Linear programming

​ ​ ​​Jojox Steel Limited manufactures black and white sheets of steel. A black sheet requires 12 minutes for moulding, 24 minutes for galvanizing, 18 minutes for rolling and 27 minutes for packaging. A white sheet requires 8 minutes for moulding, 6 minutes for galvanizing, 12 minutes for rolling and 60 minutes for packaging. The firm’s management monitors the production process times as it has a maximum of 126 hours for moulding, 120 hours for galvanizing, 142 hours for rolling and 540 hours for packaging. The selling price per sheet is Sh.4,000 for the black sheet and Sh.3,300 for the white sheet. 

​ ​​A company owns two flour mills; A and B which have different production capabilities for high, medium and low grade flour.  The company has entered into a contract to supply flour to a firm every week with 120, 80 and 240 kilograms of high, medium and low grades flour respectively.  It costs the company Sh.10,000 and Sh.8,000 per day to run Mill A and Mill B respectively.  On a daily basis, Mill A produces 60, 20 and 40 kilograms of high, medium and low grade flour respectively while Mill B produces 20, 20 and 120 kilograms of high, medium and low grade flour respectively.  

​ ​ ​​A factory produces two products; product A and product B. The company wants to determine how many units of each product to produce in order to maximise profit. The production of each product is subject to two resource constraints; machine hours and labour hours. 

​​Enumerate FOUR assumptions of linear programming model.

​ ​ ​​Bobino Electricals Limited manufactures electric kettle and electric cookers. 

​ ​ ​​World K Tours has Sh.6 million that may be used to purchase new rental minibuses for hire during the coming holidays. The minibuses may be purchased from two different manufacturers. 

​ ​ ​ ​ ​ ​ ​​A manufacturer of dresses makes two types of dresses; Standard and Executive. Each Standard dress requires 10 labour hours from the cutting department and 30 labour hours from the sewing department. Each Executive dress requires 20 labour hours from the cutting department and 40 labour hours from the sewing department. The maximum labour hours available in the cutting department and the sewing department are 320 and 540 respectively. The company makes a profit of Sh.500 on each Standard dress and Sh.800 on each Executive dress. 

​ ​ ​​A firm manufactures two models of bicycles; mountain bike and BMX. The firm earns profit of Sh.5,000 and Sh.6,000 on mountain bikes and BMX respectively. Both models are produced in three departments; assembly, fitting and painting. The time required per model produced and the time available per week (in hours) are given in the table below:

​ ​ ​ ​ ​​A linear programming problem has been formulated as below:

​ ​​Majux Limited manufactures two types of fruit juices; yellow juice and red juice. 1 packet of yellow juice requires 3 minutes for cutting of fruits, 6 minutes for blending, 7 minutes for cooling and 2 minutes for packaging. 1 packet of red juice requires 5 minutes for cutting of fruits, 4 minutes for blending, 10 minutes for cooling and 5 minutes for packaging. 

​​TMA Company produces three products; Standard, Deluxe and Luxury in three of its departments which are Cutting, Assembly and Finishing. The total available labour hours per week for Cutting, Assembly and Finishing departments are 180, 300 and 240 respectively.

To produce two units of Standard requires 240 minutes in the Cutting department, half the amount of time in the Assembly department and same amount of time in the Finishing department as in the Cutting department.

To produce one unit of Deluxe requires 60 minutes, 180 minutes and 60 minutes in Cutting, Assembly and Finishing departments respectively.

To produce three units of Luxury requires 180 minutes in Cutting department and twice the amount of time in both Assembly and Finishing departments.

The contribution per unit from Standard, Deluxe and Luxury is Sh.6, Sh.5 and Sh.2 per unit respectively.

Required:

(i) Formulate the above problem as a linear programming model.

(ii) Prepare an initial simplex tableau to solve the above model.

​ ​ ​​

​​Highlight four requirements that must be met before the linear programming model can be applied.

​ ​​A company makes two products; 1 and 2. 

​ ​ ​Write short notes on the following formulas in relation to time series analysis: 

​​Explain the process of exponential smoothing.

​​In the context of linear programming, explain each of the following: 

​ ​ ​ ​​Bantu Limited makes two types of pudding: vanilla and chocolate. Each serving of vanilla pudding requires 2 teaspoons of sugar and 25 fluid measures of water, and each serving of chocolate pudding requires 3 teaspoons of sugar and 15 fluid measures of water. Bantu Limited has available each day 3,600 teaspoons of sugar and 22,500 fluid measures of water. Bantu Limited makes no more than 600 servings of vanilla pudding because that is all that it can sell each day. Bantu Limited makes a profit of Sh.10 on each serving of vanilla pudding and Sh.7 on each serving of chocolate pudding. 

