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27 January, 16:44

Neilsen Cookie Company sells its assorted butter cookies in containers that have a net content of 1 lb. The estimated demand for the cookies is 700,000 1 lb containers. The setup cost for each production run is $546, and the manufacturing cost is $0.47 for each container of cookies. The cost of storing each container of cookies over the year is $0.35. Assuming uniformity of demand throughout the year and instantaneous production, how many containers of cookies should Neilsen produce per production run in order to minimize the production cost

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  1. 27 January, 18:32
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    46,734 units per run

    Explanation:

    total estimated demand = 700,000 containers

    setup costs per production run = $546

    manufacturing cost = $0.47 per container

    holding cost = $0.35 per container

    r = 700,000 / x

    total setup costs = 546r = 546 (700,000/x) = 382,200,000/x

    production costs = 0.47 x 700,000 = 329,000

    storage cost per unit = 1/2r x 0.35 = 0.35/2 (700,000/x) = 0.35x/1,400,000

    total storage costs = 700,000 x 0.35x/1,400,000 = 0.175x

    C (x) = 382,200,000/x + 0.175 x + 329,000

    now we find the derivative:

    C' (x) = - 382,200,000/x² + 0.175

    382,200,000/x² = 0.175

    382,200,000 = 0.175x²

    x² = 382,200,000 / 0.175 = 2,184,000,000

    x = √2,184,000,000 = 46,733.28 ≈ 46,734 units per run

    this answer is based on a continuous production process, there are 14.98 runs per year
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