A single-period production Mix: -
An electric company specializing in producing PVC electrical wire cables produces three types of electrical wires: cables for electric panels, power cables, and fire resistance cables. Wires are produced by over three production departments: drawing, coating, and packaging. The times each product needs, along with the capacity in (hours) for various departments, in addition to the selling price and the cost per unit, are shown in the table below:
Department Time per unit (hours) Capacity (hours)
Cables for electric panels Power cables Fire resistance cables
Drawing 0.55 0.35 0.45 3,000
Coating 0.5 0.7 0.6 2,500
Packaging 0.20 0.2 0.2 2,700
Unit Cost $66 $50 $145
Selling Price $110 $90 $165
The factory has some deficiencies in the drawing and coating departments. Where in each department, produced items can be good or have to be scrapped. From the previous history, the following table provides the average scrap rate per department:
Department Scrap percentage
Drawing 4%
Coating 5%
The company has received firm orders for its products. If the demands per product as required by the contract are not met, a penalty ($ / unit) is imposed. The following table shows the order quantities for each type and the penalty (in $/unit):
PVC Type Quantity required (units) Penalty ($/unit)
Cables for electric panels 1,500 $10
Power cables 1,150 $15
Fire resistance cables 2,100 $8
Requirements:-
Formulate a linear program to determine the product mix to maximize the company’s profit. In your work,
Write a ward report that shows the following;
A clear written linear program formulation with the proper notation and variable definitions.
Solve using a solver or any other computer software and provide a snapshot of the software solution.
Write a conclusion statement at the end showing the solution and any other concern related to the application of that solution
Note: Please work in pairs and submit your projects to the portal by the deadline. Clarity and readability of the work will be emphasized, and marks will be set accordingly. Finally, the rules of plagiarism will be strictly followed.
Notes about the formulation (for grading): -
The definition of the variable should include two suffix notations as for x_ij where
x_ij : is the quantity produced of product i , ( i =1, 2,3) over machine j,(j=1,2,3)
The quantities passing from machine to machine change due to the production deficiency. Therefore, there should be a clear connection between variables related to the quantities at each station.
Demand is satisfied for the good items from the packaging station. This quantity is different from the initial introduced quantity. The relations built throughout the model should take care of that
The demand can or need not be met according to the penalty. Formulate the demand constraint as an equality constraint with the penalty expressed as a slack variable and penalized in the objective function
Profit is realized from the difference between selling the good items (only good items) and the cost of producing whatever needs to be produced (at the respective cost). In addition, the penalty cost has to be added to allow for under-satisfied demand.
The company has received a firm order; no oversupply is allowed. Either you meet the order, or under supply with a penalty.
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