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School of Computing FACULTY OF ENGINEERING MJ11 (COMP1640) Modelling, Analysis & Algorithm Design Tutorial

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The Problem Statement A company introduces two new products. Each product requires some labour in the parts fabrication department, the assembly department, and the shipping department. The products are sold through a local distributor who has estimated the maximum potential sales of each product in the coming quarter. The accounting department has provided some data showing the profit contribution on each product. DepartmentProduct 1Product 2Hours Available Fabrication Assembly43900 Shipping22600 Estimated Maximum Sales Profit1020

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List All Possible Variables + Parameters +Constants FactorSymbolUnitsType Estimated Maximum Sales (product 1)E1Constant [150] Estimated Maximum Sales (product 2)E2Constant [100] ProfitP£variable, [output, objective, maximise] Profit P1P1P1 £/unitConstant [10] Profit P2P2P2 £/unitConstant [20] Number of P1 to MakeX1X1 Variable, [output, decision] Number of P2 to MakeX2X2 Variable, [output, decision] Max Hours FabricationFhConstant [1000] Max Hours AssemblyAhConstant [900] Max Hours ShippingShConstant [600] Product 1 Fabrication TimePf 1 h/unitConstant [4] Product 1 Assembly TimePa 1 h/unitConstant [4] Product 1 Shipping TimePs 1 h/unitConstant [2] Product 2 Fabrication TimePf 2 h/unitConstant [5] Product 2 Assembly TimePa 2 h/unitConstant [3] Product 2 Shipping TimePs 2 h/unitConstant [2]

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Constraints Total fabrication hours for both products should be within the available fabrication hours 4x 1 + 5x Total assembly hours for both products should be within the available assembly hours 4x 1 +3x Total shipping hours for both products should be within the available shipping hours 2x 1 + 2x The quantity of product 1 is a positive number which does not exceed the maximum estimated sale amount for product 1 0 x The quantity of product 2 is a positive number which does not exceed the maximum estimated sale amount for product 2 0 x 2 100

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Objective Function Total fabrication hours for both products should be within the available fabrication hours 4x 1 + 5x Total assembly hours for both products should be within the available assembly hours 4x 1 +3x Total shipping hours for both products should be within the available shipping hours 2x 1 + 2x The quantity of product 1 is a positive number which does not exceed the maximum estimated sale amount for product 1 0 x The quantity of product 2 is a positive number which does not exceed the maximum estimated sale amount for product 2 0 x Maximise the profit10X x 2 Within the constraints

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The Graph Max Values of Product 1 and 2 assembly shipping fabrication

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The Graph – Feasible Area

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The Objective Function

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The Objective Function – Optimal Solution

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The Final Answer 4x 1 + 5x 2 = 1000, x 2 = 100 4x = 1000 X 1 = 125

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