Joycarpap Ltd manufactures and sells three models of affordable toys: Car, Joy and Pap. Each model requires specific amount of fabrication, material worth and assembly hours, as shown in the table below:

There are 210 fabrication hours available and 170 hours of assembly. Materials in stock are worth GH¢200. Market research conducted by the company revealed that demand for the toys was such that in whatever combination of the three models produced, all of the output can be sold within a week.
Each Car contributes GH¢15 to profit, each Joy contributes GH¢20 to profit and each Pap contributes GH¢14 to profit. Using, as slack variables and P as total profit,
Required:
i) Formulate a linear programming problem.                                                                                                                      (4 marks)

ii) Set up the initial Simplex Tableau.                                                                                                                                 (4 marks)

iii) Determine the total profit in the first iteration.                                                                                                              (5 marks)

An advertising agency wishes to reach two types of audiences: Customers with annual income greater than GH¢ 15,000 (target audience A) and customers with annual income less than GH¢15,000 ( target audience B). The total advertising budget is GH¢ 200,000. One programme on TV advertising costs GH¢50,000; one programme on radio advertising costs GH¢20,000. For contract reasons, at least three programmes ought to be on TV, and the number of radio programmes must be limited to five. Surveys indicate that a single TV programme reaches 450,000 customers in target audience A and 50,000 in target audience B. One radio programme reaches 20,000 in target audience A and 80,000 in target audience B.
Required:
i) Formulate the linear programming problem.                                                                                                                  (4 marks)

ii) Construct the initial simplest tableau.                                                                                                                            (4 marks)

iii) Perform the first iteration.                                                                                                                                             (4 marks)

iv) Determine the media mix to maximize the total reach.                                                                                                 (4 marks)

v) Determine the shadow prices of the binding constraint.                                                                                                 (4 marks)

a) A particular linear programming problem is formulated as follows:
Minimize z = 2500x + 3500y
Subject to constraint:

Required:
i) Draw these constraints on the same graph paper.                                                                                                           (6 marks)

ii) Determine the optimum solution.                                                                                                                                  (6 marks)

b) Managers within a subsidiary company in a conglomerate want to know how to maximize the profit from two types of products X and Y. Each product X requires one hour of labour and six litres of molding material, whereas each product Y requires two hours of labour and five litres of molding material. The total labour hours available for each week is 40 and the total amount of molding material each week is 150. The profit contribution from product X is GH¢20 and from product Y is GH¢ 30.
Required:
Formulate, the linear programming model for the problem.                                                                                              (8 marks)

JinJin Company Limited makes two types of leather belts Type Superior and Type Standard. Type Superior is a highly quality and Type Standard is of lower quality. The respective profits are GHp 40 and GHp 30 per belt. The production of each of Type Superior requires twice as much time as a belt of Type Standard, and if all belts were of Type Standard, the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts per day (both Type Superior and Type Standard combined). Belt Type Superior requires a fancy buckle and only 400 of these are available per day. There are only 700 buckles a day available for belt Type Standard. If the aim is to determine how many units of the two types of belts the company should manufacture in order to have the maximum overall profit:

Required:
a) Formulate this problem as a Linear Programming Model.

b) Set up the initial Simplex Tableau.                                                                                                                                 (4 marks)

c) Solve your Tableau in (b) above.                                                                                                                                    (8 marks)

d) Interpret your final Simplex Tableau.                                                                                                                            (4 marks)