Question

1. Fifth Avenue Industries, a nationally known manufacturer of menswear, produces four varieties of ties. One is an expensive, all-silk tie, one is an all-polyester tie and two are blends of polyester and cotton. Table 1 below illustrates the cost and availability (per monthly productions planning period) of the three materials used in the production process TABLE 1: MATERIAL DATA FOR FIFTH AVENJUE INDUSTRIES MATERIAL | COST PER YARD | MATERIAL AVAILABLE PER MONTH YARDS) Silk Polyester Sh. 6 Cotton 1,000 2,000 1,250 Sh. 20 Sh. 9 The firm has fixed contracts with several major departmental store chains to supply ties each month. The contracts require that Fifth Avenue Industries supply a minimum quantity of each tie but allow for a larger demand if Fifth Avenue chooses to meet that demand. (Most of the ties are not shipped with the name Fifth Avenue on their label, incidentally, but with "private stock" labels supplied by the stores). Table 2 summarizes the contract for each of the four styles of ties the selling price per tie, and the fabric requirements of each variety. The production process for all ties is almost fully automated, and Fifth Avenue uses a standard labour cost of Sh. 0.75 per tie (for any variety) TABLE 2: PRODUCT DATA FOR FIFTH AVENUE INDUSTRIES MONTHLY | MONTHLY | TOTAL VARIETY OF SELLING TIE MATERIAL REQUIREMENTS PRICE PER CONTRACT DEMAND MATERIAL TIE REQUIRED PER TIE (YARDS) 0.125 MINIMUM 7,000 16,000 8,500 All silk Sh. 6.70 6,000 100% silk All Polyester Sh. 3.55 10,000 14,000 0.08 I 100% polyester Poly-cotton Sh. 4.31 13,000 blend 1 50% polyester 50% cotton 30% polyester 70% cotton 0.10 6,000 0.10 Poly-cotton Sh. 4.81 blend 2

Answer 1

Fifth Avenue’s goal is to maximize monthly profit given the following decision variables:

x₁ = number of all-silk ties produced per month

x₂ = number polyester ties

x₃ = number of blend 1 poly-cotton ties

x₄ = number of blend 2 poly-cotton ties

To determine the objective function, we have to find the profit per ties for fifth avenue.

Variety of tie	Selling price/tie	Material required per tie	Material cost per yard	Cost per tie	Profit per tie
All silk	6.70	.125	21	2.62	4.08
All polyester	3.55	.08	6	0.48	3.07
Poly cotton blend 1	4.31	.05 .05	6 9	0.30 0.45	3.56
Poly cotton blend 1	4.81	.03 .07	6 9	0.18 0.63	4.00

Thus, LPP model for Fifth Avenue Industries will be:

Objective function

Maximize profit = $4.08x₁ + $3.07x₂ + $3.56x₃ + $4.00x₄

Subject to

0.125 x₁ ? 1000 (yards of silk)

0.08 x₂ + 0.05 x₃ + 0.03 x₄ ? 2,000 (yards of polyester)

0.05 x₃ + 0.07 x₄ ? 12500 (yards of cotton)

x₁? 6,000 (contract minimum for silk)

x₁ ? 7,000 (contract maximum)

x₂ ? 10,000(contract minimum for all polyester)

x₂ ? 14,000 (contract maximum)

x₃ ? 13,000 (contract minimum for blend 1)

x₃ ? 16,000 (contract maximum)

x₄ ? 6,000 (contract minimum for blend 2)

x₄ ? 8,500 (contract maximum)

X₁ , x₂ , x₃, x₄ ? 0