Tutorial, linear programming

Tutorial (6)

Linear Programming

Question (1) VAC LTD

Vac Ltd proposes to choose as its production plan for the coming year the one which will yield the largest possible contribution (contribution being defined as selling price less the costs of direct labour and raw materials).

Details of the available products are:

Direct labour is paid at the rate of £0.50 per hour.

Material X costs £0.20 per Kg.   Material Y costs £0.70 per Kg.

The cost of machine time is considered to be negligible.

Question A

        Vac Ltd has available during the year 40,000 direct labour hours; all other inputs are available in sufficient quantities for the firm to be able to acquire enough to meet the maximum demand for all products.

Required:

        Calculations showing what Vac Ltd. should plan to produce. Add a short note explaining why your product selection criterion should lead to the best possible result in terms of the given objective.

        Do you need LP to answer this question?

Question B

        Suppose that available direct labour hours for the year remain 40,000 but that in addition, available machine hours for the year are limited to 10,000.

Required:

        Set out the revised problem in terms of a mathematical model (objective function and constraints). Augmentation of the equations is not required.

        Calculate the optimal production plan for the year, using LINDO

Question C

        In addition to products A and B, products C and D become available

        Details of these are:

        The supplier of material X states that he cannot supply VAC Ltd with more than 50,000 Kg during the next year. VAC Ltd has no stock of this material.

        (Direct labour and machine time are constrained as before)

Required:

        Set out the mathematical model to be used in determining the optimal production plan for next year, and use LINDO to produce the solution.

        Answer the following questions relating to the interpretation of these results ( very short answers)

        How many units of products A through D should the firm produce?

        What is the total contribution from this plan?

        How many Kg of material X will be unused?

        What is the dual price (shadow price) of

                Material X

                Direct labour time

        If the selling price of A should have been printed as £14.00 rather than £14.60, would this cause you to question your optimal solution?    Why?

Question (2): XYZ

XYZ Ltd. is a company whose objective is to maximise profits. It manufactures two speciality chemical powders, gamma and delta, using three processes: heating, refining and blending. The powders can be produced and sold in infinitely divisible quantities.

The following are the estimated production hours for each process, per kilo of output for each of the two chemical powders during the period 1 June to 31 August.

During the same period, revenues and costs per kilo of output are budgeted as

It is anticipated that the company will be able to sell all it can produce at the above prices, and that at any level of output fixed costs for the three month period will total £36000.

The company’s management accountant is under the impression that there will be only one scarce factor during the budget period, namely blending hours, which cannot exceed a total of 1050 hours during the period from 1 June to 31 August. He therefore correctly draws up an optimum production plan on this basis.

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