Text of 1 Pesquisa Operacional Aplicada à Logística Prof. Fernando Augusto Silva Marins...
1 Pesquisa Operacional Aplicada Logstica Prof. Fernando Augusto Silva Marins www.feg.unesp.br/~fmarins email@example.com www.feg.unesp.br/~fmarins firstname.lastname@example.org
2 Pesquisa Operacional faz diferena no desempenho de organizaes?
3 Finalistas do Prmio Edelman INFORMS 2007
4 Questes Logsticas (Pesquisa Operacional)
5 Delta Hardware Stores Problem Statement Delta Hardware Stores is a regional retailer with warehouses in three cities in California San Jose Fresno Azusa
6 Each month, Delta restocks its warehouses with its own brand of paint. Delta has its own paint manufacturing plant in Phoenix, Arizona. San Jose Fresno Azusa Phoenix
7 Although the plants production capacity is sometime inefficient to meet monthly demand, a recent feasibility study commissioned by Delta found that it was not cost effective to expand production capacity at this time. To meet demand, Delta subcontracts with a national paint manufacturer to produce paint under the Delta label and deliver it (at a higher cost) to any of its three California warehouses. Delta Hardware Stores Problem Statement
8 Given that there is to be no expansion of plant capacity, the problem is: To determine a least cost distribution scheme of paint produced at its manufacturing plant and shipments from the subcontractor to meet the demands of its California warehouses. Delta Hardware Stores Problem Statement
9 Decision maker has no control over demand, production capacities, or unit costs. The decision maker is simply being asked: How much paint should be shipped this month (note the time frame) from the plant in Phoenix to San Jose, Fresno, and Asuza and How much extra should be purchased from the subcontractor and sent to each of the three cities to satisfy their orders? Delta Hardware Stores Variable Definition
10 X 1 : amount of paint shipped this month from Phoenix to San Jose X 2 : amount of paint shipped this month from Phoenix to Fresno X 3 : amount of paint shipped this month from Phoenix to Azusa X 4 : amount of paint subcontracted this month for San Jose X 5 : amount of paint subcontracted this month for Fresno X 6 : amount of paint subcontracted this month for Azusa Decision/Control Variables:
11 National Subcontractor X4X4 X5X5 X6X6 X1X1 X2X2 X3X3 San Jose Fresno Azusa Phoenix Network Model
12 The objective is to minimize the total overall monthly costs of manufacturing, transporting and subcontracting paint, The constraints are (subject to): The Phoenix plant cannot operate beyond its capacity; The amount ordered from subcontractor cannot exceed a maximum limit; The orders for paint at each warehouse will be fulfilled. Mathematical Model
13 To determine the overall costs: The manufacturing cost per 1000 gallons of paint at the plant in Phoenix - (M) The procurement cost per 1000 gallons of paint from National Subcontractor - (C) The respective truckload shipping costs form Phoenix to San Jose, Fresno, and Azusa - (T 1, T 2, T 3 ) The fixed purchase cost per 1000 gallons from the subcontractor to San Jose, Fresno, and Azusa - (S 1, S 2, S 3 ) Mathematical Model
14 Minimize(M + T 1 ) X 1 + (M + T 2 ) X 2 + (M + T 3 ) X 3 + (C + S 1 ) X 4 + (C + S 2 ) X 5 + (C + S 3 ) X 6 Mathematical Model: Objective Function Where: Manufacturing cost at the plant in Phoenix: M Procurement cost from National Subcontractor: C Truckload shipping costs from Phoenix to San Jose, Fresno, and Azusa: T 1, T 2, T 3 Fixed purchase cost from the subcontractor to San Jose, Fresno, and Azusa: S 1, S 2, S 3 X 1 : amount of paint shipped this month from Phoenix to San Jose X 2 : amount of paint shipped this month from Phoenix to Fresno X 3 : amount of paint shipped this month from Phoenix to Azusa X 4 : amount of paint subcontracted this month for San Jose X 5 : amount of paint subcontracted this month for Fresno X 6 : amount of paint subcontracted this month for Azusa
15 To write to constraints, we need to know: The capacity of the Phoenix plant (Q 1 ) The maximum number of gallons available from the subcontractor (Q 2 ) The respective orders for paint at the warehouses in San Jose, Fresno, and Azusa (R 1, R 2, R 3 ) Mathematical Model: Constraints
