2.1. The Blending Model. Example 2.2.1. To feed her stock a farmer can purchase two kinds of feed. He determined that the herd requires 60, 84, and 72 units of the nutritional elements A,B, and C, respectively, per day. Units/lb A B C Cost( cents/lb ) Feed 1 3 7 3 10 Feed 2 2 2 6 4 If x=number of pounds of feed 1 and y=number of pounds of feed 2. The problem is thus The solution of the problem occurs at (6,21) Example 2.2.2 A landscaper with two grass seed blends. Blend I contains 60 % bluegrass and 10% fescue and costs 80 cents/lb; Blend II contains 20% bluegrass seed and 50 % fescue and costs 60 cents/lb. The field requires a composition seed with at least 30%bluegrass and 26% fescue. What is the least expensive combination of the two blends that meets the requirements? Draw a table Bluegrass Fescue Cost ( cents/lb ) Blend I 0.60 0.10 80 Blend II 0.20 0.50 60 Min. Req. 0.30 0.26
Consider a composite of 100 lb. Let x be the number of pounds of blend I and y the number of pounds of blend II. The feasible region is just a segment of the line x + y = 100 between the points (25,75) and (60,40). The minimum is obtained when x = 25 and y = 75 which means that 25 lb of Blend I needs to be mixed with 75 lb of Blend II. 2.3. The Production Model. Objective is to maximize profit using limited resources or to minimize costs subjected to specified production requirements where inputs are commodities (raw material, capital, labor, etc) and outputs are goods produced. Example 2.3.1 A manufacturer produces two boats: family rowboat and a sports canoe. The boats are molded from aluminum and are finished by hand labor. A rowboat requires 50 lb of aluminum, 6 minutes of machine time, and 3 hours of finishing labor; a canoe requires 30 lb of aluminum, 5 minutes of machine time, and 5 hours of finishing labor. For the next three (3) months the company can commit up to 1 ton of aluminum, 5 hours of machine time, and 200 hours of labor. The company realizes a $50 profit on the sale of a rowboat and a $60 profit on the sale of a canoe. (i). Assuming that all boats can be sold, how many of each type should be manufactured in the next 3 months in order to maximize profits? Let R=number of rowboats and C=number of canoes. The problem is
Can you find the solution of the prob.? (ii). Suppose now the cost to the manufacturer of the 1 ton of aluminum is not fixed: the price per pound of the last 500 lb of aluminum is 20 cents/lb more than the first 1500 lb and the price of the first 1500 lb is the cost used in determining the $50 and $60 profit estimates. What is the new optimal production schedule? Example 2.3.3 A cabinet shop makes and sells two types of cabinets, type 1 for the kitchen, and type 2, for the bathroom. Manufacture of cabinets consists of frames and drawers and assembly and finish. Labor requirements in hours/unit are given in the tableau Cabinet Frame/Drawers (hr) Assembly/Finish ( hr ) Type 1(Kitchen) 2.60 2.10 Type 2 (Bathroom) 1.50 1.80 Each week 480 hours of labor available for cabinets. To conserve labor, frames and drawers completed and ready for assembly and finish are bought from a local dealer at a cost of $200 for a kitchen frame/drawer and $110 for a bathroom frame/drawer. The kitchen cabinets sell for $350 each; the first 70 bathroom cabinets sell for $250 per unit, but any more produced sell only for $225 per unit. Assume all units produced will be sold. How many of each type of cabinet to produce and how to generate the associated frames and drawers. Define the variables ti = total number of cabinets of type i produced,i = 1,2 mi = number of frames/drawres of type i made,i = 1,2 bi = number of frames/drawres of type i bought,i = 1,2 u = number of bathroom cabinets sold up to 70
v = number of bathroom cabinets sold over 70 Thus the problem is to Example 2.3.4 Consider the operation of one division in a large plant. The manager had 4 processes to manufacture two parts of the plant s final product. Input Input Input Output Output Process Labor (workerhrs) Raw Mat. A (lb) Raw Mat. B (lb) Units-Part 1 Units-Part 2 1 20 160 30 35 55 2 30 100 35 45 42 3 10 200 60 70 0 4 25 75 80 0 90 Each week 2100 units of part i and 1800 units of part 2 should be produced using 4 tons of raw material A and 2 tons of raw material B and 1000 hours of labor. One pound of raw material A costs $3, and one pound of raw material B costs $7. Because of labor contracts, employees are paid weekly regardless they are working or not ( cost of 1000 hours of labor is fixed). However, the manager can request workers to work up to an extra 200 hours/week in overtime at a cost of $30/hr. Can the firm meet its weekly requirements with material available without using overtime? If not how much money can be saved using overtime? Case 1. No overtime. Problem is to
Case 2. Overtime used. Let 0 x5 200 be the number of overtime hours used. Hence the problem becomes to 2.4. The Transportation Model. Example (Prob. 2) A canned goods supplier has two warehouses serving four outlets. The East Coast Warehouse has 600 cases on hand and the West Coast Warehouse has 1000 cases on hand. The shipping costs, in cents per case, and the requirements for the four outlets, all located east of the Mississippi, are given in the following table. Outlet 1 Outlet 2 Outlet 3 Outlet 4 Shipping Costs East Coast Warehouse 20 16 30 20 West Coast Warehouse 45 39 50 44 Requirements 300 350 400 450 (a) Determine a shipping schedule that minimizes transportation costs. (b) Assume that there are truck rental fees of $50 if units are shipped from the East Coast Warehouse to Outlet 1 and $60 if units are shipped to Outlet 2. Determine a shipping schedule that minimizes transportation costs.
2.5. The Dynamic Planning model. Example 2.5.2 A shop must deliver 500 units of a commodity Q in Period 1, 650 in Period 2, and 625 in Period 3. The shop has two different processes that can be used to produce the commodity Q, each process using raw material M and labor. Input and output for 1 hour of operation of each are: M (units) Labor (hr) Units of Q Process X 8 3 25 Process Y 4 5 20 Each period the shop has available up to 175 units of M, but the material deteriorates quickly. Any units of M unused in one period cannot be saved for later use. The shop also has available each period up to 120 hours regular-time labor at $30/hr and another 45 hours of overtime labor at $45/hr. Surplus units of Q made in one period may be stored for later delivery at a cost of $25/unit-period, but space limitations restrict the number stored to be no more than 50/period, and no units are to remain in stock after the three periods. The shop pays only for the labor and storage space used. The cost of the raw material is fixed by other constraints. Determine a production and storage schedule that minimizes the labor and storage costs. Denote by xi = number of hours process X is used in Period i,i = 1,2,3 yi = number of hours process Y is used in Period i,i = 1,2,3 qi = number of units Q made in Period i,i = 1,2,3 si = number of units Q stored in Period i,i = 1,2,3 ui vi = = number of hours of regular time labor used in Period i,i = 1,2,3 number of hours of overtime labor used in Period i,i = 1,2,3 The problem is then to