A1_10BM60005

IT Lab Assignment – 10BM60005

IT Lab Assignment [LP models and Formulations]

1.

WYNDOR GLASS CO. produces high quality glass products, including windows and glass doors. It has three plants. Aluminum frames and hardware are made in Plant 1, wood frames are made in Plan Plantt 2 and and Plan Plantt 3 prod produc uces es the the glas glass s and and asse assemb mble les s the the products. Becau Because se decl declin inin ing g earn earnin ings gs,, top top mana manage geme ment nt has has deci decide ded d to revamp revamp the company company’s ’s produc productt line. line. Unprof Unprofita itable ble produc products ts are being being dis discon contin tinued ued,, releasi releasing ng produc productio tion n capaci capacity ty to lunch lunch two new products having large sales potential : Product 1: An 8-foot glass door with aluminum framing Product 2: A 4 × 6 foot double-hung wood framed window. Product 1 requires some of the production capacity in Plants 1 and 3, but none in plant 2. Product 2 needs only plant 2 and 3. The marketing division has concluded that the company could sell as much of either product as could be produced by these plants. However, because both the products would be competing for the same production capacity in Plant 3, it is not clear which mix of the two products would be most profitable. Find out. Data for Wyndor Glass Co problem Plant Production time per batch, Hours Product 1 2 1 1 0 2 0 2 3 3 2 Profit per batch $3,000 $ 5,000

Prod Produc ucti tion on time time availab available le per week, week, Hours 4 12 18

Ans. Let x1 be the number of windows and x2 be the number of doors Objective Function: Maximize Z = 3000*x1+5000*x2 3000*x1+5000*x2


Subject to constraints: 1*x1+0*x2<=4 0*x1+2*x2<=12 3*x1+2*x2<=18 2.

ICICI prudential life insurance co. Ltd is introducing two product lines: special risk insurance and mortgages. The expected profit is Rs 50/- per unit on special risk insurance and Rs 20/- per unit on mortgages. Management wishes to establish sales quotas for the new product lines to maximize total expected profit. The work requirements are as follows: Department Work Z- Hours per unit Work – Hours Available Sp. Risk Mortgages Underwriting 3 2 2400 Administration

0

1

800

Claims

2

0

1200

(a) Formulate LPP model for this problem and what is the maximum profit. (b) Use graphical method to solve this. Ans. Let x2 be the no of mortgage and x1 be the number of special risk Objective Function: Maximize Z = 50*x1+20*x2

Subject to constraints: 3*x1+2*x2<=2400 0*x1+1*x2<=800 2*x1+0*x2<=1200 3.

Wild West produces two types of cowboy hats. Type 1 hat requires twice as much labor time as type 2. If all the available labor time is dedicated to type 2 alone, the company can produce a total of 400 type 2 hats a day. The respective market limits for the two types are 150 and 200 per day. The profit is $ 8 per type 1 hat and $ 5 per type 2 hat. Determine the number of hats of each type to be produced to maximize the profit.

Ans. Let x1 be the type 1 hat and x2 be the type 2 hat Objective function: Maximize Z= 8*x1+5*x2


Subject to Constraints: 2*x1-x2=0 x1<=150 x2<=200 4. Two products require three sequential processes. The time available for each procedure is 10 hour a day. The following table summarizes the data of the table: Minutes per unit Product Process 1 1 10 2 5

Process 2 6 20

Process 3 8 10

Unit Profit $2 $3

Determine the optimal mix of the two products.

Ans. Let x1 be the no of units of product 1 and x2 be the number of units of product2 Objective Function: Maximize Z = 2x1*+3*x2 Subject to constraints: 10*x1+5*x2<=600 6*x1+20*x2<=600 8*x1+10*x3<=600 5. Berger Paints produces both interior and exterior paints from three raw materials, M1, M2, M3. The following table provides the basic data of the problem: Tons of Raw material per ton of Exterior paint Interior paint Max. daily availability Raw 6 4 24 material,M1 1 2 6 Raw material 3 2 3 5 4 M2 Raw material M3 Profit per ton($ 1000) The market survey indicates that the daily demand for interior paint cannot exceed that of exterior paint by more than 1 ton. Also the maximum daily demand of interior paint is 2 ton. Determine the


optimum product mix of interior and exterior paints that maximizes the total daily profit.

Ans. Let x2 be the no of tons of interior paint and x1 be the number of tons of exterior paints Objective function: Maximize Z = 5*x1+4*x2 Subject to constraints: 6*x1+4*x2<=24 1*x1+2*x2<=6 3*x1+2*x2<=3 x2-x1<=1 x2<=2 6.

