1. The use of linear programming for the allocation of scarce resources
LEARNING NOTE 11.1
The use of linear programming for the allocation of scarce resources

2. LN11.1 (1a)
Linear programming

3. LN11.1 (1b)
Example (cont.)

4. LN11.1 (2) Materials constraint (8Y + 4Z  3,440 (When Y= 0, Z = 860;
when Z= 0, Y = 430

5. LN11.1 (3) Labour constraint 6Y + 8Z  2,880 (when Z = 0, Y = 480;
when Y = 0, Z = 360)

6. LN11.1 (4) Machine capacity constraint 4Y + 6Z  2,760 (when Z = 0,
Y = 690; when y = 0, Z = 460)

7. LN11.1 (5)
Sales limitation Y  420

8. LN11.1 (6) Optimum solution Feasible production combination =
Area ABCDE

9. LN11.1 (7a)
Optimum solution
1. The optimum output can be determined by solving the simultaneous equations that intersect at point C:
8Y + 4Z = 3 440
6Y + 8Z = 2 880
so that Y = 400 and Z = 60

10. LN11.1 (7b)
Optimum solution
2. The materials and labour constraints are binding and therefore have opportunity costs. The marginal contribution from obtaining one extra unit of materials can be calculated by solving the following equations:
8Y + 4Z = (revised materials constraint)
6Y + 8Z = (unchanged labour constraint)
Y = units, Z = units
Therefore the planned output of Y would be increased by 0.2 units and Z reduced by 0.15 units and contribution will increase by £0.40 (the opportunity cost).

11. LN11.1 (7c)
Optimum solution
3. The marginal contribution from obtaining one extra labour hour can be found in a similar way:
8Y + 4Z = (unchanged materials constraint)
6Y + 8Z = (revised labour constraint)
Y = and Z = 60.2
Marginal contribution = £1.80