Question Tag: Linear Programming

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  • 30 Marks

PM – Nov 2024 – L2 – Q1 – Decision-Making Techniques

Optimization of Oshimiri Nigeria Limited's production plan to maximize profits under resource constraints using linear programming.

Oshimiri Nigeria Limited, a company based in Aba, produces two grades of industrial vanish. The selling price and associated unit variable costs for vanish Grade A and Grade B are shown below:

Particulars Grade A Grade B
Selling Price N2,100 N1,500
Material X (N240/kg) N480 N240
Skilled Labour (N144/hr) N720 N288
Unskilled Labour (N60/hr) N120 N180
Variable Overhead (N84/machine hr) N168 N336

The fixed overhead costs are N2,600,000 per month. The company plans to maximize profits.

The availability of resources for the following month is as follows:

  • Material X: 25,000 Kg
  • Skilled Labour: 48,000 hours
  • Unskilled Labour: 39,000 hours
  • Machine hours: 50,000 hours

Required:

a. Identify the objective function and the constraints of the model to be used in determining the optimum production plan for the following month. (5 Marks)

b. Determine the optimum production plan for the month and the associated profit. (5 Marks)

c. Explain the concept and significance of dual prices and slack variables in the context of the model used by the company in this scenario. (4 Marks)

d. Calculate the dual prices for constraints identified in this scenario. (10 Marks)

e. Suggest ways in which the management can overcome the capacity constraints identified above during the month and the cost implications. (6 Marks)

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  • 40 Marks

PM – May 2021 – L2 – Q1 – Costing Systems and Techniques

Analyze linear programming application in production and pricing strategy for maximizing scarce resources.

The Managing Director of NTAMS Manufacturing Company Limited, located in Lagos, attended a seminar titled “Optimizing scarce resource utility in a manufacturing setting with particular reference to linear programming.” Upon his return, he initiated a management meeting to discuss key insights, prompted by the board’s decision to prioritize two primary products.

The following are cost data for the anticipated products “Biggi” and “Smalli”:

Costs Biggi (₦) Smalli (₦)
Material Costs (5kg @ ₦50/kg) 250 (3kg @ ₦50/kg) 150
Labour Costs:
Machining Time (4 hours @ ₦15/hr) 60 (2 hours @ ₦15/hr) 30
Processing Time (4 hours @ ₦10/hr) 40 (5 hours @ ₦10/hr) 50

The company adheres to a pricing policy where total cost of production is marked up by 20%. Annual overhead is ₦10,000,000, allocated on a 3:2 basis between Biggi and Smalli, with a projected production of 200,000 Biggis and 100,000 Smallis.

Available resources for the upcoming year:

  • Materials: 1,800,000 kg
  • Machine Time: 800,000 hours
  • Other Processing Time: 1,400,000 hours

Required:

As the management accountant:

  1. Explain briefly the concept of linear programming and its usefulness.
    (5 Marks)
  2. Compute the Prices for Biggi and Smalli using the company’s pricing policy.
    (5 Marks)
  3. Advise the company on the output levels needed to maximize total profit, with full financial analysis support.
    (10 Marks)
  4. Explain the meaning and limitations of “shadow prices” and calculate them for constraints.
    (12 Marks)
  5. Assuming consistent conditions for three years with an investment cost of ₦45,000,000 and a 15% cost of capital:
    • Determine if this venture is justified.
      (4 Marks)
    • Find the breakeven discount factor for this project.
      (4 Marks)

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  • 1 Marks

QTB – Nov 2014 – L1 – SA – Q7 – Operations Research

Identifies the components necessary for a linear programming problem.