A gambler plays a game that requires dividing bet money among four choices. The game has three outcomes. The following table gives the corresponding gain or loss per dollar for different options of the game.

Return per dollar deposited in given choice Out come 1 2 3 1 -3 4 -7 2 5 -3 9 3 3 -9 10

4 15 4 -8

The gambler has total of $500, which may be played only once. The exact outcome of the game is not known. Because of this uncertainty, the gambler’s strategy is to maximize the minimum return produced by the three outcomes. How should the gambler allocate the $500 among the four choices? (Hint: his net return may be positive, negative or zero)

Ans.

Out come

Return per dollar deposited in given choice 1 2 3 4

1

-3

4

-7

15

2

5

-3

9

4

3 Amount invested

3 $0.00

-9 $0.0

10 $288.0

-8 $212.0

Return $1,164. 00 $3,440. 00 $1,184. 00

IT Lab Assignment – 10BM60005 0

0

0

Total Amount $500.0 Available 0 Total Amount $500.0 Invested 0

7. Solve the LPP Max Z = 2 x1 subject to

3 x1

x1 x1

+ 5 x 2 + 7 x3 + 2 x 2 + 4 x 3 ≤ 100

+ 4 x 2 + 2 x3 ≤ 100 +

x2

+ 3 x3 ≤ 100

x1 , x 2 , x3 ≥ 0

Ans.

Objective function: Maximize Z = 2*x1+5*x2+7*x3 Subject to constraints: 3x1+2x2+4x3<=100 1x1+4x2+2x3<=100 1x1+1x2+3x3<=100 x1+x2+x3>=0

8.

Web Mercantile sells many household products through an online catalogue. The company needs substantial warehouse space for storing its goods. Plans now are being made for leasing warehouse storage space over the next 5 months. However, since these space requirements are quite different, it may be most economical to lease only the amount needed each month on a month-bymonth basis. On the other hand, the additional cost for leasing space for additional months is much less than for the first month, so it may be less expensive to lease the maximum amount needed for the entire months. Another option is the intermediate approach for changing of changing the total amount of space leased at least once but not every month. The space requirement and the leasing costs for the various leasing periods are as follows: Month

1 2 3 4 5

Require d space (sq. ft) 30,000 20,000 40,000 10,000 50,000

Leasing period(month s) 1 2 3 4 5

Cost per sq. ft. leased $ $ $ $ $

65 100 135 160 190


Solve the model to minimize the total leasing cost for meeting the space requirement. Ans. Let x11,x12,x13,x14,x21,x23,x24,x31.x32.x33,x41,x42, ,x51,be the variables

Optimal Solution Minimize Z = 65*(x11+x21+x31+x41+x51)+100*(x12+x22+x32+x42)+135*(x 13+x23+x33)+ 160*(x14+x24)+190*x15 Subject to Constraints x11+x12+x13+x14+x15>=30000 x21+x22+x23+x24+x12+x13+x14+x15>=20000 x31+x32+x33+x22+x23+x24+x13+x14+x15>=40000 x41+x42+x32+x33+x23+x24+x14+x15>=10000 x51+x42+x33+x24+x15>=50000 9.

The Northern Airplane Company builds commercial airplanes for various airplane companies around the world. The last stage in the production process is to produce the jet engines and then to install them in the completed airplane frame. The company has been working under some contacts to deliver a considerable number of airplanes in the near future, and the production of the jet engines for these planes must now be scheduled for next four months. To met the contracted dates for delivery, company must supply engines for installation in the quantities indicated in the table (2 nd column). Thus the cumulative number engines produced by the end of the months 1,2,3,4 must be at least 10,25,50,70. The facilities that will be available for producing the engines vary according to other production, maintenance, the renovation work scheduled during this period. The resulting monthly differences in the maximum number that can be produced and the cost (in Million dollars) of producing each one are given in the 3rd column of table. Because of variations in production costs, it may well be worthwhile to produce some of the engines a month or more before they are scheduled for installation. The drawback is that the engines must be stored until the scheduled installation at storage cost of $15000 pm for each engine shown in the rightmost column of the table. Develop a schedule for the number of engines to be produced in each of the four months so that the total of the production and storage cost will be minimized.

Production scheduling data for Northern Airplane company Month Scheduled Maximum Unit cost of Unit cost of Installation production production the storage


1 2 3 4

10 15 25 20

25 35 30 10

1.08 1.11 1.10 1.13

0.015 0.015 0.015

Ans. Let x11,x12,x13,x14,,x22,x23,x24,x33,x34, x44 be the number of machines produced in each period.. Objective Function: Minimize Z =1.08*x11+1.095x12+1.11*x13+1.25*x14+1.11*x21+1.125*x22+1.1 4*x23+1.1*x31+ 1.15*x32+1.13*x41 Subject to constraints: x11+x12+x13+x14<=25 x24+x22+x23<=35 x33+x32<=30 x44<=10 x11=10 x12+x22=15 x13+x23+x33=25 x14+x24+x34+x44=10

10.