​ ​​The marketing department of Jacuzi Ltd. estimates the daily demand function of one of its products to be linear in nature. If the price was fixed at Sh.570, the daily demand would be 400 units. If the price was increased to Sh.820, the daily sales would drop to 200 units. 

​​Explain the following terms as used in Markov analysis: 

​ ​ ​ ​ ​ ​ ​​Zakem Ltd. produces two products namely; "Alkon" and "Zenon". The products pass through three departments namely; Assembly, Finishing and Packaging. There is a maximum of 200 hours in each of the Assembly and Finishing departments. 

​ ​ ​​Kazi na Bidii Ltd. sells four types of products. The resources needed to produce one unit of each product and thie sales prices are given as follows: 

​ ​​Explain the following terms as used in linear programming:

​ ​ ​ ​ ​​A training institution has four lecturers represented as L1, L2, L3 and L4. The Head of department wishes to assign them to handle three topics in quantitative analysis; T1, T2 and T3. This will be done based on competency which is measured in terms of mastery of subject matter and personal preference on the time schedule while satisfying policies and provisions of the institution.

​​A manufacturing firm produces two products, X and Y. The standard revenues and costs per unit of the products are as follows:

​​ Highlight two differences between "transportation" and "assignment" models of linear programming.

​​ Summarise three applications of shadow prices in decision making.

​​Outline three assumptions of the transportation model.

​ ​​A firm manufactures two products, X and Y, subject to constraints on three raw materials, RM1, RM2 and RM3. The objective of the firm is to select a product mix that will maximise weekly profit. Each unit of product X earns a profit of Sh.2 whereas each unit of product Y earns a profit of Sh.1.

Enumerate four assumptions that are implied in the application of the linear programming model.​​

​​Green Furniture Limited manufactures two models of plastic chairs, ​\(C_1\)​and ​\(C_2\)​ from plastic waste, using two automated machines, X and Y. The following information relates to the production of the two models of chairs for the coming year:

​​Explanation of the following terms as used in Linear programming

(i) Infeasibility.
(ii) Unboundedness.
(iii) Alternate optimality.

​ ​​TOC Limited, an oil prospecting company, intends to set up two oil refineries, refinery I and refinery II.

​​Enumerate four limitations of linear programming models.

​​ Farm Produce Limited is a producer and distributor of maize flour. The company owns milling plants in Eldoret, Nanyuki and Narok towns. The milling plants have not been able to meet the demand orders of the company’s distribution offices located in Mombasa, Kisumu, Nairobi and Isiolo towns. The company is considering the construction of a new milling plant either in Nakuru town or Meru town, in order to expand its production capacity. The data below relate to the company’s production and demand requirements.



Additional information:

1. The estimated unit production costs in Nakuru and Meru towns are Sh.98 and Sh.106 respectively.
2. The unit transportation costs (in shillings) from each milling plant to each distribution office are given as follows
   
   

   To
   From	Eldoret Nanyuki Narok	Mombasa 64 56 58	Kisumu 36 52 42	Nairobi 52 44 36	Isiolo 58 32 50

   
   
3. The estimated unit transportation costs (in shillings) from each of the proposed milling plants to each distribution office are as follows:
   
   

   To
   From	Nakuru Meru	Mombasa 60 62	Kisumu 46 56	Nairobi 40 46	Isiolo 52 28

4. Assume that the construction of one of the proposed milling plants would satisfy the demand deficiency

Required: 

​​Highlight four applications of linear programming in business.

​​Lanex Company specialises in the production of an industrial dye. The firm manufacturers two types of dyes; light and dark. The selling price and the unit variable costs for the dyes are shown below:


Each litre of light dye requires 6 minutes of skilled labour and each litre of dark dye requires 12 minutes of skilled labour.

In a given day, there are 400 man hours of skilled labour available. There are also 100 grammes of an important blending chemical available each day, where each litre of light dye requires 0.05 grammes of the blending chemical and each litre of dark dye requires 0.02 grammes of the chemical

The processing capacity at the plant is limited to 3,000 litres of dye per day.

The company is committed to supply a leading retailer with 5.000 litres of light dye and 2,500 litres of dark dye each working week (consisting of five days). In addition, there is an agreement with the unions that at least 2,000 litres should be produced each day.

Lanex company's management would like to determine the daily production volume for each of the two dyes that will maximise total contribution.

Required:
(i) A linear programming model of the production problem facing Lanex company.
(ii) Using the graphical approach, determine the optimum daily production plan and consequent contribution