The building blocks for a linear programming problem are:
A. A linear objective function and equality of constraints
B. A linear objective function and inequality constraints
C. A linear objective function, structural linear inequality constraints, and non-negativity constraints on the decision variables
D. A linear objective function, structural linear inequality constraints, and negativity constraints on the decision variables
E. A linear objective function and structural non-linear inequality constraints

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  • 10 Marks

QTB – May 2016 – L1 – SB – Q2b – Mathematics

This question involves deriving the marginal revenue function and various constraints related to product demand and labor hours.

i. The Students Union of a University plans to have an end-of-year party for all its members. The party is scheduled to take place at a major hotel that can accommodate up to 500 persons. The hotel was to charge N800 per person. At this gate fee, the hotel expects to sell 400 tickets. Market research postulates that for every N40 increase or reduction in the ticket price, the demand will fall or increase by 16 tickets.

Required:
If the variable cost per student for the dinner is N210, determine the Marginal Revenue (MR) function.
(6 marks)

ii. JANG PLC produces and sells two products G and H. G and H respectively make contributions of N10 and N15. The company wishes to maximize its profit. 4,150 units and 3,175 units of G and H, respectively, are to be sold. Direct labor hours per unit are 2.5 hours and 1.5 hours for G and H, while machine hours per unit are 45 minutes and 3 hours for G and H, respectively. Total direct labor hours available and total machine hours available are, respectively, 18,000 hours and 10,000 hours.

Required:
Derive the following constraints: the direct labor, machine time, sales demand for G, sales demand for H, and non-negativity constraints. (4marks)

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  • 20 Marks

QTB – Nov 2015 – L1 – SB – Q6 – Operations Research

Formulate a linear programming problem for optimizing investment in conservative and speculative inventories under given constraints.

Assume that the management of Community Bank Limited wants to invest up to N100,000 in inventory considered to be either conservative or speculative. The company’s board-approved investment policy is that the investment in conservative inventory should be at most N80,000, while the investment in the speculative inventory should be at least N12,000. Assume further that N1.6 return is expected on each naira invested in the conservative inventory, N2.0 return is expected on each investment in the speculative inventory, and that monetary policy regulations require that investment in the speculative inventory should be at most one-third of the investment in the conservative inventory.

Required:

a. State the type of Operations Research problem described above. (2 Marks)

b. Formulate mathematically the:

i. Objective function. (4 Marks)

ii. Constraint inequalities. (8 Marks)

iii. Investment problem. (6 Marks)

(Total: 20 Marks)

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  • 1 Marks

QTB – May 2016 – L1 – SA – Q19 – Operations Research

Expressing a production time constraint for two products given limited direct labour hours.

A company makes two products X and Y. It takes 2¼ hours to make one unit of X and 3¾ hours to make one unit of Y. If only 18,000 direct labour hours are available, then the production time constraint is expressed as

A. 3.75x + 2.25y ≤ 18,000
B. 2.25x + 3.75y ≤ 18,000
C. 3.75x + 2.25y ≥ 18,000
D. 2.75x + 3.75y ≥ 18,000
E. 3.75x + 2.25y = 18,000

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  • 20 Marks

QT – Nov 2017 – L1 – Q2 – Linear Programming

Formulate and solve a linear programming problem for media advertising to maximize customer reach.

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 simplex 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)

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  • 13 Marks

QT – May 2019 – L1 – Q7a – Linear Programming

Formulate a linear programming model for maximizing profit using decision variables and constraints.

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

There are 210 fabrication hours available, 170 hours of assembly available, and materials worth GH¢200 in stock. Market research conducted by the company revealed that demand for the toys is 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 decision variables, and , ,  as slack variables, and  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)

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  • 8 Marks

QT – May 2017 – L1 – Q4b – Linear Programming

Formulate a linear programming model for maximizing profit.

Managers within a subsidiary company in a conglomerate want to know how to maximize profit from two types of products, X and Y. Each product X requires one hour of labor and six liters of molding material, whereas each product Y requires two hours of labor and five liters of molding material. The total labor hours available for each week is 40, and the total amount of molding material each week is 150 liters. 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.

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  • 12 Marks

QT – May 2017 – L1 – Q4a – Linear Programming

Graphically represent constraints and determine the optimum solution.

A particular linear programming problem is formulated as follows:

Subject to the constraints:

Required:

i) Draw these constraints on the same graph paper.

ii) Determine the optimum solution.

