Presented By: Brian Murphy and Scott Lichtor Solving Model 7 Presented By: Brian Murphy and Scott Lichtor

Background Dairy plant produces various flavors of ice creams in 3 gallon containers and distributes them to a chain of ice cream parlors. Dairy plant knows exact demand from parlors for the next twelve months (January-December). Plant has huge freezer-warehouse that can store surplus in any one month for sale in future months. Plant can produce ice cream in regular time or overtime. Production capabilities vary from month to month due to variations in demand. Question: How many gallons should the dairy plant produce in each month in order to reduce total costs?

Key Figures *Cost of storage is 10 cents per month for each month Demand: Regular time production: Overtime production: Month Jan. Feb. Mar. Apr. May June Jul. Aug. Sept. Oct. Nov. Dec. No. of containers ordered 75 80 85 95 120 140 175 190 130 110 90 Month Jan. Feb. Mar. Apr. May June Jul. Aug. Sept. Oct. Nov. Dec. Production Capacities 100 90 80 Cost per 3 gallons (cents) 65 70 75 Month Jan. Feb. Mar. Apr. May June Jul. Aug. Sept. Oct. Nov. Dec. Production Capacities 40 50 60 Cost per 3 gallons (cents) 85 90 95

Assumptions and Variables Dairy plant has no finite storage capacity. Ice cream kept in storage does not go bad or lose value (ice cream produced in January can be sold in December). All ice cream produced is eventually sold (no leftovers) Variables: xij = containers produced during regular time in month j for use in month i. yij = containers produced during overtime in month j for use in month i.

Objective Function … Minimize z = total costs Broken down into production and storage costs for each month. January: 0.65∑xi,1 + 0.85∑yi,1 + 0.1∑(i-1)(xi,1+yi,1) +February: 0.65∑xi,2 + 0.85∑yi,2 + 0.1∑(i-2)(xi,2+yi,2) +December: .65xi,12 + .85yi,12 Regular Production Cost Overtime Production Cost Storage Cost 12 12 12 i=1 i=1 i=1 i=2 i=2 i=2 …

Constraints … Demand Constraints: January: x1,1 + y1,1 = 75 February: ∑(x2,j + y2,j) = 80 December: ∑(x12,j + y12,j) = 90 12 j=1 12 … j=1

Constraints (Cont.) … Production Constraints: January: ∑xi,1 ≤ 100, ∑yi,1 ≤ 40 February: ∑xi,2 ≤ 100, ∑yi,2 ≤ 40 December: x12,12 ≤ 90, y12,12 ≤ 50 12 12 i=1 i=1 i=2 i=2 …

Solution See Excel Sheet