Mathematics for Management Science Notes 05 prepared by Professor Jenny Baglivo

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1 Mathematics for Management Science Notes 05 prepared by Professor Jenny Baglivo Jenny A. Baglivo All rights reserved. Transportation and assignment problems Transportation/assignment problems arise frequently in planning for the distribution of goods and services from several supply locations (origins) to several demand locations (destinations). Transportation/assignment is to be made at minimum cost. Origins are indexed using i=1, 2, etc. Destinations are indexed using j=1, 2, etc. Typical notation is as follows: For the decision variables, we let x ij equal the number of units shipped from origin i to destination j. The supply at origin i is denoted by s i. The demand at destination j is denoted by d j. If the total supply is greater than or equal to the total demand, then For each origin, the constraint x i+ s i is added to the model and For each destination, the constraint x +j = d j is added to the model. (Not all products are shipped. All demands are satisfied.) If the total supply is less than the total demand, then For each origin, the constraint x i+ = s i is added to the model and For each destination, the constraint x +j d j is added to the model. (All products are shipped. Not all demands are satisfied.) page 1 of 19

2 Exercise 1: Tropicsun is a leading grower and distributor of fresh citrus products with three large citrus groves scattered around central Florida in the cities of Mt. Dora, Eustis, and Clermont. Tropicsun currently has 275,000 bushels of citrus at the grove in Mt. Dora, 400,000 bushels at the grove in Eustis, and 300,000 at the grove in Clermont. Tropicsun has citrus processing plants in Ocala, Orlando, and Lessburg with processing capacities to handle 200,000, 600,000, and 225,000 bushels, respectively. Tropicsun contracts with a local trucking company to transport its fruit from the groves to the processing plants. The trucking company charges a flat rate for every mile that each bushel of fruit must be transported. Each mile a bushel of fruit travels is known as a bushel-mile. The following table summarizes the distances (in miles) between the groves and processing plants: to Ocala to Orlando to Leesburg from Mt. Dora from Eustis from Clermont Tropicsun wants to determine how many bushels to ship from each grove to each processing plant in order to minimize the total number of bushel-miles. Note: The problem can be visualized using a transportation network, where each source and destination is represented as a node, and each route from source to destination as a link. Ocala Mt Dora Eustis Orlando Clermont Leesburg page 2 of 19

3 Define the decision variables precisely Completely specify the LP model page 3 of 19

4 Clearly state the optimal solution. How would the total cost change if Leesburg could only process 200,000 bushels? Interpret the reduced cost for shipping from Eustis to Ocala. page 4 of 19

5 Exercise 1 solution sheet: A B C D E F G H Tropicsun Distance (miles) to Ocala to Orlando to Leesburg Supply: from Mt. Dora from Eustis from Clermont M O D E L Capacity: Minimize Total B u s h e l - M i l e s Decision Variables to Ocala to Orlando to Leesburg L H S R H S from Mt. Dora = from Eustis = from Clermont = L H S <= <= <= R H S Minimize B14 By Changing B18:D20 Subject to F18:F20 = H18:H20 B22:D22 <= B24:D24 Assume: linear model, non-negative page 5 of 19