16 The number of truckloads shipped out from Phoenix cannot exceed the plant capacity: X1 + X2 + X3 Q1 The number of thousands of gallons ordered from the subcontrator cannot exceed the order limit: X4 + X5 + X6 Q2 The number of thousands of gallons received at each warehouse equals the total orders of the warehouse: X1 + X4 = R1 X2 + X5 = R2 X3 + X6 = R3 All shipments must be nonnegative and integer: X1, X2, X3, X4, X5, X6 0 X1, X2, X3, X4, X5, X6 integer Mathematical Model: Constraints
17 Orders: R 1 = 4000, R 2 = 2000, R 3 = 5000 (gallons) Capacity: Q 1 = 8000, Q 2 = 5000 (gallons) Subcontractor price per 1000 gallons: C = $5000 Cost of production per 1000 gallons: M = $3000 Mathematical Model: Data Collection
18 Transportation costs $ per 1000 gallons Subcontractor: S 1 =$1200; S 2 =$1400; S 3 = $1100 Phoenix Plant: T 1 = $1050;T 2 = $750; T 3 = $650 Mathematical Model: Data Collection
19 Min (3000+1050)X 1 +(3000+750)X 2 +(3000+650)X 3 +(5000+1200)X 4 + + (5000+1400)X 5 + (5000+1100)X 6 Ou Min 4050 X 1 + 3750 X 2 + 3650 X 3 + 6200 X 4 + 6400 X 5 + 6100 X 6 Subject to: X 1 + X 2 + X 3 8000 (Plant Capacity) X 4 + X 5 + X 6 5000 (Upper Bound order from subcont.) X 1 + X 4 = 4000 (Demand in San Jose) X 2 + X 5 = 2000 (Demand in Fresno) X 3 + X 6 = 5000 (Demand in Azusa) X 1, X 2, X 3, X 4, X 5, X 6 0 (nonnegativity) X 1, X 2, X 3, X 4, X 5, X 6 integer Delta Hardware Stores Operations Research Model
20 X 1 = 1,000 gallons X 2 = 2,000 gallons X 3 = 5,000 gallons X 4 = 3,000 gallons X 5 = 0 X 6 = 0 Optimum Total Cost = $48,400 Delta Hardware Stores Solutions
21 CARLTON PHARMACEUTICALS Carlton Pharmaceuticals supplies drugs and other medical supplies. It has three plants in: Cleveland, Detroit, Greensboro. It has four distribution centers in: Boston, Richmond, Atlanta, St. Louis. Management at Carlton would like to ship cases of a certain vaccine as economically as possible.
22 Data Unit shipping cost, supply, and demand Assumptions Unit shipping costs are constant. All the shipping occurs simultaneously. The only transportation considered is between sources and destinations. Total supply equals total demand. To FromBostonRichmondAtlantaSt. LouisSupply Cleveland $353040321200 Detroit 374042251000 Greensboro 40152028800 Demand 1100400750 CARLTON PHARMACEUTICALS
23 CARLTON PHARMACEUTICALS Network presentation
24 Boston Richmond Atlanta St.Louis Destinations Sources Cleveland Detroit Greensboro S 1 =1200 S 2 =1000 S 3 = 800 D 1 =1100 D 2 =400 D 3 =750 D 4 =750 37 40 42 32 35 40 30 25 35 15 20 28
25 The structure of the model is: Minimize Total Shipping Cost ST [Amount shipped from a source] [Supply at that source] [Amount received at a destination] = [Demand at that destination] Decision variables X ij = the number of cases shipped from plant i to warehouse j. where: i=1 (Cleveland), 2 (Detroit), 3 (Greensboro) j=1 (Boston), 2 (Richmond), 3 (Atlanta), 4(St.Louis) CARLTON PHARMACEUTICALS Linear Programming Model
26 Boston Richmond Atlanta St.Louis D 1 =1100 D 2 =400 D 3 =750 D 4 =750 The supply constraints Cleveland S 1 =1200 X11 X12 X13 X14 Supply from Cleveland X11+X12+X13+X14 1200 Detroit S 2 =1000 X21 X22 X23 X24 Supply from Detroit X21+X22+X23+X24 1000 Greensboro S 3 = 800 X31 X32 X33 X34 Supply from Greensboro X31+X32+X33+X34 800
27 CARLTON PHARMACEUTICAL The complete mathematical model Minimize 35X11 + 30X12 + 40X13 + 32X14 + 37X21 + 40X22 + 42X23 + 25X24+ + 40X31+15X32 + 20X33 + 38X34 ST Supply constraints: X11+X12+X13+X141200 X21+X22+X23+X241000 X31+X32+X33+X34800 Demand constraints: X11+X21+X311100 X12+ X22+X32400 X13+ X23+X33750 X14+X24+X34750 All Xij are nonnegative = Total shipment out of a supply node cannot exceed the supply at the node. Total shipment received at a destination node, must equal the demand at that node.
28 CARLTON PHARMACEUTICALS Spreadsheet =SUM(B7:B9) Drag to cells C11:E11 =SUMPRODUCT(B7:E9,B15:E17) =SUM(B7:E7) Drag to cells G8:G9
29 MINIMIZE Total Cost SHIPMENTS Demands are met Supplies are not exceeded CARLTON PHARMACEUTICALS Spreadsheet
30 CARLTON PHARMACEUTICALS Spreadsheet - solution
31 CARLTON PHARMACEUTICALS Sensitivity Report Reduced costs The unit shipment cost between Cleveland and Atlanta must be reduced by at least $5, before it would become economically feasible to utilize it If this route is used, the total cost will increase by $5 for each case shipped between the two cities.