Optimization of Production and Inventory Management for Acme Manufacturing Using Linear Programming

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Optimization of Production and Inventory Management for AcmeManufacturing Using Linear ProgrammingAcme Manufacturing makes a variety of household appliances at a single manufacturing facility.Theexpected demand for one of these appliancesduring the next four months is shown in thefollowing tablealong with the expected production costs and the expected capacity for producingthese items.Month1 2 3 4Demand 420 580 310 540Production Cost $49.00 $45.00 $46.00 $47.00Production Capacity 500 520 450 550Acme estimates it costs $1.50 per month for each unit of this appliance carried in inventory(estimated byaveraging the beginning and ending inventory levels each month). Currently,Acme has 120 units ininventory on hand for this product. To maintain a level workforce, thecompany wants to produce at least 400units per month. They also want to maintain a safetystock of at least 50 units per month. Acme wants todetermine how many of each appliance tomanufacture during each of next four months to meet the expecteddemand at the lowest possibletotal cost.1.Formulate aLinear Programming model for this problemLet,𝑃𝑖=𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛𝑜𝑛𝑖𝑡ℎ𝑚𝑜𝑛𝑡ℎ𝑆𝑖=𝑆𝑎𝑓𝑒𝑡𝑦𝑠𝑡𝑜𝑐𝑘𝑜𝑓𝑖𝑡ℎ𝑚𝑜𝑛𝑡ℎThen the formulated Linear programing is,Minimize Z =$49𝑃1+$45𝑃2+$46𝑃3+$47𝑃4+$1.50∗ቀ120+𝑆12+𝑆1+𝑆22+𝑆2+𝑆32+𝑆3+𝑆42ቁSubject to,𝑃1≤500𝑃1+120−𝑆1=420𝑃2≤520𝑃2+𝑆1−𝑆2=580𝑃3≤450𝑃3+𝑆2−𝑆3=310𝑃4≤550𝑃4+𝑆3−𝑆4=540𝑃𝑖≥400𝑎𝑛𝑑𝑆𝑖≥50

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