700 buckles a day available for belt B. What should be the daily production of each type of belt? Formulate this problem as an LP model and solve it by simplex method. 14. A pharmaceutical company has 100 kg of A, 180 kg of B and 120 kg of C ingredients available per month. Company can use these materials to make three basic pharmaceutical products namely 5-10-5, 5-5-10 and 20-5-10, where the numbers in each case represent the percentage of weight of A, B and C, respectively in each of the products. The cost of these raw materials is as follows.
5
Ingredient
Cost per kg (Birr)
A
80
B
20
C
50
Inert Ingredients
20
The selling prices of these products are Birr 40.5, Birr 43 and 45 per kg, respectively. There is a capacity restriction of the company for product 5-10-5, so that company cannot produce more than 30 kg per month. Determine how much of each of the products company should produce in order to maximize its monthly profit.
15. Write the dual to the following LP pr oblems a)
Maximize Z = x1 - x2 + 3x3 Subject to the constr aints: x1 + 2x2 + x3 ≤ 10 2x1 -0x2 - x3 ≤ 2 2x1 -2 x2 -3 x3 ≤ 6 x1, x2, x3 ≥ O.
b) Maximize Z = 3x 1 - 2x2 + 4x3 Subject to the constr aints: 3x1 + 5x2 + 4x3 ≥7 6x1 + x2 +3x3 ≥ 4 7x1 -2x2 - x3 ≤ 10 x1 -2x2 +5x3 ≥ 3 4x1 + 7x2 - 2x3 ≥2 x1, x2, x3 ≥ O.
c)
Maximize Z = x1 +2 x2 Subject to the constr aints: 2x1 +4x2 ≤ 160 x1 -x2 = 30 x1 ≥10 x1, x2 ≥ O. 6
16. Solve the f ollowing all-integer programming pro blem using the br anch and bound method. Minimize Z = 3x 1 + 2.5x 2 Subject to the constr aints: x1 +2x2 ≥ 20 3 x1 +2x2 ≥ 50 x1, x2 ≥ 0 and integers.
17. A department has five employees with five jobs to be performed. The time (in hours) each men will take to perform each job is given in the ef fectiveness matrix. How should the jobs be allocated, one per employee, so as to minimize the total man-hours? Employees
Jobs
I
II
III
IV
V
A
10
5
13
15
16
B
3
9
18
13
6
C
10
7
2
2
2
D
7
11
9
7
12
E
7
9
10
4
12
18. A corporation is considering four possible investment opportunities. The following table presents information about the investment (in Birr thousand) profits : Project
Present value of Expected return
Capital Required Year-wise by Projects Year 1
Year 2
Year 3
1
650
700
550
400
2
700
850
550
350
3
225
300
150
100
4
250
350
200
-
1,200
700
400
Capital available for investment
In addition, pr o jects 1 and 2 are mutually exclusive and project 4 is contingent on the prior acceptance of project 3. Formulate an integer programming model to determine which 7
projects should be accepted and which should be rejected to maximize the present value from accepted pr ojects. (Hint: use zero-one integer programming approach).
19. A dairy firm has three plants located in a state. The daily milk production at each plant is as follows: Plant 1: 6 million liters, Plant 2: 1 million liters, and Plant 3: 10 million liters Each day, the firm must fulfill the needs of its four distribution centers. Minimum requirement at each centre is as follows: Distribution centre 1: 7 million liters, Distribution centre 2: 5 million liters, Distribution centre 3: 3 million liters, and Distribution centre 4: 2 million liters Cost in hundreds of rupees of shipping one million liter from each plant to each distribution centre is given in the following table:
D1
Distribution Center D2 D3
D4
t P1 n a l P P2
2
3
11
7
1
0
6
1
P3
5
8
15
9
Find initial basic feasible solution for given problem by using: a) North-west comer method b) Least cost method if the object is to minimize the total tra nsportation cost.
20. A computer centre has thr ee expert progr ammers. The centre wants three application programs to be developed. The head of the computer centre, after studying carefu lly the programs to be developed, estimates the computer time in minutes required by the experts for the application programs as follows.