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  • 30 Marks

PM – Nov 2024 – L2 – Q1 – Decision-Making Techniques

Optimization of Oshimiri Nigeria Limited's production plan to maximize profits under resource constraints using linear programming.

Oshimiri Nigeria Limited, a company based in Aba, produces two grades of industrial vanish. The selling price and associated unit variable costs for vanish Grade A and Grade B are shown below:

Particulars Grade A Grade B
Selling Price N2,100 N1,500
Material X (N240/kg) N480 N240
Skilled Labour (N144/hr) N720 N288
Unskilled Labour (N60/hr) N120 N180
Variable Overhead (N84/machine hr) N168 N336

The fixed overhead costs are N2,600,000 per month. The company plans to maximize profits.

The availability of resources for the following month is as follows:

  • Material X: 25,000 Kg
  • Skilled Labour: 48,000 hours
  • Unskilled Labour: 39,000 hours
  • Machine hours: 50,000 hours

Required:

a. Identify the objective function and the constraints of the model to be used in determining the optimum production plan for the following month. (5 Marks)

b. Determine the optimum production plan for the month and the associated profit. (5 Marks)

c. Explain the concept and significance of dual prices and slack variables in the context of the model used by the company in this scenario. (4 Marks)

d. Calculate the dual prices for constraints identified in this scenario. (10 Marks)

e. Suggest ways in which the management can overcome the capacity constraints identified above during the month and the cost implications. (6 Marks)

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  • 40 Marks

PM – May 2021 – L2 – Q1 – Costing Systems and Techniques

Analyze linear programming application in production and pricing strategy for maximizing scarce resources.

The Managing Director of NTAMS Manufacturing Company Limited, located in Lagos, attended a seminar titled “Optimizing scarce resource utility in a manufacturing setting with particular reference to linear programming.” Upon his return, he initiated a management meeting to discuss key insights, prompted by the board’s decision to prioritize two primary products.

The following are cost data for the anticipated products “Biggi” and “Smalli”:

Costs Biggi (₦) Smalli (₦)
Material Costs (5kg @ ₦50/kg) 250 (3kg @ ₦50/kg) 150
Labour Costs:
Machining Time (4 hours @ ₦15/hr) 60 (2 hours @ ₦15/hr) 30
Processing Time (4 hours @ ₦10/hr) 40 (5 hours @ ₦10/hr) 50

The company adheres to a pricing policy where total cost of production is marked up by 20%. Annual overhead is ₦10,000,000, allocated on a 3:2 basis between Biggi and Smalli, with a projected production of 200,000 Biggis and 100,000 Smallis.

Available resources for the upcoming year:

  • Materials: 1,800,000 kg
  • Machine Time: 800,000 hours
  • Other Processing Time: 1,400,000 hours

Required:

As the management accountant:

  1. Explain briefly the concept of linear programming and its usefulness.
    (5 Marks)
  2. Compute the Prices for Biggi and Smalli using the company’s pricing policy.
    (5 Marks)
  3. Advise the company on the output levels needed to maximize total profit, with full financial analysis support.
    (10 Marks)
  4. Explain the meaning and limitations of “shadow prices” and calculate them for constraints.
    (12 Marks)
  5. Assuming consistent conditions for three years with an investment cost of ₦45,000,000 and a 15% cost of capital:
    • Determine if this venture is justified.
      (4 Marks)
    • Find the breakeven discount factor for this project.
      (4 Marks)

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  • 1 Marks

QTB – Nov 2014 – L1 – SA – Q7 – Operations Research

Identifies the components necessary for a linear programming problem.

