Question

Please formulate and solve each of the following problems. For each problem, you should include the final SOLVER printout (either your final spreadsheet or an answer report), as well as (1) clear and precise definitions for all decision variable; (2) your objective function indicating whether it is to be maximized and minimized; (3) all constraints, including non-negativity and integrality (if necessary); and (4) what the optimal decision is (in words) and what outcome will be produced.

1. A manufacturer of stereos has plants in Atlanta, New Haven and Dallas, and distributions centers in San Francisco, Boston, Washington, D.C., and Cleveland. The tables show weekly production capacities, demand requirements, and unit transportation costs (in dollars).

ORIGIN	PRODUCTION	DESTINATION	REQUIREMENTS
Atlanta	65	San Francisco	50
New Haven	75	Boston	35
Dallas	45	Washington, D.C.	35

                                                                                Cleveland                                       65

UNIT TRANSPORTATION COSTS
	San Francisco	Boston	Washington, D.C.	Cleveland
Atlanta	13	9	6	5
New Haven	11	6	7	4
Dallas	7	8	15	10

The goal is to minimize total transportation costs.

3. A company has five jobs, each of which must be assigned to a single machine. The table shows the dollar costs for each possible job-machine assignment:

JOB                                                    MACHINE

                                    A                     B                     C                     D                     E

1                                  138                  127                  118                  121                  143

2                                  157                  138                  129                  132                  160

3                                  143                  129                  131                  130                  172

4                                  111                  119                  123                  107                  120

5                                  102                  120                  100                  119                  100     

Find the set of assignments with the lowest possible total cost.

Answer 1

We will be using LINGO OPTIMIZATION TOOL to solve the above problems :

The transportation cost matrix is as follows :

	SAN FRANSISCO	BOSTON	WASHINGTON D.C.	CLEVELAND	ORIGIN
ATLANTA	13	9	6	5	65
NEW HEAVEN	11	6	7	4	75
DALLAS	7	8	15	10	45
DESTINATION	50	35	35	65

Summation of ORIGIN = Summation of DESTINATION = 185.

Therefore, this transportation problem is balanced.

Our objective is the minimization of TRANSPORTATION COST.

VARIABLES DECLARATION :

x11: Number of units from Atlanta to San Fransisco.

x12: Number of units from Atlanta to Boston.

x13: Number of units from Atlanta to Washington D.C.

x14: Number of units from Atlanta to Cleveland.

x21: Number of units from New Haven to San Fransisco.

x22: Number of units from New Haven to Boston.

x23: Number of units from New Haven to Washington D.C.

x24: Number of units from New Haven to Cleveland.

x31: Number of units from Dallas to San Fransisco.

x32: Number of units from Dallas to Boston.

x33: Number of units from Dallas to Washington D.C.

x34: Number of units  from Dallas to Cleveland.

OBJECTIVE FUNCTION :

Minimize Z = 13x11+9x12+6x13+5x14+11x21+6x22+7x23+4x24+7x31+8x32+15x33+10x34;

SUBJECT TO THE CONSTRAINTS :

x11+x12+x13+x14 = 65;

x21+x22+x23+x24 = 75;

x31+x32+x33+x34 = 45;

x11+x21+x31 =50;

x12+x22+x32 = 35;

x13+x23+x33 = 35;

x14+x24+x34 = 65;

x11 >=0;

x12>=0;

x13 >=0;

x14>=0;

x21 >=0;

x22>=0;

x23 >=0;

x24 >=0;

x31 >=0;

x32>=0;

x33 >=0;

x34>=0;

LINGO CODE IS GIVEN BY :

min = 13*x11+9*x12+6*x13+5*x14+11*x21+6*x22+7*x23+4*x24+7*x31+8*x32+15*x33+10*x34;

x11+x12+x13+x14 = 65;
x21+x22+x23+x24 = 75;
x31+x32+x33+x34 = 45;

x11+x21+x31 = 50;
x12+x22+x32 = 35;
x13+x23+x33 = 35;
x14+x24+x34 = 65;

x11 >=0;
x12 >=0;
x13 >=0;
x14 >=0;
x21 >=0;
x22 >=0;
x23 >=0;
x24 >=0;
x31 >=0;
x32 >=0;
x33 >=0;
x34 >=0;

THE OUTPUT IS GIVEN BY :

Objective value: 1080.000

Variable Value
  X11 0.000000   
X12 0.000000
  X13 35.00000
  X14 30.00000
  X21 5.000000   
  X22 35.00000
X23 0.000000
  X24 35.00000   
X31 45.00000
X32 0.000000
  X33 0.000000   
X34 0.000000   

i.e. The optimal set it (x13,x14,x21,x22,x24,x31)

x13=35, x14=30, x21=5, x22=35, x24=35, x31=45;

