LINEAR PROGRAMMING

FORMULATION

Blending Problem

The Kayan Construction Co. is building roads on the side of South Mountain. It is necessary to use explosives to blow up the underground boulders to make the surface level. There are three ingredients (A, B, C) in the explosive used. It is known that at least 10 ounces of the explosive must be used to get results. If more than 20 ounces is used, the explosion will be too damaging. Also, for an explosion, at least 1/4 ounce of ingredient C must be used for every ounce of ingredient A, and at least one ounce of ingredient B must be used for every ounce of ingredient C. The costs of ingredients A, B, and C are $6, $18, and $20 per ounce, respectively.

LP Formulation

Let Xa = oz of A, Xb = oz of B, Xc = oz of C

Min C = $6Xa + $18Xb + $20Xc

Subject to:Xa + Xb + Xc ≥ 10

Xa + Xb + Xc ≤ 20

Xc ≥ 1/4 Xa

   Xb ≥ Xc

Xa, Xb, Xc ≥  0

Product Mix Problem

Lara Valves produces valves. Two alternative production lines are available. The company just received an order for producing 1000 Mark I valves. Line 1 can produce the valves at a rate of 15 minutes for each valve. The production capacity on Line 2 is 5 valves per hour. Line 1 is available, for this order, for not more than 200 hours at a cost of $8 per hour. Line 2 is available, for this order, for not more than 170 hours at a cost of $5 per hour. (This can be formulated in two different ways.)

LP Formulation

METHOD 1

X1 = hrs on line 1, X2 = hrs on line 2

Min Z = $8X1 + $5X2

Subject to:

4X1 + 5X2 = 1000

X1  ≤   200

X2  ≤  170

X1, X2 ≥  0

 

METHOD 2

Y1 = # valves line 1, Y2 = # valves line 2

Min Z = $2Y1 + $1Y2

Subject to:

Y1 + Y2 = 1000

Y1  ≤  800

Y2  ≤  850

Y1, Y2 ≥  0

Budget ProblemVoice Fair Co. advertises its weekly sales in newspapers,

television, and radio. Each dollar spent in advertising in newspapers is estimated to reach an exposure of 12 buying customers; each dollar in TV reaches an exposure of 15 buying customers; and each dollar in radio reaches an exposure of 10 buying customers. The company has an agreement with all three media services according to which it will spend not less than 20% of its total money actually expended in each medium. Further, it is agreed that the combined newspaper and TV budget will not be larger than three times the radio budget. The company has just decided to spend no more than $17,000 on advertising. How much should the company budget for each medium if it is interested in reaching as many buying customers as possible?

LP Formulation

Let X1 = $ newspaper, X2 = $ television, X3 = $ radio

Max Z = 12X1 + 15X2 + 10X3

Subject to:X1 + X2 + X3 ≤ 17,000

X1 ≥ .2(X1 + X2 + X3)

X2 ≥ .2(X1 + X2 + X3)

X3 ≥ .2(X1 + X2 + X3)

X1 + X2  ≤ 3X3

X1, X2, X3 ≥  0

A CARGO PLANE

A cargo plane has three compartments for storing cargo: front, centre and rear. These compartments have the following limits on both weight and space:Compartment Weight capacity (tonnes) Space capacity (cubic metres)

Front 10 6800Centre 16 8700Rear 8 5300

Furthermore, the weight of the cargo in the respective compartments must be the same proportion of that compartment's weight capacity to maintain the balance of the plane.

The following four cargoes are available for shipment on the next flight:Cargo Available Weight (tonnes) Volume (cubic metres/tonne) Profit ($/tonne) C1 18 480 310 C2 15 650 380 C3 23 580 350 C4 12 390 285

Any proportion of these cargoes can be accepted. The objective is to determine how much (if any) of each cargo C1, C2, C3 and C4 should be accepted and how to distribute each among the compartments so that the total profit for the flight is maximised

Variables

We need to decide how much of each of the four cargoes to put in each of the three compartments. Hence let:

xij be the number of tonnes of cargo i (i=1,2,3,4 for C1, C2, C3 and C4 respectively) that is put into compartment j (j=1 for Front, j=2 for Centre and j=3 for Rear) where xij >=0 i=1,2,3,4; j=1,2,3

Objective The objective is to maximise total profit, i.e.

maximize Z= 310[x11+ x12+x13] + 380[x21+ x22+x23] + 350[x31+ x32+x33] + 285[x41+ x42+x43]

Constraints

cannot pack more of each of the four cargoes than we have available

x11 + x12 + x13 <= 18

x21 + x22 + x23 <= 15

x31 + x32 + x33 <= 23

x41 + x42 + x43 <= 12

the weight capacity of each compartment must be respected

x11 + x21 + x31 + x41 <= 10

x12 + x22 + x32 + x42 <= 16

x13 + x23 + x33 + x43 <= 8

the volume (space) capacity of each compartment must be respected

480x11 + 650x21 + 580x31 + 390x41 <= 6800

480x12 + 650x22 + 580x32 + 390x42 <= 8700

480x13 + 650x23 + 580x33 + 390x43 <= 5300

the weight of the cargo in the respective compartments must be the same proportion of that compartment's weight capacity to maintain the balance of the plane

[x11 + x21 + x31 + x41]/10 = [x12 + x22 + x32 + x42]/16 = [x13 + x23 + x33 + x43]/8

Production planning problem

A company manufactures four variants of the same product and in the final part of the manufacturing process there are assembly, polishing and packing operations. For each variant the time required for these operations is shown below (in minutes) as is the profit per unit sold.

Assembly Polish Pack Profit (£)

Variant 1 2 3 2 1.50

2 4 2 3 2.50

3 3 3 2 3.00

4 7 4 5 4.50

Given the current state of the labour force the company estimate that, each year, they have 100000 minutes of assembly time, 50000 minutes of polishing time and 60000 minutes of packing time available. How many of each variant should the company make per year and what is the associated profit?

Suppose now that the company is free to decide how much time to devote to each of the three operations (assembly, polishing and packing) within the total allowable time of 210000 (= 100000 + 50000 + 60000) minutes. How many of each variant should the company make per year and what is the associated profit?

Variables/Objective

Let:

xi be the number of units of variant i (i=1,2,3,4) made per year

Tass be the number of minutes used in assembly per year Tpol be the number of minutes used in polishing per year Tpac be the number of minutes used in packing per year

where xi >= 0 i=1,2,3,4 and Tass, Tpol, Tpac >= 0

Objective

Presumably to maximise profit - hence we have

Maximize Z = 1.5x1 + 2.5x2 + 3.0x3 + 4.5x4

Constraints

(a) operation time definition

Tass = 2x1 + 4x2 + 3x3 + 7x4 (assembly) Tpol = 3x1 + 2x2 + 3x3 + 4x4 (polish) Tpac = 2x1 + 3x2 + 2x3 + 5x4 (pack)

(b) operation time limits

The operation time limits depend upon the situation being considered. In the first situation, where the maximum time that can be spent on each operation is specified, we simply have:

Tass <= 100000 (assembly) Tpol <= 50000 (polish) Tpac <= 60000 (pack)

In the second situation, where the only limitation is on the total time spent on all operations, we simply have:

Tass + Tpol + Tpac <= 210000 (total time)