The building blocks for a linear programming problem are:
A. A linear objective function and equality of constraints
B. A linear objective function and inequality constraints
C. A linear objective function, structural linear inequality constraints, and non-negativity constraints on the decision variables
D. A linear objective function, structural linear inequality constraints, and negativity constraints on the decision variables
E. A linear objective function and structural non-linear inequality constraints

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  • 10 Marks

QTB – May 2016 – L1 – SB – Q2b – Mathematics

This question involves deriving the marginal revenue function and various constraints related to product demand and labor hours.

i. The Students Union of a University plans to have an end-of-year party for all its members. The party is scheduled to take place at a major hotel that can accommodate up to 500 persons. The hotel was to charge N800 per person. At this gate fee, the hotel expects to sell 400 tickets. Market research postulates that for every N40 increase or reduction in the ticket price, the demand will fall or increase by 16 tickets.

Required:
If the variable cost per student for the dinner is N210, determine the Marginal Revenue (MR) function.
(6 marks)

ii. JANG PLC produces and sells two products G and H. G and H respectively make contributions of N10 and N15. The company wishes to maximize its profit. 4,150 units and 3,175 units of G and H, respectively, are to be sold. Direct labor hours per unit are 2.5 hours and 1.5 hours for G and H, while machine hours per unit are 45 minutes and 3 hours for G and H, respectively. Total direct labor hours available and total machine hours available are, respectively, 18,000 hours and 10,000 hours.

Required:
Derive the following constraints: the direct labor, machine time, sales demand for G, sales demand for H, and non-negativity constraints. (4marks)

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  • 20 Marks

QTB – Nov 2015 – L1 – SB – Q6 – Operations Research

Formulate a linear programming problem for optimizing investment in conservative and speculative inventories under given constraints.

Assume that the management of Community Bank Limited wants to invest up to N100,000 in inventory considered to be either conservative or speculative. The company’s board-approved investment policy is that the investment in conservative inventory should be at most N80,000, while the investment in the speculative inventory should be at least N12,000. Assume further that N1.6 return is expected on each naira invested in the conservative inventory, N2.0 return is expected on each investment in the speculative inventory, and that monetary policy regulations require that investment in the speculative inventory should be at most one-third of the investment in the conservative inventory.

Required:

a. State the type of Operations Research problem described above. (2 Marks)

b. Formulate mathematically the:

i. Objective function. (4 Marks)

ii. Constraint inequalities. (8 Marks)

iii. Investment problem. (6 Marks)

(Total: 20 Marks)

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  • 1 Marks

QTB – May 2016 – L1 – SA – Q19 – Operations Research

Expressing a production time constraint for two products given limited direct labour hours.

A company makes two products X and Y. It takes 2¼ hours to make one unit of X and 3¾ hours to make one unit of Y. If only 18,000 direct labour hours are available, then the production time constraint is expressed as

A. 3.75x + 2.25y ≤ 18,000
B. 2.25x + 3.75y ≤ 18,000
C. 3.75x + 2.25y ≥ 18,000
D. 2.75x + 3.75y ≥ 18,000
E. 3.75x + 2.25y = 18,000

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  • 20 Marks

QT – Nov 2017 – L1 – Q2 – Linear Programming

Formulate and solve a linear programming problem for media advertising to maximize customer reach.

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 simplex 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)

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  • 13 Marks

QT – May 2019 – L1 – Q7a – Linear Programming

Formulate a linear programming model for maximizing profit using decision variables and constraints.

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

There are 210 fabrication hours available, 170 hours of assembly available, and materials worth GH¢200 in stock. Market research conducted by the company revealed that demand for the toys is 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 decision variables, and , ,  as slack variables, and  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)

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  • 8 Marks

QT – May 2017 – L1 – Q4b – Linear Programming

Formulate a linear programming model for maximizing profit.

Managers within a subsidiary company in a conglomerate want to know how to maximize profit from two types of products, X and Y. Each product X requires one hour of labor and six liters of molding material, whereas each product Y requires two hours of labor and five liters of molding material. The total labor hours available for each week is 40, and the total amount of molding material each week is 150 liters. 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.

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  • 12 Marks

QT – May 2017 – L1 – Q4a – Linear Programming

Graphically represent constraints and determine the optimum solution.

A particular linear programming problem is formulated as follows:

Subject to the constraints:

Required:

i) Draw these constraints on the same graph paper.

ii) Determine the optimum solution.

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