6 Sensitivity and formulas sheets: Adjustable Cells F i n a l Reduced O b j e c t i v e A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e C o s t C o e f f i c i e n t I n c r e a s e D e c r e a s e $B$18 from Mt. Dora to Ocala E+30 $C$18 from Mt. Dora to Orlando E+30 2 $D$18 from Mt. Dora to Leesburg $B$19 from Eustis to Ocala E $C$19 from Eustis to Orlando $D$19 from Eustis to Leesburg $B$20 from Clermont to Ocala E $C$20 from Clermont to Orlando E+30 $D$20 from Clermont to Leesburg E Constraints F i n a l S h a d o w C o n s t r a i n t A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e P r i c e R.H. Side I n c r e a s e D e c r e a s e $F$18 from Mt. Dora LHS $F$19 from Eustis LHS $F$20 from Clermont LHS $B$22 LHS to Ocala $C$22 LHS to Orlando E $D$22 LHS to Leesburg A B C D E F G H T r o p i c s u n Distance (miles) to Ocala to Orlando to Leesburg Supply: from Mt. Dora from Eustis from Clermont M O D E L Capacity: Minimize Total B u s h e l - M i l e s =SUMPRODUCT(B5:D7,B18:D20) Decision Variables to Ocala to Orlando to Leesburg L H S R H S from Mt. Dora =SUM(B18:D18) = =F5 from Eustis =SUM(B19:D19) = =F6 from Clermont =SUM(B20:D20) = =F7 L H S =SUM(B18:B20) =SUM(C18:C20) =SUM(D18:D20) <= <= <= R H S =B9 =C9 =D9 page 6 of 19

7 Assignment problems are special cases of transportation problems where s i = d j = 1. Exercise 2: Fowle Marketing Research Company has just received requests for market research studies from three new clients. The company faces the task of assigning a project leader to each client. Currently, four individuals have no other commitments and are available for the project leader assignments. Fowle s management realizes, however, that the time required to complete each study will depend on the experience and ability of the project leader assigned. Estimated project completion times in days are given in the following table: Project Leader: Client 1 Client 2 Client 3 1: Terry : Carle : McClymonds : Higley The three projects have approximately the same priority, and the company wants to assign project leaders to minimize the total number of days required to complete all three projects. If a project leader is to be assigned to one client only, what assignments should be made? This problem could be solved by enumerating all 24 possible assignments of three of the four project leaders to the clients. page 7 of 19

8 Define the decision variables precisely Completely specify the LP model Clearly state the optimal solution page 8 of 19

9 Solution and sensitivity sheets: Fowle A B C D E F G H Marketing Completion Time (days) to Client 1 to Client 2 to Client 3 Terry assigned Carle assigned McClymonds assigned Higley assigned M O D E L Minimize Total Completion Time: 26 Decision Variables to Client 1 to Client 2 to Client 3 L H S R H S Terry assigned <= 1 Carle assigned <= 1 McClymonds assigned <= 1 Higley assigned <= 1 L H S = = = R H S Adjustable Cells F i n a l Reduced O b j e c t i v e A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e C o s t C o e f f i c i e n t I n c r e a s e D e c r e a s e $B$18 Terry assigned to Client E+30 2 $C$18 Terry assigned to Client E+30 $D$18 Terry assigned to Client E+30 4 $B$19 Carle assigned to Client E+30 1 $C$19 Carle assigned to Client E+30 3 $D$19 Carle assigned to Client E+30 $B$20 McClymonds assigned to Client $C$20 McClymonds assigned to Client E+30 1 $D$20 McClymonds assigned to Client $B$21 Higley assigned to Client $C$21 Higley assigned to Client E+30 1 $D$21 Higley assigned to Client E+30 1 Constraints F i n a l S h a d o w C o n s t r a i n t A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e P r i c e R.H. Side I n c r e a s e D e c r e a s e $B$23 LHS to Client $C$23 LHS to Client $D$23 LHS to Client $F$18 Terry assigned LHS E+30 0 $F$19 Carle assigned LHS $F$20 McClymonds assigned LHS $F$21 Higley assigned LHS E+30 1 page 9 of 19

10 Transshipment problems: Transshipment problems have the same basic goals as transportation problems (in particular, there is a need to ship goods from origins to destinations at minimum cost), but Goods may travel from a source through an intermediate location (known as a transshipment node) to a destination, Some destinations may also serve as transshipment points to other destinations, etcetera. The locations (sources, destinations, and transshipment points) are visualized as nodes in a network, and are numbered consecutively. If nodes i and j are connected by a direct route in the network (a link), then the decision variable x ij is used to represent the number of items shipped from node i to node j. For example, if a company has 1. two plants (sources), 2. a warehouse (serving as a pure transshipment point), and 3. two retail outlets (destinations), a transportation network could look like the following: 1: Plant 1 4: Outlet 1 2: Plant 2 5: Outlet 2 3: Warehouse The links correspond to (i, j) equal to (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5). page 10 of 19

