STATS ASSIGNMENT

Stats Assignment



Stats Assignment

Question 1

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 6800

Centre 16 8700

Rear 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 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.

Formulate the above problem as a linear program

What assumptions are made in formulating this problem as a linear program?

Briefly describe the advantages of using a software package to solve the above linear program, over a judgemental approach to this problem.

Solution

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

Note here that we are explicitly told we can split the cargoes into any proportions (fractions) that we like.

Constraints

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

x11 + x12 + x13 <= 18x21 + 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

Objective

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

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

The basic assumptions are:

that each cargo can be split into whatever proportions/fractions we desire

that each cargo can be split between two or more compartments if we so desire

that the cargo can be packed into each compartment (for example if the cargo was spherical it would not be possible to pack a compartment ...