8
Programmers s 1 m a r 2 o r P
3
A
B
C
120
100
80
80
90
110
110
140
120
Assign the programmers to the programs in such a way that the total computer time is minimum. (Hint: Use Hungarian Method). 21. A Shoe manufacturing firm employs typists on hourly piece-rate basis for their daily work. There are five typists and their charges and speed are different . According to an earlier understanding, only one job is given to one typist and the typist is paid for a full hour even if she works for a fraction of an hour. Find the least cost allocation based on the following data. Typist
Rate per hour
Number of pages typed/hour
Jobs
Number of pages
A
5
12
P
199
B
6
14
Q
175
C
3
8
R
145
D
4
10
S
298
E
4
11
T
178
22. A firm manufactures three types of products. The fixed and variable costs are given below: Fixed Cost (Birr)
Variable Cost per Unit (Birr)
Product A:
25,000
12
Product B:
35,000
9
Product C:
53,000
7
The likely demand (units) of the products is given below: Poor demand:
3,000
Moderate demand:
7,000
High demand:
11,000
If the sale price of each type of product is Birr 25, then, prepare the payoff matrix. 9
23. A food products company is contemplating the introduction of a revol utionary new product with new packaging or replace the existing product at much higher price (S1) or a moderate change in the composition of the existing product with a new packaging at a small incr ease in price (S2) or a small change in the composition of the existi ng pr oduct exce pt the wor d 'New' with a negligi ble increase in price (S3). The three possi ble states of nature or events are: (i) high increase in sales ( N1), (ii) no change in sales (N2) and (iii) decrease. in sales (N3). The marketing department of the company worked out the payoffs in terms of year ly net pr ofits for each of the str ategies of three events (expected sales). This is re presented in the following ta ble: Strategies
States of nature N1
N2
N3
S1
700,000
300,000
150,000
S2
500,000
450,000
0
S3
300,000
300,000
300,000
Which strategy should the concerned executive choose on the basis of: i) Maximin criterion
ii) Maximax criterion
iii) Minimax regret criterion
iv) Laplace criterion?
24. A com pany manuf actur es goods for a market in which the technology of the product is changing ra pidly. The resear ch and development de partment has produced a new product which appears to have potential for commercial ex ploitation. A further Birr 60,000 is required for development testing. The company has 100 customers and each customer might purchase at the most one unit of the product. Market resear ch suggests that a selling price of Birr 6000 for each unit with total variable costs of manufactur ing and selling estimate ar e Birr 2,000 for each unit. From previous exper ience, it has been possible to derive a probability distribution relating to the pr oportion of customer s who will buy the product as follows: Proportion of customers: Probability:
0.04 0.10
0.08 0.10
0.12 0.20
0.16 0.40
0.20 0.20
10
Determine the ex pected opportunity losses, given no other information than that stated above, and state whether or not the company should d evelop the product. 25. A company is considering the introduction of a new product to its existing product range . It has defined two levels, of sales as 'high' and ' low' on which to base its decision and has estimated the changes that each market level will occur, together with their costs and consequent profits or losses. The information is summarized below. State of nature
Probability
High sales
0.3
Low sales
0.7
Course of Action Market the product (Birr ‘000) 150
Do not market the product (Birr ‘000) 0
-40
0
The company's marketing manager suggests that a market resea .rch survey may be undertaken to provide further information on which to base the decision. On past experience with a certain market research organization , the marketing manager assesses its ability to give good information in the light of subsequent actual sales achievements as follows:
Market research (survey customer) ‘High’ sales forecast
Actual Sales Market ‘high’ 0.5
Market ‘low’ 0.1
Indecisive survey report
0.3
0.4
‘Low’ sales forecast
0.2
0.5
The market research survey will cost Birr 20,000, state whether or not there is a case for employing the market research organizat ion.
26. A glass factory specializing in crystal is developing a substantial backlog and the firm's management is considering three courses of action: Arrange for sub-contracting (S 1), begin overtime production (S2), and construct new facilities (S3). The correct choice depends largely upon future demand which may be low, medium, or high. By consensus, management ranks the respective probabilities as 0.10, 0.50 and 0.40. A cost analysis reveals effect upon the profits that is shown in the table below :
11
Demand
Probability
Course of Action S1
S2
(subcontracting)
S3
(Begin Overtime)
(Construct Facilities)
Low (L)
0.10
10
-20
-150
Medium (M)
0.50
50
60
20
High (H)
0.40
50
100
200
Show this decision situation in the form of a decision tree and indicate the most preferred decision and corresponding expected value.