11 For each node in the network, there is a flow constraint, where the left and right hand sides are as follows: (1) LHS = OUTFLOW INFLOW (2) RHS = supply with a plus sign or demand with a negative sign or zero if it is a pure transshipment node. The outflow corresponds to the number of units leaving a given node. The inflow corresponds to the number of units entering a given node. You can use the same relational symbol ( =,, ) for all flow constraints: Total Supply = Total Demand Total Supply > Total Demand Total Supply < Total Demand LHS = RHS LHS RHS LHS RHS Exercise 3: A company has two plants (at nodes 1 and 2), one regional warehouse (at node 3), and two retail outlets (at nodes 4 and 5). In the next production cycle: 1. Plants 1 and 2 can produce 400 and 600 units of product, respectively, 2. Outlets 1 and 2 have a demand for 750 and 250 units of product, respectively, and 3. The costs for shipping each unit of the product are as follows: From1 to 3 $4 From 2 to 4 $9 From 1 to 4 $10 From 2 to 5 $6 From 1 to 5 $8 From 3 to 4 $2 From 2 to 3 $4 From 3 to 5 $3 At most 500 units can be shipped between the warehouse and the outlet at node 4. The company wants to determine the least costly way to ship the units. page 11 of 19

12 Construct the transshipment network, including supplies and demands (with appropriate signs), and costs along each route. Define the decision variables precisely Completely specify the LP model page 12 of 19

13 Clearly state the optimal solution Interpret the shadow price and range of feasibility for the last constraint. page 13 of 19

14 Exercise 3 solution sheet: A B C D E F Company with W a r e h o u s e Supply(+)/ Nodes: Demand(-) 1: Plant Warehouse Max 2: Plant to Outlet 1: 500 3: Whs 0 4: Outlet : Outlet D e c i s i o n V a r i a b l e s : Unit Cost #Units 1 to to to to to to to to M i n i m i z e Total Cost: 6750 Subject to Outflow Inflow L H S R H S Node 1 flow = 400 Node 2 flow = 600 Node 3 flow = 0 Node 4 flow = -750 Node 5 flow = -250 Whs to Outlet <= 500 Minimize B24 By Changing D14:D21 Subject to: D28:D32 = F22:F32 D34 <= F34 Assume: linear model, non-negative page 14 of 19

15 Sensitivity and formulas sheets: Adjustable Cells F i n a l Reduced O b j e c t i v e A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e C o s t C o e f f i c i e n t I n c r e a s e D e c r e a s e $D$14 1 to 3 #Units E+30 $D$15 1 to 4 #Units E+30 1 $D$16 1 to 5 #Units E+30 2 $D$17 2 to 3 #Units $D$18 2 to 4 #Units $D$19 2 to 5 #Units E+30 $D$20 3 to 4 #Units E+30 $D$21 3 to 5 #Units E+30 1 Constraints F i n a l S h a d o w C o n s t r a i n t A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e P r i c e R.H. Side I n c r e a s e D e c r e a s e $D$28 Node 1 flow LHS $D$29 Node 2 flow LHS E+30 $D$30 Node 3 flow LHS $D$31 Node 4 flow LHS $D$32 Node 5 flow LHS $D$34 Whs to Outlet 1 LHS A B C D E F Company with W a r e h o u s e Supply(+)/ Nodes: Demand(-) 1: Plant Warehouse Max 2: Plant to Outlet 1: 500 3: Whs 0 4: Outlet : Outlet D e c i s i o n V a r i a b l e s : Unit Cost #Units 1 to to to to to to to to M i n i m i z e Total Cost: =SUMPRODUCT(B14:B21,D14:D21) Subject to Outflow Inflow L H S R H S Node 1 flow =SUM(D14:D16) =B28-C28 = =B5 Node 2 flow =SUM(D17:D19) =B29-C29 = =B6 Node 3 flow =SUM(D20:D21) =D14+D17 =B30-C30 = =B7 Node 4 flow =D15+D18+D20 =B31-C31 = =B8 Node 5 flow =D16+D19+D21 =B32-C32 = =B9 Whs to Outlet 1 =D20 <= =E6 page 15 of 19

