Advanced: Formulation of an initial tableau and interpretation of a final tableau using the simplex method Hint: Reverse the signs and ignore entries of 0 and 1.

The Kaolene Co. Ltd has six different products all made from fabricated steel. Each product passes through a combination of five production operations: cutting, forming, drilling, welding and coating.

Steel is cut to the length required, formed into the appropriate shapes, drilled if necessary, welded together if the product is made up of more than one part, and then passed through the coating machine. Each operation is separate and independent, except for the cutting and forming operations, when, if needed, forming follows continuously after cutting. Some products do not require every production operation.

The output rates from each production operations. based on a standard measure for each product, are set out 1n the tableau below, along with the total hours of work available for each operation. The contribution per unit of each product ts also given. It is estimated that three of the products have sales ceilings and these are also given at the top of page 722

The production and sales for the year were found using a linear programming algorithm. The final tableau is given below:

Variables X7 to X11 are the slack variables relating to the production constraints, expressed in the order of production. Variables X12 to X14 are the slack variables relating to the sales ceilings of X3, X5 and X6 respectively.
After analysis of the above results, the production manager believes that further mechanical work on the cutting and forming machines costing £200 can improve their hourly output rates as follows:

The optimal solution to the new situation indicates the shadow prices of the cutting, drilling and welding sections to be £59.3, £14.2 and £71.5 per hour respectively.
Requirements:

(a) Explain the meaning of the seven items ringed in the final tableau. (9 marks)

(b) Show the range of values within which the following variables or resources can change without changing the optimal mix indicated in the final tableau (i) c4: contribution of X4 (ii) bs: available coating time. (4 marks)

(c) Formulate the revised linear programming problem taking note of the revised output rates for cutting and forming.
(5 marks)

(d) Determine whether the changes in the cutting and forming rates will increase profitability.

(e) Using the above information discuss the usefulness of linear programming to managers in solving this type of problem.

Transcribed Image Text:

Products Contribution per unit () Output rate per hour: X X X3 Xa X5 X6 5.7 10.1 12.3 9.8 17.2 14.0 Cutting Forming 6550 650 700 450 450 Drilling Welding Coating 500 Maximum sales units (000) 88111 370 450 300 420 520 180 380 200 380 300 380 670 400 720 540 480 600 450 150 20 70 Cutting Forming Drilling Welding Coating Production hours available 12000 16000 4000 4000 16000