Optimal Transportation cost = (6*35)+(5*30)+(11*5)+(6*35)+(4*35)+(7*45)   = 1080;

3. ASSIGNMENT PROBLEM :

COST MATRIX IS GIVEN BY :

	A	B	C	D	E
1	138	127	118	121	143
2	157	138	129	132	160
3	143	129	131	130	172
4	111	119	123	107	120
5	102	120	100	119	100

VARIABLES DECLARATION :

x11: Job 1 is assigned to machine A.

x12: Job 1 is assigned to machine B.

x13: Job 1 is assigned to machine C.

x14: Job 1 is assigned to machine D.

x15: Job 1 is assigned to machine E.

x21: Job 2 is assigned to machine A.

x22: Job 2 is assigned to machine B.

x23: Job 2 is assigned to machine C.

x24: Job 2 is assigned to machine D.

x25: Job 2 is assigned to machine E.

x31: Job 3 is assigned to machine A.

x32: Job 3 is assigned to machine B.

x33: Job 3 is assigned to machine C.

x34: Job 3 is assigned to machine D.

x35: Job 3 is assigned to machine E.

x41: Job 4 is assigned to machine A.

x42: Job 4 is assigned to machine B.

x43: Job 4 is assigned to machine C.

x44: Job 4 is assigned to machine D.

x45: Job 4 is assigned to machine E.

x51: Job 5 is assigned to machine A.

x52: Job 5 is assigned to machine B.

x53: Job 5 is assigned to machine C.

x54: Job 5 is assigned to machine D.

x55: Job 5 is assigned to machine E.

OBJECTIVE FUNCTION :

Minimize Z = 138x11+127x12+118x13+121x14+143x15+157x21+138x22+129x23+132x24+160x25+

143x31+129x32+131x33+130x34+172x35+111x41+119x42+123x43+107x44+120x45+

102x51+120x52+100x53+119x54+100x55;

SUBJECT TO THE CONSTRAINTS :

x11+x12+x13+x14+x15=1;

x21+x22+x23+x24+x25 =1;

x31+x32+x33+x34+x35=1;

x41+x42+x43+x44+x45=1;

x51+x52+x53+x54+x55=1;

x11+x21+x31+x41+x51=1;

x12+x22+x32+x42+x52=1;

x13+x23+x33+x43+x53=1;

x14+x24+x34+x44+x54=1;

x15+x25+x35+x45+x55=1;

xij >=0; i=1,2,3,4,5; j=1,2,3,4,5

These constraints show that each machine is assigned one job and each job is assigned to one machine.

LINGO CODE :

min = 138*x11+127*x12+118*x13+121*x14+143*x15+
157*x21+138*x22+129*x23+132*x24+160*x25+
143*x31+129*x32+131*x33+130*x34+172*x35+
111*x41+119*x42+123*x43+107*x44+120*x45+
102*x51+120*x52+100*x53+119*x54+100*x55;

x11+x12+x13+x14+x15=1;
x21+x22+x23+x24+x25 =1;
x31+x32+x33+x34+x35=1;
x41+x42+x43+x44+x45=1;
x51+x52+x53+x54+x55=1;

x11+x21+x31+x41+x51=1;
x12+x22+x32+x42+x52=1;
x13+x23+x33+x43+x53=1;
x14+x24+x34+x44+x54=1;
x15+x25+x35+x45+x55=1;

x11 >=0;
x12 >=0;
x13 >=0;
x14 >=0;
x15 >=0;
x21 >=0;
x22 >=0;
x23 >=0;
x24 >=0;
x25 >=0;
x31 >=0;
x32 >=0;
x33 >=0;
x34 >=0;
x35 >=0;
x41 >=0;
x42 >=0;
x43 >=0;
x44 >=0;
x45 >=0;
x51 >=0;
x52 >=0;
x53 >=0;
x54 >=0;
x55 >=0;

THE OUTPUT IS GIVEN BY :

Objective value: 590.0000

Variable Value

X11 0.000000   
X12 0.000000
X13 0.000000

X14 1.000000
X15 0.000000   
X21 0.000000   
X22 0.000000   
X23 1.000000   
X24 0.000000   
X25 0.000000   
X31 0.000000
X32 1.000000   
X33 0.000000   
X34 0.000000   
X35 0.000000   
X41 1.000000   
X42 0.000000   
X43 0.000000   
X44 0.000000   
X45 0.000000   
X51 0.000000   
X52 0.000000
X53 0.000000   
X54 0.000000   
X55 1.000000

The optimal set is (x14,x23,x32,x41,x55)

i.e.

Job 1 is assigned to machine D.

Job 2 is assigned to machine C.

Job 3 is assigned to machine B.

Job 4 is assigned to machine A.

Job 5 is assigned to machine E.

The optimal cost = 121+129+129+111+100 = 590.