16 Exercise 4: The Bavarian Motor Company (BMC) manufactures expensive luxury cars in Hamburg, Germany, and exports cars to sell in the United States. The exported cars are shipped from Hamburg to ports in Newark, New Jersey, and Jacksonville, Florida. From these ports, the cars are transported by rail or truck to distributors located in Boston, Columbus, Atlanta, Richmond, and Mobile. 1. There are 200 cars in Newark and 300 in Jacksonville. 2. The numbers of cars needed in Boston, Columbus, Richmond, Atlanta, and Mobile are 100, 60, 80, 170, and 70, respectively. 3. Per car shipping costs are as follows (see the graph for numbering): From 1 to 2 $30 From 5 to 6 $35 From 1 to 4 $40 From 6 to 5 $25 From 2 to 3 $50 From 7 to 4 $50 From 3 to 5 $35 From 7 to 5 $45 From 5 to 3 $40 From 7 to 6 $50 From 5 to 4 $30 BMC would like to determine the least costly way of transporting the cars from the ports to the locations where they are needed. Complete the following transshipment network by filling in the supplies and demands (with appropriate signs) and the unit costs along each link. 2: Boston 3: Columbus 1: Newark 5: Atlanta 4: Richmond 6: Mobile 7: Jacksonville page 16 of 19

17 Define the decision variables precisely Completely specify the LP model page 17 of 19

18 Clearly state the optimal solution. Interpret the shadow price of the first constraint, and its range of feasibility. page 18 of 19

19 Exercise 4 solution and sensitivity reports: A B C D E F Bavarian Motor Works Supply(+)/ Decision Demand(-) V a r i a b l e s : City Unit Cost: #Cars City to City to City to City to City to City to to to to to to M i n i m i z e Total Cost: Subject to Outflow Inflow L H S R H S City 1 flow <= 200 City 2 flow <= -100 City 3 flow <= -60 City 4 flow <= -80 City 5 flow <= -170 City 6 flow <= -70 City 7 flow <= 300 Adjustable Cells F i n a l Reduced O b j e c t i v e A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e C o s t C o e f f i c i e n t I n c r e a s e D e c r e a s e $F$6 1 to 2 #Cars $F$7 1 to 4 #Cars $F$8 2 to 3 #Cars $F$9 3 to 5 #Cars E $F$10 5 to 3 #Cars $F$11 5 to 4 #Cars E $F$12 5 to 6 #Cars E $F$13 6 to 5 #Cars E $F$14 7 to 4 #Cars E+30 5 $F$15 7 to 5 #Cars $F$16 7 to 6 #Cars Constraints F i n a l S h a d o w C o n s t r a i n t A l l o w a b l e A l l o w a b l e C e l l N a m e V a l u e P r i c e R.H. Side I n c r e a s e D e c r e a s e $D$22 City 1 flow LHS $D$23 City 2 flow LHS $D$24 City 3 flow LHS $D$25 City 4 flow LHS $D$26 City 5 flow LHS $D$27 City 6 flow LHS $D$28 City 7 flow LHS E page 19 of 19

Logistics. Lecture notes. Maria Grazia Scutellà. Dipartimento di Informatica Università di Pisa. September 2015

Logistics. Lecture notes. Maria Grazia Scutellà. Dipartimento di Informatica Università di Pisa. September 2015 Logistics Lecture notes Maria Grazia Scutellà Dipartimento di Informatica Università di Pisa September 2015 These notes are related to the course of Logistics held by the author at the University of Pisa.