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Mother's Novelty Cakes (Using Linear Programming)

984 words - 4 pages

My mother loves to make cakes and pastries whenever she is off from work. She makes really exquisite gourmet cakes that other people, friends and the likes, start to order them from her whenever there is an occasion. For that reason, she bought a dual purpose oven that can bake different kinds of pastries and at the same time be used to age gourmet cakes such as fruit cakes. My mother, among her network, is famous for her orange rum cake and fruit cake. Since then, she always prepared both for her friends. I wonder now which between the two is most profitable given her limited time to prepare such orders. My mother makes her orders from Friday noon up to Saturday noon. Together with her house help, they work from Friday noon until midnight and start again 7 in the morning the following day up to noon time.Ingredients, costing, and procedure for making Orange Rum Cake and Fruit cake can be seen in Annex A and B.The time requirements for each of the two cakes, their suggested selling prices, their variable costs, and their contribution margins appear in the table below.Orange Rum CakeFruit CakeSuggested Retail PriceLess Variable Costs to Produce One Unit of CakeContribution Margin per Unit P 200117.88P82.12P 350201.19P148.81Labor TimeOven Time (baking and aging).5 hour per unit1 hour per unit.33 hour per unit4 hours per unitUsing Linear Programming formulation, I can now know which between the two cakes is more profitable given the constraints.Decision Variables:ORC= number of Orange Rum Cake to be producedFC= number of Fruit Cake to be producedObjective Function:Maximize Total Contribution Margin = P82.12 ORC + P148.81 FCSubject to:ORC, FC= 0(non negativity constraint).5 ORC + .33 FC= < 17 Labor Hours(labor hours constraint)1 ORC + 4 FC= < 24 Oven Hours(oven hours constraint)Linear Programming Results:ORCFC RHSDualMaximize 82.1149Labor Hours0.50.3240 P 1970.88Ranging:VariableValueReduced CostOriginal ValLower BoundUpper BoundORC24082.1237.2InfinityFC0179.67148.81Infinity328.48ConstraintDual ValueSlack/SurplusOriginal ValLower BoundUpper BoundLabor Hours051712 InfinityOven Hours (baking and aging)82.12024034Solution:VariableStatusValueORCBasic24FCNONBasic0slack 1Basic5slack 2NONBasic0Optimal Value (Z) 1970.88Iterations:CjBasic Variables 82.12 ORC 148.81 FC 0 slack 1 0 slack 2 QuantityIteration 10slack 10.50.3310170slack 2140124Zj00000cj-zj82.12148.8100Iteration 20slack 10.417501-0.082515.02148.81FC0.25100.256Zj37.2025148.81037.2025892.86cj-zj44.917500-37.203Iteration 30slack 10-1.671-0.5582.12ORC140124Zj82.12328.48082.121,970.88cj-zj0-179.670-82.12It may appear at first that Fruit Cake has the larger contribution margin. However, this is not so because Fruit Cake takes longer oven hours to produce as compared to producing Fruit Cake. Given the constraints, producing 24 pieces of Orange Rum Cake gives the optimal contribution of P 1970.88. However, although producing just Orange Rum cakes gives my mother the optimal value in terms of...

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