Using the general-purpose nonlinear programming solver used in this blog many times for over a decade instead of a dynamic programming approach used in Ecker and Kupferschmid [29], second edition

Jsun Yui Wong

A loading problem is presented in Ecker and Kupferschmid [29; please see p. 353]. Their problem is briefly summarized as follows:

weight value

In tons in thousands of $

k w(k) v(k)

1 2 4

2 8 10

3 3 6

4 4 8

And the ship has a total weight capacity of ten tons.

Maximize v(1)x(1)+…+v(4)x(4)

subject to

w(1)x(1)+…+w(4)x(4)<=10

x(k)>=0, x(k) integer k = 1,…, 4,

where x(k) = number of crates of product k selected.

“Thus, the branch-and-bound method of Chapter 8 could be used to solve this problem. An alternate approach is to use dynamic programming,” Ecker and Kupferschmid [29, p. 353].

Because there is a chance that the long constraint above is binding, it is worthy to explore line 244, which is 244 X(1) = (-(8 * X(2) + 3 * X(3) + 4 * X(4)) + 10) / 2.

Instead of the dynamic programming approach used in Ecker and Kupferchmid [29, pp. 353-356], the following computer program, which is modeled after the general-purpose nonlinear programming solver used in this blog many times for over a decade, attempts to solve this loading problem in Ecker and Kupferschmid [29, p. 353].

0 REM DEFDBL A-Z

1 DEFINT K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)

81 FOR JJJJ = -32000 TO 32000

85 RANDOMIZE JJJJ

86 M = -3E+50

118 FOR J44 = 1 TO 4

119 A(J44) = 0 + (RND * 10)

120 NEXT J44

128 FOR I = 1 TO 3000

129 FOR KKQQ = 1 TO 4

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

151 FOR IPP = 1 TO FIX(1 + RND * 5)

153 J = 1 + FIX(RND * 4)

154 IF J < 5 THEN GOTO 156 ELSE GOTO 156

156 r = (1 - RND * 2) * A(J)

158 X(J) = A(J) + (RND ^ (RND * 15)) * r

161 GOTO 169

163 IF RND < .5 THEN X(J) = A(J) - INT(RND * 15) ELSE X(J) = A(J) + INT(RND * 15)

164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0

169 NEXT IPP

172 X(1) = INT(X(1))

174 X(2) = INT(X(2))

176 X(3) = INT(X(3))

178 X(4) = INT(X(4))

239 REM IF 2 * X(1) + 8 * X(2) + 3 * X(3) + 4 * X(4) > 10 THEN 1670

244 X(1) = (-(8 * X(2) + 3 * X(3) + 4 * X(4)) + 10) / 2

249 X(1) = INT(X(1))

326 FOR J44 = 1 TO 4

328 IF X(J44) < 0 THEN 1670

330 NEXT J44

471 P = 4 * X(1) + 10 * X(2) + 6 * X(3) + 8 * X(4)

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 4

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1557 GOTO 128

1670 NEXT I

1889 IF M < 18 THEN 1999

1933 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [102]. Its complete output of one run through JJJJ=-31972 is shown below:

5 0 0 0 20

-31998

2 0 2 0 20

-31994

2 0 2 0 20

-31991

1 0 0 2 20

-31987

3 0 0 1 20

-31984

3 0 0 1 20

-31982

2 0 2 0 20

-31979

5 0 0 0 20

-31978

0 0 2 1 20

-31972

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. Five distinct solutions are shown above, same as the five solutions in Ecker and Kupferschmid [29, p. 356]. By using the following computer system and QB64v1000-win [102], the wall-clock time (**not CPU time**) for obtaining the output through JJJJ = -31972 was 2 seconds, counting from "Starting program...". One can compare the computational procedure above with that in Ecker and Kupferschmid [29, pp. 353-356].

The computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System.

**Acknowledgement**

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

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