27. A business man has two independent investment portfolios A and B available to him, but he lacks the capital to undertake both of them simultaneously. He can choose A first and then stop, or if A is not successful, then take B or vice-versa. The probability of success of A is 0.6, while for B it is 0.4 . Both investment schemes require an initial capital outlay of Birr 10,000 and both return nothing if the venture is unsuccessful. Successful completion of A will return Birr 20,000 (over cost) and successful completion of B will return Birr 24,000 (over cost). Draw decision tree and determine the best strategy.
28. [CA, May 1988; Delhi Univ., MBA, 1997]For the game with payoff matrix: Player A
Player B B1
B2
B3
A1
-1
2
-2
A2
6
4
-6
determine the best strategies for players A and B. Also determine the value of game. Is this game:
i) Fair? ii) Strictly determinable?
29. A company management and the labor union are negotiating a new three year settlement. Each of these has 4 strategies:
I - Hard and aggressive bargaining
II - Reasoning and logical approach
III - Legalistic strategy
IV - Conciliatory approach
The costs to the company are given for every pair of strategy choice. 12
Union Strategies
I
II
III
IV
I II
20 25
15 14
12 8
35 10
III
40
2
10
5
IV
-5
4
11
0
Company Strategies
What strategy will the two sides adopt? Also determine the value of the game. 30. A television repairman finds that the time spent on his jobs has an exponential distribution with a mean of 30 minutes. If he repairs sets in the order in which they came in, and if the arrival of sets f ollows a Poisson distribution approximately with an average rate of 10 per 8-hour day, what is the r epair man's expected idle time each day? How many jobs are a head of the average set just brought in? 31. In a railway marshalling yard, goods trains arrive at a rate of 30 trains per day. A ssuming that the inter-ar ri val time follows an exponential distribution and the service time (the time taken to hump a train) distribution is also exponential with an average of 36 minutes . Calculate a) Expected queue size (line length) b) Probability that the queue size exceeds 10. If the input of trains increases to an average of 33 per da y, what will be the change in ‘a’ and ‘b’? 32. A warehouse has only one loading dock manned by a three person crew. Trucks arrive at the loading dock at an average rate of 4 trucks per hour and the arrival rate is Poisson distributed. The loading of a truck takes 10 minutes on an average and can be assumed to be exponentially distributed. The operating cost of a truck is Birr 20 per hour and the members of the loading crew are paid at Birr 6 each per hour. Would you advise the truck owner to add another crew of three persons? 33. Two manufacturers A and B are competing with each other in a restricted market. Over the year, A's customers have exhibited a high degree of loyalty as measured by the fact that customers are using A's product 80 per cent of the time. Also former customers purchasing the product from B have switched back to A's product 60 per cent of the time. a) Construct and interpret the state transition matrix in terms of: i) Retention and loss, and ii) Retention and gain. 13
b) Calculate the probability of a customer purchasing A's product at the end of the second period. 34. A manufacturing company has a certain piece of equipment that is inspected at the end of each day and classified as just overhauled , good, fair or inoperative. If the item is inoperative it is overhauled, a procedure that takes one day. Let us denote the four classifications as states 1, 2, 3, and 4, respectively. Assume that the working condition of the equipment follows a Markov chain with the following transition matrix: Tomorrow
1 P = Today
1
2
3
4
0
¾
¼
0
2
0
½
½
0
3
0
0
½
½
4
1
0
0
0
If it costs Birr 125 to overhaul a machine (including lost time) on the average and Birr 75 as production lost if a machine is found inoperative. Using the steady-state probabilities, compute the expected cost of maintenance per day.
35. A bakery keeps stock of a popular brand of cake. Previous experience shows the daily demand pattern for the item with associated probabilities, as given below: Daily demand (number) Probability
0 0.01
10 020
20 0.15
30
40
50
0.50
0.12
0.02
Use the following sequence of random numbers to simulate the demand for next 10 days. Random numbers:
25,
39,
65,
76,
12,
O5,
73 ,
89,
19,
49.
Also estimate the daily average demand for the cakes on the basis of simulated data.
36. Salesman located in a city A decided to travel to city B. He knew the distances of alternative routes from city A to city B. He then drew a highway network map as shown in the figure below. The city of origin, A is city 1. The destination city B, is city 10. Other cities through which the salesman will have to pass through are numbered 2 to 9. The arrow representing routes between cities and distances in kilometers are indicated on each 14
route. The salesman’s problem is to find the shortest route that covers all the selected cities from A to B.
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