Question

2. The Queen City Style Company makes several lines of skirts, dresses, and sport coats. Recently, a consultant suggested that the company reevaluate its South Islander line and allocate its resources to products that would maximize contribution to profits and to overhead. Each product requires the same polyester fabric and must pass through the cutting and sewing departments. The following data were collected for the study: Product Processing Time (hr) Material (yd) Max. Demand Cutting Sewing Skirt 1 1 1 - Dress 3 4 1 - Sport coat 4 6 4 150 The cutting department has 1000 hours of capacity, sewing has 1600 hours of capacity, and 800 yards of material are available. Each skirt contributes $8 to profits and overhead; each dress, $10; and each sport coat $35.

d. How would you interpret the shadow price = 3 and the allowable increase = 25, which you obtain for the maximum demand constraint in the sensitivity report? Interpret this with the marketing effort to increase demand.

e. How does the shadow price of a resource change if the limited resource decreases by the allowable decrease limit? If the resource increases by the allowable increase limit?

Answer 1

We tabulate the given data as shown below:

Based on the above, let no. of skirts be x, no. of dress be y and no. of Sport coats be z

Total profit = 8x + 10y + 35z

We have to maximize this profit

Subject to constraints:

1x + 3y + 4z <= 1000..............Constraint for the capacity of Cutting

1x + 4y + 6z <= 1600...............Constraint for capacity of Sewing

1x + 1y + 4z <= 800.................Constraint for the capacity of Material

z <= 150...................................Constraint for max. demand for sport coat

x, y, z >= 0................................Non-negativity constraint

We solve the above problem in excel using excel solver and generate a sensitivity report as shown below:

The above solution in the form of formulas along with Excel Solver extract is shown below for better understanding and reference:

We have generated the Sensitivity Report as shown below:

d. Constraint related to max. demand for sports coat is shown in Cell B13. It has a shadow price of 3 and an allowable increase of 25. This means that the increase with 1 unit of increase in RHS constraint, the objective function will increase by $3. However, this is valid upto the value of 150 + 25 = 175 only.

Hence, Marketing efforts should be expended to increase the max. demand upto 175 nos. of sports coats.

e. The shadow price does not change upto allowable increase or allowable decrease. It remains constant. However, if the increase of decrease goes beyond these limits, the problem needs to be solved again as, beyond these limits, the constraint under discussion does not remain limited. Other constraint becomes binding which governs the shadow price.

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