The Versatech Corporation has decided to produce three new products. Five branch plants now have excess product capacity. The unit manufacturing cost of the first product would be $31,$29,$32,$28, and $29 in plants 1,2,3,4 and 5 respectively. The unit manufacturing of the second product would be $45, $41, $46, $ 42, and $43 in plants 1,2,3,4 and 5 respectively. The unit manufacturing of the third product would be $38, $35, $40, in plants 1, 2, 3 respectively, whereas Plants 4 and 5 don’t have capability for producing this product. Sales forecast indicates that 600, 1,000, 800 units of products 1, 2 and 3 respectively should be produced per day. Plants 1,2,3, respectively should be produced per day. Plants 1,2,3,4,and 5 have the capacity to produce 400,600, 400, 600, and 1,000 units daily., respectively, regardless of the product or combination of products involved. Assume that any plants having the capability and capacity to produce them can produce any combination of the products in any quantity. How to allocate the new products to the plants to minimize total manufacturing cost? (a)Formulate the transportation problem by constructing the appropriate parameter table. (b)Obtain the optimal solution.


Ans. Let x11, x12,x13,x14,x15 be the number of units produced of product 1 in plants1,2,3,4 and 5 respectively. Similarly x21,x22,x23,x24,x25 be the number of units produced of product 2 in plants1,2,3,4 and 5 respectively. and x31,x32,x33,x34,x35 be the number of units produced of product 2 in plants1,2,3,4 and 5 respectively. Objective Function: Minimize Z = 31*x11+29*x12+32*x13+28*x14+29*x15+45*x21+41*x22+46*x23+ 42*x24+43*x25+38*x31+35*x32+40*x33 Subject to Constraints: x 11+x12+x13+x14+x15=600 x 21+x22+x23+x24+x25=1000 x 31+x32+x33=800 x 11+x21+x31<=400 x 12+x22+x32<=600 x 13+x23+x33<=400 x 14+x24<=600 x15 +x25<=1000

Transportation Problems 11. The shipping cost ($) per truck load of a company is given below. Formulate a transportation problem by constructing appropriate parameters and obtain the optimal solution using Excel Solver shipping cost($) Ware house(destination) 1 2 4 Sources 1 2 3

Allocation

464 867 352 791 995 685 80 85

3

513

654

416

690

682

388

65

70

Output(total shipped)

75 125 100


Ans. Let x11,x12,x13,x14 be the transportation cost from source 1 to destinations 1,2,3 and 4 respectively. Similarly x21,x22,x23,x24 be the transportation cost from source 2 to destinations 1,2,3 and 4 respectively and x31,x32,x33,x34 be the transportation cost from source 3 to destinations 1,2,3 and 4 respectively. Objective Function: Minimize Z = (464*x11+513*x12+654*x13+867*x14+352*x21+416*x22+690*x23 +791*x24+995*x31+682*x32+388*x33+685*x34)

Subject to constraints x11+x12+x13+x14=75 x21+x22+x23+x24=125 x31+x32+x33+x34=100 x11+x21+x31=80 x12+x22+x32=65 x13+x23+x33=70

12. The Versatech Corporation has decided to produce three new products. Five branch plants now have excess product capacity. The unit manufacturing cost of the first product would be $31,$29,$32,$28, and $29 in plants 1,2,3,4 and 5 respectively. The unit manufacturing of the second product would be $45, $41, $46, $ 42, and $43 in plants 1,2,3,4 and 5 respectively. The unit manufacturing of the third product would be $38, $35, $40, in plants 1, 2, 3 respectively, whereas Plants 4 and 5 don’t have capability for producing this product. Sales forecast indicates that 600, 1,000, 800 units of products 1, 2 and 3 respectively should be produced per day. Plants 1,2,3, respectively should be produced per day. Plants 1,2,3,4,and 5 have the capacity to produce 400,600, 400, 600, and 1,000 units daily., respectively, regardless of the product or combination of products involved. Assume that any plants having the capability and capacity to produce them can produce any combination of the products in any quantity. How to allocate the new products to the plants to minimize total manufacturing cost? (a) Formulate the transportation problem by constructing the appropriate parameter table. (b) Obtain the optimal solution using Excel Solver.

Ans. Let x11, x12,x13,x14,x15 be the number of units produced of product 1 in plants1,2,3,4 and 5 respectively.


Similarly x21,x22,x23,x24,x25 be the number of units produced of product 2 in plants1,2,3,4 and 5 respectively. and x31,x32,x33,x34,x35 be the number of units produced of product 2 in plants1,2,3,4 and 5 respectively.

Objective Function: Minimize Z = 31*x11+29*x12+32*x13+28*x14+29*x15+45*x21+41*x22+46*x23+ 42*x24+43*x25+38*x31+35*x32+40*x33 Subject to Constraints: x 11+x12+x13+x14+x15=600 x 21+x22+x23+x24+x25=1000 x 31+x32+x33=800 x 11+x21+x31<=400 x 12+x22+x32<=600 x 13+x23+x33<=400 x 14+x24<=600 x15 +x25<=1000 13. The move-it company has two plants producing fork lift trucks that then shipped to three distribution centers. The production costs are same at two plants, and the cost of shipping of each truck is shown for each combination of plants and distribution centre.

Plant A

Distribution Centre 1 2 $800 $400

3 $700

B $600 $500

$800

A total of 60 forklifts are produced and shipped per week. Each plant can produce and ship any amount up to a maximum of 50 trucks per weeks, so there is no considerable flexibility on how to divide the total production between the two plants so as to reduce shipping costs. However each distribution centre must receive exactly 20 trucks per week. Managers objective is to determine how many forklift trucks should be produced at each plant, and then what is the over all shipping pattern should be to minimize total shipping costs. Formulate the transportation problem and solve by excel solver to find optimal solution

Ans. Let x11,x12 and x13 be the transportation cost from plant A to destinations 1,2 and 3 respectively.


Similarly x21,x22 and x23be the transportation cost from plant Bto destinations 1,2 and 3 respectively

Objective Function Minimize Z = 11*800+x12*700+x13*400+x21*600+x22*800+x23*500

x

Subject to Constraints x 11+x12+x13<=50 x 21+x22+x23<=50 x 11+x21=20 x 12+x22=20 x 13+x32=20

14. The Cost-Less Corp. supplies its four retail outlets from its four plants. The shipping cost per shipment from each plant to each retail outlet is given below.

Plant 2 3 4

1

Unit shipping cost retail Outlet($) 1 2 3 4 500 600 400 200 200 900 100 300 300 400 200 100 200 100 300 200

Plants 1,2 3, and 4 make 10,20,20, and 10 shipments per month, respectively. Retail outlets 1,2, 3 and 4 need to receive 20,10,,10 and 20 shipments per month, respectively. The objective is to minimize the total shipping cost. Formulate the transportation problem and find the optimal solution.

Ans. Let x11,x12,x13,x14 be the shipping cost from plant 1 to retail outlet 1,2,3 and 4 respectively. Similarly x21,x22,x23,x24 be the shipping cost from plant 2 to retail outlet 1,2, 3 and 4 respectively and x31,x32,x33,x34 be the shipping cost from plant 3 to retail outlet 1,2 3 and 4 respectively. Objective Function:


Minimize Z = x11*500+x12*600+x13*400+x14*200+x21*200+x22*900+x23*100+ x24*300+ x31*300+x32*400+x33*200+x34*100+x41*200+x42*100+x43*300+ x44*200

Subject to Constraints: x x x x x x x x

11+x12+x31+x41=10 21+x22+x32+x42=20 31+x32+x33+x43=20 41+x42+x43+x44=10 11+x21+x31+x41=20 21+x22+x23+x24=10 31+x32+x33+x43=10 41+x42+x43+x44=20

15. Consider the transportation parameters and solve it for optimality.

2 3

Destination 1 2 3 7 2 4 4 3

3 6 3 8

4 4 2 5

Demand

3

2

2

Source

1

3

problem

having

following

Supply 4 3 2

Ans. Let x11, x12, x13, x14 be the transportation cost from source 1 to destinations 1,2,3 and 4 respectively. Similarly x21,x22,x23,x24 be the transportation cost from source 2 to destinations 1,2,3 and 4 respectively and x31,x32,x33,x34 be the transportation cost from source 3 to destinations 1,2,3 and 4 respectively. Objective Function: Minimize Z = x11*3+x12*7+x13*6+x14*4+x21*2+x22*4+x23*3+x24*2+x31*4+x3 2*3+x33*8+x34

Subject to Constraints


x 11+x12+x13+x14=4 x 21+x22+x23+x24=3 x 31+x32+x33+x34=2 x 11+x21+x31=3 x 12+x22+x32=3 x 13+x23+x33=2 x 14+x24+x34=2

A1_10BM60005

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